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S. 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Proof locators refer to arXiv v2, not necessarily final journal numbering. v1 had four authors and the title Unambiguous DNFs from Hex.", "journal": "SIAM Journal on Computing", "pages": "FOCS21-157--FOCS21-173", "doi": "10.1137/22M1480616", "snapshot": { "sha256": "5915f76cb7ab3d6d76a81b9c3ac5acbecbc4e6a26fec200c7a83af643e1367dd", "url": "https://arxiv.org/pdf/2102.08348", "pages": 15, "retrieved": "2026-10-17" }, "inspected_versions": [ { "snapshot_id": "BGJK21", "sha256": "5915f76cb7ab3d6d76a81b9c3ac5acbecbc4e6a26fec200c7a83af643e1367dd", "url": "https://arxiv.org/pdf/2102.08348", "pages": 15, "retrieved": "2026-10-17" } ] }, { "id": "Pab26", "authors": [ "Chirag Pabbaraju" ], "title": "Optimal Unambiguous DNFs and Alon-Saks-Seymour", "year": 2026, "kind": "preprint", "url": "https://arxiv.org/abs/2608.02533v2", "eprint": "2608.02533", "read_version": "v2, 15 October 2026 (manuscript dated 16 October), 41 PDF pages", "accessed": "2026-10-17", "publication_status": "SODA_2027_camera_ready_as_reported_by_author", "metadata_note": "Cited as a 2026 preprint; do not invent future proceedings metadata.", "snapshot": { "sha256": "e682d8ae2ff6693200d06f887f5867f72e2137bc33378bc0a9cc0b847349fe0d", "url": "https://arxiv.org/pdf/2608.02533", "pages": 41, "retrieved": "2026-10-17" }, "inspected_versions": [ { "snapshot_id": "Bal26UC", "sha256": "e682d8ae2ff6693200d06f887f5867f72e2137bc33378bc0a9cc0b847349fe0d", "url": "https://arxiv.org/pdf/2608.02533", "pages": 41, "retrieved": "2026-10-17" } ] }, { "id": "Bal26", "authors": [ "Kaspars Balodis" ], "title": "An Optimal Separation Between Certificate Complexity and Approximate Degree", "year": 2026, "kind": "preprint", "url": "https://arxiv.org/abs/2608.30351v1", "eprint": "2608.30351", "read_version": "v1, 31 August 2026, 17 PDF pages", "accessed": "2026-10-17", "publication_status": "arxiv_preprint", "snapshot": { "sha256": "67f1d332ba19a2e8a6d474bbad2695c7edb456fc281d1c5838c10045edde0827", "url": "https://arxiv.org/pdf/2608.30351", "pages": 17, "retrieved": "2026-10-17" }, "inspected_versions": [ { "snapshot_id": "Bal26adeg", "sha256": "67f1d332ba19a2e8a6d474bbad2695c7edb456fc281d1c5838c10045edde0827", "url": "https://arxiv.org/pdf/2608.30351", "pages": 17, "retrieved": "2026-10-17" } ] }, { "id": "SW86", "authors": [ "Michael Saks", "Avi Wigderson" ], "title": "Probabilistic Boolean Decision Trees and the Complexity of Evaluating Game Trees", "year": 1986, "kind": "inproceedings", "booktitle": "27th Annual Symposium on Foundations of Computer Science (FOCS 1986)", "pages": "29--38", "publisher": "IEEE", "doi": "10.1109/SFCS.1986.44", "url": "https://www.math.ias.edu/~avi/PUBLICATIONS/MYPAPERS/SW86/SW86.pdf", "read_version": "Original scanned proceedings; pp.29–31 visually inspected", "accessed": "2026-10-17", "snapshot": { "sha256": "1dc6c6ad7faf874843443711ee78f0a6d219d85c26707f9191727acd48062df5", "url": "https://www.math.ias.edu/~avi/PUBLICATIONS/MYPAPERS/SW86/SW86.pdf", "pages": 10, "retrieved": "2026-10-17" }, "inspected_versions": [ { "snapshot_id": "SW86", "sha256": "1dc6c6ad7faf874843443711ee78f0a6d219d85c26707f9191727acd48062df5", "url": "https://www.math.ias.edu/~avi/PUBLICATIONS/MYPAPERS/SW86/SW86.pdf", "pages": 10, "retrieved": "2026-10-17" } ] }, { "id": "Nis89", "authors": [ "Noam Nisan" ], "title": "CREW PRAMS and Decision Trees", "year": 1989, "kind": "inproceedings", "booktitle": "Proceedings of the Twenty-First Annual ACM Symposium on Theory of Computing (STOC 1989)", "pages": "327--335", "publisher": "Association for Computing Machinery", "doi": "10.1145/73007.73038", "url": "https://doi.org/10.1145/73007.73038", "accessed": "2026-10-17", "metadata_note": "Original obtained from user. R_1 is expected query cost with false positives forbidden and acceptance at least 1/2 on 1-inputs; R_2 is expected query cost with two-sided error at most 1/4. Do not identify its constants with default capped R at error 1/3.", "read_version": "User-supplied original STOC 1989 pp.327–335; Section 1.2, Lemmas 2.4–2.5 and 4.1–4.2, Theorems 3–4; pp.329,333–334 visually checked.", "related_publication": "Expanded journal version Nis91, SIAM Journal on Computing 20(6), 999–1007 (1991).", "inspected_versions": [ { "id": "Nis89original", "source_file": "user-supplied/73007.73038.pdf", "sha256": "fef3d08ddcd8e1f57292001267ca63c581a04211c223b236f42c57859ee10d93", "pages": 9, "retrieved": "2026-10-17", "provenance": "user_supplied_original_pdf", "snapshot_id": "Nis89original" } ], "snapshot": { "sha256": "fef3d08ddcd8e1f57292001267ca63c581a04211c223b236f42c57859ee10d93", "pages": 9, "retrieved": "2026-10-17", "provenance": "user_supplied_original_pdf" } }, { "id": "Kothari15", "authors": [ "Robin Kothari" ], "title": "Nearly optimal separations between communication (or query) complexity and partitions", "year": 2015, "kind": "preprint", "report": "ECCC TR15-195", "url": "https://eccc.weizmann.ac.il/report/2015/195/", "read_version": "Original single-author report, 13 PDF pages", "accessed": "2026-10-17", "metadata_note": "Predecessor of the three-author CCC 2016 paper AKK16; do not merge author lists.", "snapshot": { "sha256": "049aa21c361c70b636363f720a4b023fbb40be75d9dad621c66abd67f50df2f5", "url": "https://eccc.weizmann.ac.il/report/2015/195/download/", "pages": 13, "retrieved": "2026-10-17" }, "inspected_versions": [ { "snapshot_id": "AKK16", "sha256": "049aa21c361c70b636363f720a4b023fbb40be75d9dad621c66abd67f50df2f5", "url": "https://eccc.weizmann.ac.il/report/2015/195/download/", "pages": 13, "retrieved": "2026-10-17" } ] }, { "id": "HLS07", "authors": [ "Peter Høyer", "Troy Lee", "Robert Špalek" ], "title": "Negative Weights Make Adversaries Stronger", "year": 2007, "kind": "inproceedings", "booktitle": "Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing (STOC 2007)", "pages": "526--535", "publisher": "Association for Computing Machinery", "doi": "10.1145/1250790.1250867", "url": "https://arxiv.org/abs/quant-ph/0611054", "eprint": "quant-ph/0611054", "read_version": "arXiv author manuscript, 29 PDF pages; Definitions 1–2 and Theorem 2", "accessed": "2026-10-17", "snapshot": { "sha256": "8c705081dc3243baa1c83a64ed92870c7cd42fa91dd872c469621158af2c37bb", "url": "https://arxiv.org/pdf/quant-ph/0611054", "pages": 29, "retrieved": "2026-10-17" }, "inspected_versions": [ { "snapshot_id": "HLS07", "sha256": "8c705081dc3243baa1c83a64ed92870c7cd42fa91dd872c469621158af2c37bb", "url": "https://arxiv.org/pdf/quant-ph/0611054", "pages": 29, "retrieved": "2026-10-17" } ] }, { "id": "LMRSS11", "authors": [ "Troy Lee", "Rajat Mittal", "Ben W. Reichardt", "Robert Špalek", "Mario Szegedy" ], "title": "Quantum Query Complexity of State Conversion", "year": 2011, "kind": "inproceedings", "booktitle": "52nd Annual IEEE Symposium on Foundations of Computer Science (FOCS 2011)", "pages": "344--353", "publisher": "IEEE", "doi": "10.1109/FOCS.2011.75", "url": "https://arxiv.org/abs/1011.3020v2", "eprint": "1011.3020", "read_version": "v2, 23 August 2011; 19 PDF pages", "accessed": "2026-10-17", "snapshot": { "sha256": "edb6a9db14e47b34a18a5b54610041c680bbe090084e81ca1822eddd990228f1", "url": "https://arxiv.org/pdf/1011.3020", "pages": 19, "retrieved": "2026-10-17" }, "inspected_versions": [ { "snapshot_id": "LMRSS11", "sha256": "edb6a9db14e47b34a18a5b54610041c680bbe090084e81ca1822eddd990228f1", "url": "https://arxiv.org/pdf/1011.3020", "pages": 19, "retrieved": "2026-10-17" } ] }, { "id": "LS21survey", "authors": [ "Lily Li", "Morgan Shirley" ], "title": "The General Adversary Bound: A Survey", "year": 2021, "kind": "survey", "url": "https://arxiv.org/abs/2104.06380", "eprint": "2104.06380", "accessed": "2026-10-17", "annotation": "Focused exposition of the general adversary characterization: lower-bound progress arguments, semidefinite duality, span-program witness size and the upper-bound algorithm. Useful proof-level companion to HLS07 and LMRSS11; cite original results for priority.", "metadata_note": "Cites the arXiv survey manuscript; abstract and contents inspected, not used to certify a primary theorem." }, { "id": "AB23", "authors": [ "Andris Ambainis", "Aleksandrs Belovs" ], "title": "An Exponential Separation Between Quantum Query Complexity and the Polynomial Degree", "year": 2023, "url": "https://doi.org/10.4230/LIPIcs.CCC.2023.24", "kind": "research", "accessed": "2026-10-17", "booktitle": "38th Computational Complexity Conference (CCC 2023)", "series": "LIPIcs", "volume": "264", "pages": "24:1--24:13", "doi": "10.4230/LIPIcs.CCC.2023.24", "eprint": "2301.09218", "publisher": "Schloss Dagstuhl -- Leibniz-Zentrum für Informatik", "snapshot": { "sha256": "f718a04c79b578dde01b2f0ce98eb6e5e3e7d30d7b653a8a5f6357149b9e7c13", "url": "https://drops.dagstuhl.de/storage/00lipics/lipics-vol264-ccc2023/LIPIcs.CCC.2023.24/LIPIcs.CCC.2023.24.pdf", "pages": 13, "retrieved": "2026-10-17" }, "inspected_versions": [ { "snapshot_id": "AB23", "sha256": "f718a04c79b578dde01b2f0ce98eb6e5e3e7d30d7b653a8a5f6357149b9e7c13", "url": "https://drops.dagstuhl.de/storage/00lipics/lipics-vol264-ccc2023/LIPIcs.CCC.2023.24/LIPIcs.CCC.2023.24.pdf", "pages": 13, "retrieved": "2026-10-17" } ], "related_publication": "Journal version AB25: computational complexity 34(2), article 14, 1–25 (2025), doi:10.1007/s00037-025-00277-4; journal proof text not inspected." }, { "id": "ABK21", "authors": [ "Anurag Anshu", "Shalev Ben-David", "Srijita Kundu" ], "title": "On Query-To-Communication Lifting for Adversary Bounds", "year": 2021, "url": "https://doi.org/10.4230/LIPIcs.CCC.2021.30", "kind": "research", "accessed": "2026-10-17", "booktitle": "36th Computational Complexity Conference (CCC 2021)", "series": "LIPIcs", "volume": "200", "pages": "30:1--30:39", "doi": "10.4230/LIPIcs.CCC.2021.30", "eprint": "2012.03415", "read_version": "CCC 2021 final proceedings, 39 pages. Full preprint v1 also inspected; final version corrects the spurious factor 2 in the weight construction in the proof of Lemma 27.", "metadata_note": "Theorem and lemma locators used here also occur in the final proceedings. Final Lemma 27 constructs q(x,i)=q_x(i), not 2q_x(i) as in arXiv v1; the published constant-one inequality is retained.", "publisher": "Schloss Dagstuhl -- Leibniz-Zentrum für Informatik", "snapshot": { "sha256": "62110b0c53e26e8cccf332743f693769b230340509ea5e69b5c53501b5ac781b", "url": "https://drops.dagstuhl.de/storage/00lipics/lipics-vol200-ccc2021/LIPIcs.CCC.2021.30/LIPIcs.CCC.2021.30.pdf", "pages": 39, "retrieved": "2026-10-17" }, "inspected_versions": [ { "snapshot_id": "ABK21", "sha256": "6a9cad1335639b2c5b4edb252e22c32903724fc2b268bb23431008b3f80df44b", "url": "https://arxiv.org/pdf/2012.03415", "pages": 39, "retrieved": "2026-10-17" }, { "snapshot_id": "ABK21final", "sha256": "62110b0c53e26e8cccf332743f693769b230340509ea5e69b5c53501b5ac781b", "url": "https://drops.dagstuhl.de/storage/00lipics/lipics-vol200-ccc2021/LIPIcs.CCC.2021.30/LIPIcs.CCC.2021.30.pdf", "pages": 39, "retrieved": "2026-10-17" } ] }, { "id": "AKPV18", "authors": [ "Andris Ambainis", "Martins Kokainis", "Krišjānis Prūsis", "Jevgēnijs Vihrovs" ], "title": "All Classical Adversary Methods are Equivalent for Total Functions", "year": 2018, "url": "https://doi.org/10.4230/LIPIcs.STACS.2018.8", "kind": "research", "accessed": "2026-10-17", "booktitle": "35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)", "series": "LIPIcs", "volume": "96", "pages": "8:1--8:14", "doi": "10.4230/LIPIcs.STACS.2018.8", "metadata_note": "Four-author proceedings version. Claims checked in the later five-author full preprint are attributed separately to AKPVZ18, not silently to this version.", "publisher": "Schloss Dagstuhl -- Leibniz-Zentrum für Informatik", "snapshot": { "sha256": "dbaca5f94003db356f3c381194224ae4d9f7a61bc683bbacb52a462d3cdedf7c", "url": "https://drops.dagstuhl.de/storage/00lipics/lipics-vol096-stacs2018/LIPIcs.STACS.2018.8/LIPIcs.STACS.2018.8.pdf", "pages": 14, "retrieved": "2026-10-17" }, "inspected_versions": [ { "snapshot_id": "AKPV18final", "sha256": "dbaca5f94003db356f3c381194224ae4d9f7a61bc683bbacb52a462d3cdedf7c", "url": "https://drops.dagstuhl.de/storage/00lipics/lipics-vol096-stacs2018/LIPIcs.STACS.2018.8/LIPIcs.STACS.2018.8.pdf", "pages": 14, "retrieved": "2026-10-17" } ] }, { "id": "AKPVZ18", "authors": [ "Andris Ambainis", "Martins Kokainis", "Krišjānis Prūsis", "Jevgēnijs Vihrovs", "Aleksejs Zajakins" ], "title": "All Classical Adversary Methods are Equivalent for Total Functions", "year": 2018, "url": "https://arxiv.org/abs/1709.08985v3", "kind": "research", "accessed": "2026-10-17", "eprint": "1709.08985", "read_version": "arXiv:1709.08985v3, 12 November 2018, 22 pages", "related_publication": "Four-author STACS 2018 version, DOI 10.4230/LIPIcs.STACS.2018.8; author list and theorem locators are version-dependent", "snapshot": { "sha256": "b21bf94143e94c17f5a61181c6fe91b5afa659ea8aa81346935e3e55b37b3f65", "url": "https://arxiv.org/pdf/1709.08985", "pages": 22, "retrieved": "2026-10-17" }, "inspected_versions": [ { "snapshot_id": "AKPV18", "sha256": "b21bf94143e94c17f5a61181c6fe91b5afa659ea8aa81346935e3e55b37b3f65", "url": "https://arxiv.org/pdf/1709.08985", "pages": 22, "retrieved": "2026-10-17" } ] }, { "id": "GIPS21", "authors": [ "Shafi Goldwasser", "Russell Impagliazzo", "Toniann Pitassi", "Rahul Santhanam" ], "title": "On the Pseudo-Deterministic Query Complexity of NP Search Problems", "year": 2021, "url": "https://doi.org/10.4230/LIPIcs.CCC.2021.36", "kind": "research", "accessed": "2026-10-17", "booktitle": "36th Computational Complexity Conference (CCC 2021)", "series": "LIPIcs", "volume": "200", "pages": "36:1--36:22", "doi": "10.4230/LIPIcs.CCC.2021.36", "publisher": "Schloss Dagstuhl -- Leibniz-Zentrum für Informatik", "snapshot": { "sha256": "d79091562a42393bea363263515e4f525b202b0eda2a844d3e8363ad135df41c", "url": "https://drops.dagstuhl.de/storage/00lipics/lipics-vol200-ccc2021/LIPIcs.CCC.2021.36/LIPIcs.CCC.2021.36.pdf", "pages": 22, "retrieved": "2026-10-17" }, "inspected_versions": [ { "snapshot_id": "GIPS21", "sha256": "d79091562a42393bea363263515e4f525b202b0eda2a844d3e8363ad135df41c", "url": "https://drops.dagstuhl.de/storage/00lipics/lipics-vol200-ccc2021/LIPIcs.CCC.2021.36/LIPIcs.CCC.2021.36.pdf", "pages": 22, "retrieved": "2026-10-17" } ] }, { "id": "GS26", "authors": [ "Uma Girish", "Rocco Servedio" ], "title": "Forrelation Is Extremally Hard", "year": 2026, "url": "https://doi.org/10.4230/LIPIcs.ITCS.2026.72", "kind": "research", "accessed": "2026-10-17", "booktitle": "17th Innovations in Theoretical Computer Science Conference (ITCS 2026)", "series": "LIPIcs", "volume": "362", "pages": "72:1--72:22", "doi": "10.4230/LIPIcs.ITCS.2026.72", "eprint": "2508.02514", "publisher": "Schloss Dagstuhl -- Leibniz-Zentrum für Informatik", "snapshot": { "sha256": "6ddca758ad889faed006ef129078304b710326bd1a860e6128bbad9743ef8426", "url": "https://drops.dagstuhl.de/storage/00lipics/lipics-vol362-itcs2026/LIPIcs.ITCS.2026.72/LIPIcs.ITCS.2026.72.pdf", "pages": 22, "retrieved": "2026-10-17" }, "inspected_versions": [ { "snapshot_id": "GS26", "sha256": "6ddca758ad889faed006ef129078304b710326bd1a860e6128bbad9743ef8426", "url": "https://drops.dagstuhl.de/storage/00lipics/lipics-vol362-itcs2026/LIPIcs.ITCS.2026.72/LIPIcs.ITCS.2026.72.pdf", "pages": 22, "retrieved": "2026-10-17" } ], "question_status_version_recheck": { "source": "GS26", "url": "https://arxiv.org/abs/2508.02514", "html_sha256": "b5abd1054508220a33c586d604220c39ed9e4d3f46da0360fed13ef31d78a20c", "versions": [ "[v1] " ], "checked": "2026-10-23" } }, { "id": "CDM23full", "authors": [ "Arkadev Chattopadhyay", "Yogesh Dahiya", "Meena Mahajan" ], "title": "Query Complexity of Search Problems", "year": 2023, "url": "https://eccc.weizmann.ac.il/report/2023/039/", "kind": "research", "accessed": "2026-10-17", "report": "Electronic Colloquium on Computational Complexity, Report TR23-039", "related_publication": "MFCS 2023, DOI 10.4230/LIPIcs.MFCS.2023.34; full report contains proofs omitted from the proceedings", "snapshot": { "sha256": "00f2e6c2c268f37e761d8da1beec66ba26916960d3d5e936df2743bb27bff4a7", "url": "https://eccc.weizmann.ac.il/report/2023/039/download/", "pages": 23, "retrieved": "2026-10-17" }, "inspected_versions": [ { "snapshot_id": "CDM23full", "sha256": "00f2e6c2c268f37e761d8da1beec66ba26916960d3d5e936df2743bb27bff4a7", "url": "https://eccc.weizmann.ac.il/report/2023/039/download/", "pages": 23, "retrieved": "2026-10-17" } ] }, { "id": "BHMT02", "authors": [ "Gilles Brassard", "Peter Høyer", "Michele Mosca", "Alain Tapp" ], "title": "Quantum Amplitude Amplification and Estimation", "year": 2002, "url": "https://arxiv.org/abs/quant-ph/0005055", "kind": "research", "booktitle": "Quantum Computation and Information (Washington, DC, 2000)", "series": "Contemporary Mathematics", "volume": "305", "pages": "53--74", "publisher": "American Mathematical Society", "doi": "10.1090/conm/305/05215", "eprint": "quant-ph/0005055", "accessed": "2026-10-17", "read_version": "arXiv:quant-ph/0005055v1, 15 May 2000; Section 2.1, Theorem 4", "snapshot": { "sha256": "badedce80b0fe53e6cd93f915ce79c9680e375bb6565f83d2f572bc4114b0278", "url": "https://arxiv.org/pdf/quant-ph/0005055", "pages": 32, "retrieved": "2026-10-17" }, "inspected_versions": [ { "snapshot_id": "BHMT02", "sha256": "badedce80b0fe53e6cd93f915ce79c9680e375bb6565f83d2f572bc4114b0278", "url": "https://arxiv.org/pdf/quant-ph/0005055", "pages": 32, "retrieved": "2026-10-17" } ] }, { "id": "DJ92", "authors": [ "David Deutsch", "Richard Jozsa" ], "title": "Rapid Solution of Problems by Quantum Computation", "year": 1992, "url": "https://doi.org/10.1098/rspa.1992.0167", "kind": "research", "journal": "Proceedings of the Royal Society of London. 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On 22 October 2026 the strongest completed target has exact exponent log(14055^8)/log(3738277295413674), displayed as 2.1308255793978239; 2.15 remains open. Historical proposal prose may retain obsolete open wording.", "read_version": "HTML snapshot 2026-10-22; SHA-256 72048a3086aab11326884e6337cccf17d7e482032be59ee8b54c412b2c4fce64", "metadata_note": "Platform-reported formalization status is not an independent Lean build or a journal publication. 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See the witness card for proof-access limits." }, { "id": "P2M200726", "authors": [ "Prove2Me contributors" ], "title": "Block Sensitivity vs. Sensitivity: Reach 2.007", "year": 2026, "kind": "online_resource", "url": "https://prove2.me/missions/Block_Sensitivity_vs._Sensitivity%3A_Reach_2.007", "accessed": "2026-10-17", "date": "2026-10-15", "read_version": "Public goal page and dated campaign entry, inspected 17 October 2026", "annotation": "The campaign records the asymptotic goal as proved. Its stated route is to certify a finite seed with s<=A, bs at a chosen input >=B, and A^2.007<=B, then use the shared normalization/composition theorem. The accepted proof body was not recovered here; this is a summary of the published certificate framework, not an assertion of its exact implementation.", "metadata_note": "Platform attests this goal proved. Acceptance/solve dates are distinguished from proposal text. No independent Lean rebuild or completeness of contributor attribution is claimed. See the witness card for proof-access limits." }, { "id": "P2M2020526", "authors": [ "Prove2Me contributors" ], "title": "Block Sensitivity vs. Sensitivity: Reach 2.0205", "year": 2026, "kind": "online_resource", "url": "https://prove2.me/missions/Block_Sensitivity_vs._Sensitivity%3A_Reach_2.0205", "accessed": "2026-10-17", "date": "2026-10-16", "read_version": "Public goal page and dated campaign entry, inspected 17 October 2026", "annotation": "Retain four profiles, reduce the target-row count to fifteen, and split the gate estimate by whether the candidate tournament has a sink. The resulting finite certificate satisfies A^4041=B, A^2.0391<=B. The accepted proof body and actual seed parameters were not inspected; no more specific construction mechanism is asserted.", "metadata_note": "Platform attests this goal proved. Acceptance/solve dates are distinguished from proposal text. No independent Lean rebuild or completeness of contributor attribution is claimed. See the witness card for proof-access limits." }, { "id": "P2M20526", "authors": [ "Prove2Me contributors" ], "title": "Block Sensitivity vs. Sensitivity: Reach 2.05", "year": 2026, "kind": "online_resource", "url": "https://prove2.me/missions/Block_Sensitivity_vs._Sensitivity%3A_Reach_2.05", "accessed": "2026-10-17", "date": "2026-10-16", "read_version": "Public goal page and dated campaign entry, inspected 17 October 2026", "annotation": "The inspected mission marks reach_205 proved and displays accepted activity attributed to Robin Kothari. The stated certificate criterion is s<=A, bs at a chosen input >=B, and A^(41/20)<=B, followed by the shared amplification theorem. The accepted proof body was not retrieved, so its exact seed and proof route remain source-extraction gaps.", "metadata_note": "Platform attests this goal proved. Acceptance/solve dates are distinguished from proposal text. No independent Lean rebuild or completeness of contributor attribution is claimed. See the witness card for proof-access limits." }, { "id": "P2M2063526", "authors": [ "abhishekdpseok" ], "title": "Campaign exponent 2.0635: separation exponent 2.0635", "year": 2026, "kind": "online_resource", "url": "https://prove2.me/theorems/f4a6abf1-69b0-4785-a7d9-cd45649231b2", "accessed": "2026-10-22", "read_version": "Public primary statement, construction summary and status, inspected 22 October 2026; HTML SHA-256 88408d3e4f02ea03dfc7c3e5617032d2ef10ae9c750a9fd7acacaea3bb9aff4f", "annotation": "The tracker now records the exact target 4127/2000 proved. Later Kothari/Codex sources identify accepted submission f80a3f37-72fa-4d56-a094-3e4d351764f3 as their foundation. The theorem provenance still contains stale prose saying full verification is outstanding; the dated proved badge and tracker supersede that status wording. Actual seed and proof details remain unextracted.", "metadata_note": "Attribution follows the public source; usernames are retained where no verified full name is supplied. Platform verification is not an independent Lean replay by this atlas. Accepted solution downloads require authentication (HTTP 401).", "related_urls": [ "https://prove2.me/missions/Block_Sensitivity_vs._Sensitivity%3A_Reach_2.0635" ], "date": "2026-10-18" }, { "id": "P2M207526", "authors": [ "Robin Kothari", "Codex" ], "title": "Seven-level nine-predicate certificate: separation exponent 2.075", "year": 2026, "kind": "online_resource", "url": "https://prove2.me/theorems/49fdd176-c319-4f1e-b15f-b1fd5130f0bc", "accessed": "2026-10-22", "read_version": "Public primary statement, construction summary and status, inspected 22 October 2026; HTML SHA-256 36bb89af56451c29c1af9f81a4d678a994e17cc6b08375d2d75ddd4e729c1105", "annotation": "Joint-change bounds and finite labeling existence control the nine sensitivity profiles; construct disjoint block witnesses and certify the finite exponent 83/40 by exact integer arithmetic. The public record identifies Full2075.lean, SHA-256 b8b3d440c3c148ee8401967fc8f2270df9c8c142dfb63f8fde28dfb88bf8c152, and accepted submission 819ad4c6-aa25-48ca-82e9-c8df39c182c0. Apply the campaign finite-certificate theorem. The actual seed integers are not exposed in the public summary.", "metadata_note": "Attribution follows the public source; usernames are retained where no verified full name is supplied. Platform verification is not an independent Lean replay by this atlas. Accepted solution downloads require authentication (HTTP 401).", "related_urls": [ "https://prove2.me/missions/Block_Sensitivity_vs._Sensitivity%3A_Reach_2.075" ], "date": "2026-10-19" }, { "id": "P2M208526", "authors": [ "yousofy", "Codex" ], "title": "Eight-layer nine-predicate refinement: separation exponent 2.085", "year": 2026, "kind": "online_resource", "url": "https://prove2.me/theorems/7e54a28e-196b-40a3-a2c0-957e75db9e4f", "accessed": "2026-10-22", "read_version": "Public primary statement, construction summary and status, inspected 22 October 2026; HTML SHA-256 d0ce42a34dbb78398dfdcc4f4de067978ed14021fdef8c85c4bd835375630434", "annotation": "Refine the finite sensitivity-profile certificate and assemble actual Boolean functions, then invoke finite-seed amplification. The source names Sensitivity20855.finite_seed and achieves_20855 and accepted submission c6096517-61e7-4f51-bc68-c1521d61eb65. The public target is exactly 2.085; the internal declaration name alone is not evidence for a 2.0855 bound. Seed integers remain unavailable.", "metadata_note": "Attribution follows the public source; usernames are retained where no verified full name is supplied. Platform verification is not an independent Lean replay by this atlas. Accepted solution downloads require authentication (HTTP 401).", "related_urls": [ "https://prove2.me/missions/Block_Sensitivity_vs._Sensitivity%3A_Eight-layer_exponent_%242.085%24" ], "date": "2026-10-20" }, { "id": "P2M211926", "authors": [ "yousofy", "Codex" ], "title": "Periodic two-stage profile invariant: separation exponent 2.119", "year": 2026, "kind": "online_resource", "url": "https://prove2.me/theorems/cf005af0-93b2-44c1-991f-ddab41d85874", "accessed": "2026-10-22", "read_version": "Public primary statement, construction summary and status, inspected 22 October 2026; HTML SHA-256 ba5dcb59b06115376495e8d5a75357a90a46284a3c24247a3f0a0d8c084af2c9", "annotation": "An exact two-stage invariant improves the profile recurrence in the inherited nine-predicate framework. The source reports standalone verification and accepted submission bb3251a4-f00f-4b06-962a-97be2de5db1b. The public summary does not specify each parameter’s recurrence role, so no unprinted formula is inferred from their names.", "metadata_note": "Attribution follows the public source; usernames are retained where no verified full name is supplied. Platform verification is not an independent Lean replay by this atlas. Accepted solution downloads require authentication (HTTP 401).", "related_urls": [ "https://prove2.me/missions/Block_Sensitivity_vs._Sensitivity%3A_Periodic_exponent_%242.119%24" ], "date": "2026-10-20" }, { "id": "P2M2126", "authors": [ "Prove2Me contributors" ], "title": "Campaign checkpoint 2.10: separation exponent 2.1", "year": 2026, "kind": "online_resource", "url": "https://prove2.me/theorems/34794a98-e5bd-4623-ae65-526fc87e9ad1", "accessed": "2026-10-22", "read_version": "Public primary statement, construction summary and status, inspected 22 October 2026; HTML SHA-256 2943cda457ad6e48a1603c273a6f77c63746fb58901c6177970a31d733ae7202", "annotation": "Use the current campaign completed status for the exact formal 21/10 goal. The stronger informal 2.105 label is not imported as a verified exponent: the public formal target at this theorem remains 2.10. Accepted proof details are not public without authentication.", "metadata_note": "Attribution follows the public source; usernames are retained where no verified full name is supplied. Platform verification is not an independent Lean replay by this atlas. Accepted solution downloads require authentication (HTTP 401).", "related_urls": [ "https://prove2.me/missions/Block_Sensitivity_vs._Sensitivity%3A_Reach_2.10" ], "date": "2026-10-20" }, { "id": "P2M212729426", "authors": [ "Prove2Me contributors" ], "title": "Block Sensitivity vs. Sensitivity: Reach 2.127294", "year": 2026, "kind": "online_resource", "url": "https://prove2.me/missions/Block_Sensitivity_vs._Sensitivity%3A_Reach_2.127294", "accessed": "2026-10-22", "read_version": "Public primary statement, construction summary and status, inspected 22 October 2026; HTML SHA-256 cb9002045a2182ee2fbcb0a63b3b3fd452c41ddeb68ef51d51af322fd12ee504", "annotation": "Sharpen joint boundary estimates in the twelve-predicate recursion, then prove the campaign asymptotic goal. The tracker records completion on 20 October 2026; exact seed parameters and the accepted proof body remain unavailable in the inspected public record.", "metadata_note": "Attribution follows the public source; usernames are retained where no verified full name is supplied. Platform verification is not an independent Lean replay by this atlas. Accepted solution downloads require authentication (HTTP 401).", "date": "2026-10-20" }, { "id": "P2M2130126", "authors": [ "yousofy", "Codex" ], "title": "Coupled target and guard profiles: separation exponent 2.1301", "year": 2026, "kind": "online_resource", "url": "https://prove2.me/theorems/ff7ef69d-ca91-413b-872c-cf760dc244dc", "accessed": "2026-10-22", "read_version": "Public primary statement, construction summary and status, inspected 22 October 2026; HTML SHA-256 afe0fcf432ddadd353772d99175525551957c24c6002c2ee2303d0d2d888622e", "annotation": "Multiple defect tolerances and coupled target/guard profiles improve a periodic recursion. Verify the shared labeling conditions and actual disjoint block assembly, then obtain the finite-family bounds and the logarithmic exponent log(G)/log(R)>2.1301. The construction must bound sensitivity at every input and exhibit nonempty pairwise disjoint sensitive blocks at zero. Fixed prefactors are absorbed into an arbitrary positive exponent loss; the known polynomial relation between block sensitivity and sensitivity ensures unbounded sensitivity.", "metadata_note": "Attribution follows the public source; usernames are retained where no verified full name is supplied. Platform verification is not an independent Lean replay by this atlas. Accepted solution downloads require authentication (HTTP 401).", "related_urls": [ "https://prove2.me/missions/Block_Sensitivity_vs._Sensitivity%3A_Reach_%242.1301%24" ], "date": "2026-10-20" }, { "id": "P2M2130326", "authors": [ "yousofy", "Codex" ], "title": "Shared local-lemma labeling, four-stage cycle: separation exponent 2.1303", "year": 2026, "kind": "online_resource", "url": "https://prove2.me/theorems/637bbe88-761a-4bb9-bad9-78a8c1bb90c7", "accessed": "2026-10-22", "read_version": "Public primary statement, construction summary and status, inspected 22 October 2026; HTML SHA-256 39bf4d7e70c08122ef7de6a66e25ab6f6bdb07cccac3b62c9a4ed69f4cae0172", "annotation": "Apply the finite asymmetric Lovasz local lemma to simultaneous bad-label events, proving joint independence, dependency counts and an exact integer activity certificate. A coupled four-stage profile recurrence and actual Boolean/block assembly give the finite family. Its exponent log(G)/log(R)>2.1303. The construction must bound sensitivity at every input and exhibit nonempty pairwise disjoint sensitive blocks at zero. Fixed prefactors are absorbed into an arbitrary positive exponent loss; the known polynomial relation between block sensitivity and sensitivity ensures unbounded sensitivity. The mission description preserves a historical server-timeout note; the supporting finite-family theorem is now publicly marked Proved.", "metadata_note": "Attribution follows the public source; usernames are retained where no verified full name is supplied. Platform verification is not an independent Lean replay by this atlas. Accepted solution downloads require authentication (HTTP 401).", "related_urls": [ "https://prove2.me/missions/Block_Sensitivity_vs._Sensitivity%3A_Reach_%242.1303%24" ], "date": "2026-10-20" }, { "id": "P2M21326", "authors": [ "Prove2Me contributors" ], "title": "Block Sensitivity vs. Sensitivity: Reach 2.13", "year": 2026, "kind": "online_resource", "url": "https://prove2.me/missions/Block_Sensitivity_vs._Sensitivity%3A_Reach_2.13", "accessed": "2026-10-22", "read_version": "Public primary statement, construction summary and status, inspected 22 October 2026; HTML SHA-256 ac1525f9b931a7bc00c4656c96c8ef8338cb77cf7892e564b56ac68554f40aa4", "annotation": "The tracker records the 2.13 goal completed. Independently, AchievesExponent(2.1301) implies AchievesExponent(2.13) by replacing epsilon with epsilon+0.0001. This is a valid implication from the public stronger theorem, not a claim about the unretrieved accepted 2.13 proof implementation.", "metadata_note": "Attribution follows the public source; usernames are retained where no verified full name is supplied. Platform verification is not an independent Lean replay by this atlas. Accepted solution downloads require authentication (HTTP 401).", "date": "2026-10-20" }, { "id": "P2M2130825579397823926", "authors": [ "yousofy", "Codex" ], "title": "Six shared defect tolerances: separation exponent 2.1308255793978239", "year": 2026, "kind": "online_resource", "url": "https://prove2.me/theorems/01bb63ef-ced2-4ae9-8b81-b5ed295526c6", "accessed": "2026-10-22", "read_version": "Public primary statement, construction summary and status, inspected 22 October 2026; HTML SHA-256 b04d6d55bc6a9d1517c3236ca84983dff96e9a4835359dcdd15543e821ec7429", "annotation": "Share one labeling across six ordinary defect tolerances and a source tolerance; feed the simultaneous bounds into a recursive Boolean sensitivity profile and block amplification. The public primitive theorem explicitly expands sensitivity and block sensitivity as finite maxima, proves the full asymptotic quantifiers, and strictly improves log(14055^8)/log(3738289747988852). The campaign wrapper transfers the primitive definitions to the shared ones. Accepted construction submission: a9d25c46-43bd-4fb1-826b-0ed075b8cf0a. The exact exponent is logarithmic, not the truncated tracker decimal.", "metadata_note": "Attribution follows the public source; usernames are retained where no verified full name is supplied. Platform verification is not an independent Lean replay by this atlas. Accepted solution downloads require authentication (HTTP 401).", "related_urls": [ "https://prove2.me/missions/Block_Sensitivity_vs._Sensitivity%3A_Reach_%242.1308255793978239%5Cldots%24" ], "date": "2026-10-20" }, { "id": "P2Mcoupledexact26", "authors": [ "yousofy", "Codex" ], "title": "Coupled target/guard finite family", "year": 2026, "kind": "online_resource", "url": "https://prove2.me/theorems/ddfe4bf3-88be-4876-a196-33c20c0319ff", "accessed": "2026-10-22", "read_version": "Public primary statement, construction summary and status, inspected 22 October 2026; HTML SHA-256 9e43d4388146c10b27df1c21b585fe7cb5df694f8ca3b9fdc9e28cee0a98843c", "annotation": "Public exact logarithmic endpoint and finite-family provenance. Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas.", "metadata_note": "Attribution follows the public source; usernames are retained where no verified full name is supplied. Platform verification is not an independent Lean replay by this atlas. Accepted solution downloads require authentication (HTTP 401)." }, { "id": "P2Msharedexact26", "authors": [ "yousofy", "Codex" ], "title": "Four-stage shared-labeling exact endpoint", "year": 2026, "kind": "online_resource", "url": "https://prove2.me/theorems/2bc2b8ee-797d-4500-9dc8-9b23c457e131", "accessed": "2026-10-22", "read_version": "Public primary statement, construction summary and status, inspected 22 October 2026; HTML SHA-256 6bf422d49212e66377e42e53fddba15acf4648fd8a125cfb3e9ed3fdcb4a8cbc", "annotation": "Public exact logarithmic endpoint and finite-family provenance. Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas.", "metadata_note": "Attribution follows the public source; usernames are retained where no verified full name is supplied. Platform verification is not an independent Lean replay by this atlas. Accepted solution downloads require authentication (HTTP 401)." }, { "id": "P2Mrefinedexact26", "authors": [ "yousofy", "Codex" ], "title": "Refined four-stage exact endpoint", "year": 2026, "kind": "online_resource", "url": "https://prove2.me/theorems/62961329-4b50-48c7-a1fd-11159da9de26", "accessed": "2026-10-22", "read_version": "Public primary statement, construction summary and status, inspected 22 October 2026; HTML SHA-256 0a6b52af68f3c8d9aeb301d910503ea7bc4820fb477ba234303297b86923e2ba", "annotation": "Public exact logarithmic endpoint and finite-family provenance. Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas.", "metadata_note": "Attribution follows the public source; usernames are retained where no verified full name is supplied. Platform verification is not an independent Lean replay by this atlas. Accepted solution downloads require authentication (HTTP 401)." }, { "id": "P2Msharedfamily26", "authors": [ "yousofy", "Codex" ], "title": "Finite Boolean families from shared local-lemma labelings and a four-stage cycle", "year": 2026, "kind": "online_resource", "url": "https://prove2.me/theorems/13efa201-cabf-4776-8b79-3e67e73e02a0", "accessed": "2026-10-22", "read_version": "Public primary statement, construction summary and status, inspected 22 October 2026; HTML SHA-256 d881729152512e1cd85da802ce3cecf121e3cea08ebd4cf0192a6fa878d9c639", "annotation": "Public finite-family theorem: s<=C R^m, bs(f,0)>=G^m with the exact constants on the gallery card. Its shared finite asymmetric local lemma is proved directly in the reported development; the underlying classical lemma is not a new 2026 result. Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas.", "metadata_note": "Attribution follows the public source; usernames are retained where no verified full name is supplied. Platform verification is not an independent Lean replay by this atlas. Accepted solution downloads require authentication (HTTP 401)." }, { "id": "MT10", "authors": [ "Robin A. Moser", "Gábor Tardos" ], "title": "A constructive proof of the general Lovász Local Lemma", "year": 2010, "kind": "article", "journal": "Journal of the ACM", "volume": "57", "number": "2", "pages": "1–15", "doi": "10.1145/1667053.1667060", "url": "https://arxiv.org/abs/0903.0544v3", "eprint": "0903.0544", "accessed": "2026-10-22", "read_version": "arXiv v3 (20 May 2009), Theorem 1.1 cited by the public shared-labeling construction; journal bibliographic metadata verified separately", "annotation": "Background for the asymmetric Lovász local lemma used to choose one labeling obeying several defect constraints. The 2026 development reports proving the existential finite lemma directly; it does not claim the classical lemma as new.", "metadata_note": "Journal bibliographic fields verified against publisher-deposited Crossref record for DOI 10.1145/1667053.1667060 on 22 October 2026. The campaign cites arXiv v3 Theorem 1.1; this atlas has not independently audited the entire Moser–Tardos proof." }, { "id": "LMparallel26", "authors": [ "Qipeng Liu", "Saachi Mutreja" ], "title": "Parallel Quantum Advantage with Limited Adaptivity Requires Structure", "year": 2026, "url": "https://arxiv.org/abs/2608.20297v2", "kind": "preprint", "accessed": "2026-10-23", "read_version": "arXiv v2, 20 October 2026; Conjecture 1.1, Introduction p.4, Theorems 1.6–1.8 pp.7–8", "annotation": "Proposes the unrestricted almost-everywhere simulation question separately from the bounded-polynomial influence conjecture. Reports simulation for parallel queries, terminal-parallel algorithms with limited preprocessing, and fixed adaptive layer count. The exponent depends exponentially on the layer count; this is not the full adaptive result. Proof route: couple dense distributions to uniform oracles, apply a parallel hybrid argument, then use separate preprocessing and fixed-layer induction arguments.", "snapshot": { "id": "LMparallel26", "snapshot_id": "LMparallel26", "url": "https://arxiv.org/pdf/2608.20297v2", "sha256": "0f75a3e35b756026328f5f60ba55487ef6b32faddc65689d340bb7722522c75d", "pages": 58, "retrieved": "2026-10-23" }, "inspected_versions": [ { "id": "LMparallel26", "snapshot_id": "LMparallel26", "url": "https://arxiv.org/pdf/2608.20297v2", "sha256": "0f75a3e35b756026328f5f60ba55487ef6b32faddc65689d340bb7722522c75d", "pages": 58, "retrieved": "2026-10-23" } ], "metadata_note": "Primary version and selected question/theorem statements and stated scope inspected. This is a literature-status and proof-route check, not an independent verification of every proof in the paper.", "eprint": "2608.20297", "question_status_version_recheck": { "source": "LMparallel26", "url": "https://arxiv.org/abs/2608.20297", "html_sha256": "352b0ce46e01c5aebbcb53a1aecb7a5e1f9236380ab2ebe3a9d6e9d28e0c0cb7", "versions": [ "[v1] ", "[v2] " ], "checked": "2026-10-23" } }, { "id": "BSW22", "authors": [ "Nikhil Bansal", "Makrand Sinha", "Ronald de Wolf" ], "title": "Influence in Completely Bounded Block-multilinear Forms and Classical Simulation of Quantum Algorithms", "year": 2022, "url": "https://arxiv.org/abs/2203.00212v1", "kind": "preprint", "accessed": "2026-10-23", "read_version": "arXiv v1, 1 March 2022; Conjectures 1.1–1.2, Theorem 1.3, Section 4", "annotation": "Explicitly states both the folklore simulation and polynomial influence questions. Theorem 1.3 proves an influence lower bound for completely bounded block-multilinear forms, yielding restricted simulation. Proof route: a noncommutative root-influence inequality, using free-probability techniques. Complete boundedness is an additional hypothesis; the result is not a proof for all cube-bounded polynomials.", "snapshot": { "id": "BSW22", "snapshot_id": "BSW22", "url": "https://arxiv.org/pdf/2203.00212v1", "sha256": "e407c77f8db9ab7cbca8e08ddfbd23cf33e26e940b4e8c99034eb091f054ef59", "pages": 21, "retrieved": "2026-10-23" }, "inspected_versions": [ { "id": "BSW22", "snapshot_id": "BSW22", "url": "https://arxiv.org/pdf/2203.00212v1", "sha256": "e407c77f8db9ab7cbca8e08ddfbd23cf33e26e940b4e8c99034eb091f054ef59", "pages": 21, "retrieved": "2026-10-23" } ], "metadata_note": "Primary version and selected question/theorem statements and stated scope inspected. This is a literature-status and proof-route check, not an independent verification of every proof in the paper.", "eprint": "2203.00212", "question_status_version_recheck": { "source": "BSW22", "url": "https://arxiv.org/abs/2203.00212", "html_sha256": "611c10b2aa4ae99054074a14d51ff9b835554251dff9e8bd7a0b43fc9f995348", "versions": [ "[v1] " ], "checked": "2026-10-23" } }, { "id": "Bha26", "authors": [ "Sreejata Kishor Bhattacharya" ], "title": "Aaronson-Ambainis Conjecture Is True For Random Restrictions", "year": 2024, "url": "https://eccc.weizmann.ac.il/report/2024/035/revision/2/", "kind": "preprint", "accessed": "2026-10-23", "read_version": "ECCC TR24-035, revision 2, 4 March 2026; Conjecture 1.3 and abstract/main result", "annotation": "States the unrestricted influence conjecture and proves it for a non-negligible fraction of random restrictions, under a lower bound on variance. This does not establish the desired influential coordinate for every original unrestricted polynomial. First report 2024; the inspected revision is from 2026.", "snapshot": { "id": "Bha26", "snapshot_id": "Bha26", "url": "https://eccc.weizmann.ac.il/report/2024/035/revision/2/download/", "sha256": "07ddc523ad0f1e7e4c3dcd8f274c84601af21f318c2d1fdc8f1b7c3f3de9c120", "pages": 21, "retrieved": "2026-10-23" }, "inspected_versions": [ { "id": "Bha26", "snapshot_id": "Bha26", "url": "https://eccc.weizmann.ac.il/report/2024/035/revision/2/download/", "sha256": "07ddc523ad0f1e7e4c3dcd8f274c84601af21f318c2d1fdc8f1b7c3f3de9c120", "pages": 21, "retrieved": "2026-10-23" } ], "metadata_note": "Primary version and selected question/theorem statements and stated scope inspected. This is a literature-status and proof-route check, not an independent verification of every proof in the paper. Bibliographic year is the original 2024 report, not the 2026 revision date. The author name follows the PDF title page.", "report": "ECCC TR24-035, revision 2 (4 March 2026)", "date": "2026-03-04", "question_status_version_recheck": { "source": "Bha26", "url": "https://eccc.weizmann.ac.il/report/2024/035/", "html_sha256": "2345f17e22ef4d2c45307455e5e2880cb3962b0cb3fc3a2c4435928def4f4aae", "versions": [], "checked": "2026-10-23" }, "eprint": "2402.13952", "related_publication": "ITCS 2025, under the title Random Restrictions of Bounded Low Degree Polynomials Are Juntas; separately keyed Bha25. One underlying work for recurrence counts." }, { "id": "BBCLD26", "authors": [ "Sreejata Kishor Bhattacharya", "Farzan Byramji", "Arkadev Chattopadhyay", "Yogesh Dahiya", "Shachar Lovett" ], "title": "Quantum-Classical Equivalence for AND-Functions", "year": 2026, "url": "https://arxiv.org/abs/2606.03249v1", "kind": "preprint", "accessed": "2026-10-23", "read_version": "arXiv v1, 2 June 2026; Section 1.1 pp.3–5", "annotation": "Shows polynomial quantum/deterministic communication equivalence for AND-functions with polylogarithmic input-length losses. The inspected formulas retain log n; this is not a dimension-free AND-query bound in log sparsity. Proof route: characterize deterministic communication by De Morgan sparsity and lower bound approximate gamma_2 via structure of sparse representations.", "snapshot": { "id": "BBCLD26", "snapshot_id": "BBCLD26", "url": "https://arxiv.org/pdf/2606.03249v1", "sha256": "99de7e6451ed655ee15ad331a2003c4ebabdb944b3552206d29606158d55ad02", "pages": 20, "retrieved": "2026-10-23" }, "inspected_versions": [ { "id": "BBCLD26", "snapshot_id": "BBCLD26", "url": "https://arxiv.org/pdf/2606.03249v1", "sha256": "99de7e6451ed655ee15ad331a2003c4ebabdb944b3552206d29606158d55ad02", "pages": 20, "retrieved": "2026-10-23" } ], "metadata_note": "Primary version and selected question/theorem statements and stated scope inspected. This is a literature-status and proof-route check, not an independent verification of every proof in the paper.", "eprint": "2606.03249", "question_status_version_recheck": { "source": "BBCLD26", "url": "https://arxiv.org/abs/2606.03249", "html_sha256": "b11c68ea9c31561d3d9aa774ae3786d4ecd64a3c0c133a010d3f82d3078d35ac", "versions": [ "[v1] " ], "checked": "2026-10-23" } }, { "id": "FLM26", "authors": [ "Yuval Filmus", "Anqi Li", "Dor Minzer" ], "title": "An FKN Theorem for the Binary Grassmann Scheme", "year": 2026, "url": "https://arxiv.org/abs/2608.11320v1", "kind": "preprint", "accessed": "2026-10-23", "read_version": "arXiv v1, 11 August 2026; Theorems 1.2–1.3 and Introduction", "annotation": "Degree-one robust structure on the binary Grassmann scheme. Relevant adjacent progress, but it does not establish arbitrary-degree robust decision-tree bounds on the symmetric group, or polynomial equivalence of query measures on all q-analog domains. Those distinctions are retained in the question audit.", "snapshot": { "id": "FLM26", "snapshot_id": "FLM26", "url": "https://arxiv.org/pdf/2608.11320v1", "sha256": "4c58726423e3e7c0d6df86dcc81d03c3f85c8c2c32d0fa4dd05f99cf434ff6fe", "pages": 16, "retrieved": "2026-10-23" }, "inspected_versions": [ { "id": "FLM26", "snapshot_id": "FLM26", "url": "https://arxiv.org/pdf/2608.11320v1", "sha256": "4c58726423e3e7c0d6df86dcc81d03c3f85c8c2c32d0fa4dd05f99cf434ff6fe", "pages": 16, "retrieved": "2026-10-23" } ], "metadata_note": "Primary version and selected question/theorem statements and stated scope inspected. This is a literature-status and proof-route check, not an independent verification of every proof in the paper.", "eprint": "2608.11320", "question_status_version_recheck": { "source": "FLM26", "url": "https://arxiv.org/abs/2608.11320", "html_sha256": "9141d661ed6a2b4bb5d67338c11d287a04dad17f42f08a76176cff744cc41a77", "versions": [ "[v1] " ], "checked": "2026-10-23" } }, { "id": "NPS26", "authors": [ "Sai Soumya Nalli", "Karthikeya Polisetty", "Jayalal Sarma" ], "title": "On Condensation of Block Sensitivity, Certificate Complexity and the AND (and OR) Decision Tree Complexity", "year": 2026, "url": "https://arxiv.org/abs/2602.01042v2", "kind": "preprint", "accessed": "2026-10-23", "read_version": "arXiv v2, 16 March 2026; abstract, Sections 3–4 and Appendix A", "annotation": "Refutes hardness-condensation assertions for block sensitivity and related measures using explicit restricted-function examples. Inspected as adjacent work: hardness condensation is distinct from bounding AND-query depth by log sparsity. No resolution of the latter is inferred.", "snapshot": { "id": "NPS26", "snapshot_id": "NPS26", "url": "https://arxiv.org/pdf/2602.01042v2", "sha256": "77033f09295c2342a91188c7cad83ae3e478ceb4e2db5dbb884c0729de5ae804", "pages": 12, "retrieved": "2026-10-23" }, "inspected_versions": [ { "id": "NPS26", "snapshot_id": "NPS26", "url": "https://arxiv.org/pdf/2602.01042v2", "sha256": "77033f09295c2342a91188c7cad83ae3e478ceb4e2db5dbb884c0729de5ae804", "pages": 12, "retrieved": "2026-10-23" } ], "metadata_note": "Primary version and selected question/theorem statements and stated scope inspected. This is a literature-status and proof-route check, not an independent verification of every proof in the paper.", "eprint": "2602.01042", "question_status_version_recheck": { "source": "NPS26", "url": "https://arxiv.org/abs/2602.01042", "html_sha256": "86092fd07a457cba7eca9f6369f2ac2c0f61998a0ce7d5d952a948ba1f947696", "versions": [ "[v1] ", "[v2] " ], "checked": "2026-10-23" } }, { "id": "CPS23", "authors": [ "Rahul Chugh", "Supartha Podder", "Swagato Sanyal" ], "title": "Decision Tree Complexity versus Block Sensitivity and Degree", "year": 2023, "url": "https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2023.27", "kind": "inproceedings", "accessed": "2026-10-23", "read_version": "FSTTCS 2023 proceedings, Question 1 and discussion p.27:3, Theorems 4–5 pp.27:4–27:5", "annotation": "Explicitly discusses improving D versus block sensitivity from cubic to quadratic and proves quadratic upper bounds for restricted classes such as zebra functions and bounded alternation. The unrestricted conjecture is not settled by those theorems. Same underlying work as the indexed 2022 preprint CPS22; not a second recurrence count.", "snapshot": { "id": "CPS23", "snapshot_id": "CPS23", "url": "https://drops.dagstuhl.de/storage/00lipics/lipics-vol284-fsttcs2023/LIPIcs.FSTTCS.2023.27/LIPIcs.FSTTCS.2023.27.pdf", "sha256": "c9127ba4ded765c248e7086593d7a67c5320f5901e84cd2c71459591e7847745", "pages": 23, "retrieved": "2026-10-23" }, "inspected_versions": [ { "id": "CPS23", "snapshot_id": "CPS23", "url": "https://drops.dagstuhl.de/storage/00lipics/lipics-vol284-fsttcs2023/LIPIcs.FSTTCS.2023.27/LIPIcs.FSTTCS.2023.27.pdf", "sha256": "c9127ba4ded765c248e7086593d7a67c5320f5901e84cd2c71459591e7847745", "pages": 23, "retrieved": "2026-10-23" } ], "metadata_note": "Primary version and selected question/theorem statements and stated scope inspected. This is a literature-status and proof-route check, not an independent verification of every proof in the paper.", "doi": "10.4230/LIPIcs.FSTTCS.2023.27", "booktitle": "43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)", "series": "Leibniz International Proceedings in Informatics (LIPIcs)", "volume": "284", "pages": "27:1--27:23", "publisher": "Schloss Dagstuhl -- Leibniz-Zentrum für Informatik" }, { "id": "Bha25", "authors": [ "Sreejata Kishor Bhattacharya" ], "title": "Random Restrictions of Bounded Low Degree Polynomials Are Juntas", "year": 2025, "kind": "inproceedings", "booktitle": "16th Innovations in Theoretical Computer Science Conference (ITCS 2025)", "series": "Leibniz International Proceedings in Informatics (LIPIcs)", "volume": "325", "pages": "17:1--17:21", "publisher": "Schloss Dagstuhl -- Leibniz-Zentrum für Informatik", "doi": "10.4230/LIPIcs.ITCS.2025.17", "url": "https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.17", "accessed": "2026-10-23", "read_version": "Final ITCS 2025 paper; Conjecture 4, Theorems 5 and 30, article pp.17:2–17:3 and 17:19", "snapshot": { "id": "Bha25", "snapshot_id": "Bha25", "url": "https://drops.dagstuhl.de/storage/00lipics/lipics-vol325-itcs2025/LIPIcs.ITCS.2025.17/LIPIcs.ITCS.2025.17.pdf", "sha256": "c55f17d6e40d605dbeff024902efafd4290aaf8d16ce399fb247c7eb98779dae", "pages": 21, "retrieved": "2026-10-23" }, "inspected_versions": [ { "id": "Bha25", "snapshot_id": "Bha25", "url": "https://drops.dagstuhl.de/storage/00lipics/lipics-vol325-itcs2025/LIPIcs.ITCS.2025.17/LIPIcs.ITCS.2025.17.pdf", "sha256": "c55f17d6e40d605dbeff024902efafd4290aaf8d16ce399fb247c7eb98779dae", "pages": 21, "retrieved": "2026-10-23" } ], "annotation": "Published version of the random-restriction result, with a changed title. Explicitly states the full Aaronson–Ambainis influence conjecture and proves a restricted random-restriction theorem. Same work as the 2024 report and its March 2026 revision; not an additional distinct-paper recurrence.", "metadata_note": "Official publisher metadata and selected full-PDF conjecture/theorem statements checked. The latest arXiv page 2402.13952v2 says accepted at ITCS 2025. Full proof not independently re-proved." }, { "id": "ADGG26", "authors": [ "Ryan Anselm", "Michelle Ding", "Dar Gilboa", "Sabee Grewal" ], "title": "Optimal and Verifiable Quantum Advantages in Communication Complexity", "year": 2026, "kind": "preprint", "url": "https://arxiv.org/abs/2610.05585v1", "eprint": "2610.05585", "doi": "10.48550/arXiv.2610.05585", "read_version": "arXiv v1, 4 October 2026, 65 pages; Theorems 1.1–1.3, Sections 1.1.1, 4.1–4.2; selected statements and proof route checked, not a full proof audit", "annotation": "Pelagic Fourier Fishing gives an optimal one-query bounded-error versus linear randomized query separation for a total relation, via a two-way communication lower bound. Fourier sampling lands in the middle of squared Fourier mass with probability at least 3/4; the classical proof uses entropy/coupling and a Gaussian window robust to shifts. This does not answer BK24 Open Problem 1: the Pelagic algorithm is bounded-error. The exact protocol of Theorem 1.3 is a different communication problem and yields a fixed-degree polynomial, not exponential, separation. No exact query separation is inferred from it.", "accessed": "2026-10-23", "snapshot": { "url": "https://arxiv.org/pdf/2610.05585v1", "sha256": "897e58f3014d3102b91e51e2ad29e1728eeaa0c99b9650ed44849f2a818ec286", "pages": 65, "retrieved": "2026-10-23" }, "inspected_versions": [ { "snapshot_id": "ADGG26", "url": "https://arxiv.org/pdf/2610.05585v1", "sha256": "897e58f3014d3102b91e51e2ad29e1728eeaa0c99b9650ed44849f2a818ec286", "pages": 65, "retrieved": "2026-10-23" } ] }, { "id": "GNSTW16", "authors": [ "Parikshit Gopalan", "Noam Nisan", "Rocco A. Servedio", "Kunal Talwar", "Avi Wigderson" ], "title": "Smooth Boolean Functions are Easy: Efficient Algorithms for Low-Sensitivity Functions", "year": 2016, "kind": "inproceedings", "url": "https://www.cs.columbia.edu/~rocco/Public/GNSTW.pdf", "eprint": "1508.02420", "booktitle": "Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science (ITCS 2016)", "publisher": "Association for Computing Machinery", "pages": "59--70", "doi": "10.1145/2840728.2840738", "read_version": "12-page author-hosted proceedings PDF; Theorem 8 and Section 2.1 checked", "annotation": "Theorem 8 explicitly attributes the relevant-variable bound n(f) <= s(f)4^{s(f)} to Simon. This is historical context for Wellens Conjecture 3.17, which asks to remove the prefactor s(f); the 2016 paper is not a recurrence of that later sharper conjecture. Its structural algorithms use majority reconstruction of neighboring values; these do not prove the O(4^s) variable bound.", "accessed": "2026-10-23", "snapshot": { "url": "https://www.cs.columbia.edu/~rocco/Public/GNSTW.pdf", "sha256": "ae98487d6197f518e6e6f32997406e1e124cb95e30fa84ce8cf2bc701150d2a8", "pages": 12, "retrieved": "2026-10-23" }, "inspected_versions": [ { "snapshot_id": "GNSTW16", "url": "https://www.cs.columbia.edu/~rocco/Public/GNSTW.pdf", "sha256": "ae98487d6197f518e6e6f32997406e1e124cb95e30fa84ce8cf2bc701150d2a8", "pages": 12, "retrieved": "2026-10-23" } ] } ], "measures": [ { "id": "D", "name": "deterministic query complexity", "symbol": "D", "definition": "Minimum worst-case depth of a deterministic bit-query tree computing the required value (or a valid output for a relation).", "definition_status": "source_checked", "family": "core", "sources": [ "BdW02", "KLLPV26", "CDM23full" ], "aliases": [ { "text": "deterministic decision-tree complexity", "source": "BdW02", "status": "source_checked", "normalization": "Same measure; source-specific cost/encoding caveats remain in the definition.", "locator": "Section 3.1", "alias_kind": "name" }, { "text": "DT", "source": "LM26", "locator": "Section 2; DM23 Section 2", "status": "source_checked", "normalization": "same definition" }, { "text": "Depth", "source": "LM26", "locator": "Section 2; DM23 Section 2", "status": "source_checked", "normalization": "same definition" }, { "text": "Depth", "source": "DM23", "locator": "Section 2", "status": "source_checked", "normalization": "same deterministic query depth" }, { "text": "P^dt", "source": "GPW18", "locator": "Section 1, query notation", "status": "source_checked", "normalization": "same definition", "alias_kind": "notation" }, { "text": "D", "source": "ABK16", "locator": "Section 2.2", "status": "source_checked", "normalization": "same definition", "alias_kind": "notation" }, { "text": "D", "source": "SW86", "locator": "Section I, p.29", "status": "source_checked", "normalization": "deterministic query complexity", "alias_kind": "notation" }, { "text": "D^dt", "source": "CDM23full", "locator": "Section 2: decision-tree and pseudo-deterministic definitions", "status": "source_checked", "normalization": "Depth is worst case; psD^dt is randomized pseudo-deterministic depth despite the letter D. The notation also applies to search relations." }, { "text": "P^dt(f)", "source": "Goos15", "status": "source_checked", "locator": "Sections 1.1–2, pp.2–4", "normalization": "Total Boolean functions; superscript dt denotes query model.", "alias_kind": "symbol" } ], "notes": "", "definition_locator": "KLLPV26 Definition 14", "definition_audit": "Definition and normalization inspected; historical alias inventory remains incomplete.", "alias_attribution_history": [ { "text": "D", "source": "BdW02", "status": "seed_alias_exact_occurrence_audit_pending", "normalization": "See definition; verify exact typography during source audit.", "disposition": "replaced_by_source_checked_occurrence", "replacement_source": "ABK16" } ], "alias_candidates": [] }, { "id": "R", "name": "bounded-error randomized query complexity", "symbol": "R", "definition": "Minimum worst-case query cap with success at least 2/3 on every input in the domain.", "definition_status": "source_checked", "family": "core", "sources": [ "ABK16", "KLLPV26", "CDM23full", "GIPS21" ], "aliases": [ { "text": "R", "source": "ABK16", "status": "source_checked", "normalization": "same definition", "locator": "Section 2.2", "alias_kind": "notation" }, { "text": "BPP^dt", "source": "GJPW18", "locator": "Section 2", "status": "source_checked", "normalization": "same definition", "alias_kind": "notation" }, { "text": "R_2", "source": "Aar02", "locator": "Section 1", "status": "source_checked", "normalization": "Equivalent up to constants; inspect the source’s expected-cost and one-sided conventions.", "alias_kind": "notation" }, { "text": "Monte Carlo", "source": "ABBLS17", "locator": "Section 2, Randomized decision tree algorithms", "status": "source_checked", "normalization": "Expected query cost in this source; source error constants differ from fixed-cap R.", "alias_kind": "name" }, { "text": "R^dt", "source": "CDM23full", "locator": "Section 2: decision-tree and pseudo-deterministic definitions", "status": "source_checked", "normalization": "Depth is worst case; psD^dt is randomized pseudo-deterministic depth despite the letter D. The notation also applies to search relations." }, { "text": "BPP^dt", "source": "GIPS21", "locator": "Theorem 4 and Section 2", "status": "source_checked", "normalization": "Search relations; classical/quantum bounded-error depth and pseudo-deterministic depth, as specified by the stable ID." } ], "notes": "", "definition_locator": "KLLPV26 Definition 15", "definition_audit": "Definition and normalization inspected; historical alias inventory remains incomplete.", "alias_attribution_history": [ { "text": "R_2", "source": "ABK16", "status": "seed_alias_exact_occurrence_audit_pending", "normalization": "See definition; verify exact typography during source audit.", "disposition": "replaced_by_source_checked_occurrence", "replacement_source": "Aar02" }, { "text": "Monte Carlo query complexity", "source": "ABK16", "status": "seed_alias_exact_occurrence_audit_pending", "normalization": "See definition; verify exact typography during source audit.", "disposition": "replaced_by_source_checked_occurrence", "replacement_source": "ABBLS17" } ], "repository_display_variants": [ { "text": "R_{1/3}", "source": "ABK16", "status": "repository_display_not_a_historical_occurrence", "normalization": "See definition; verify exact typography during source audit.", "reason": "Do not present as a verified historical occurrence." } ] }, { "id": "R0", "name": "zero-error randomized query complexity", "symbol": "R_0", "definition": "Minimum worst-input expected number of queries of an always-correct randomized algorithm. Fixed-cap algorithms with constant inconclusive probability are equivalent only up to constants.", "definition_status": "source_checked", "family": "core", "sources": [ "ABK16", "KLLPV26" ], "aliases": [ { "text": "R_0", "source": "ABK16", "status": "source_checked", "normalization": "same definition", "locator": "Section 2.2", "alias_kind": "notation" }, { "text": "Las Vegas", "source": "BdW02", "locator": "Section 3.3, footnote", "status": "source_checked", "normalization": "Zero-error randomized model; numerical cost convention must be specified.", "alias_kind": "name" }, { "text": "R", "source": "SW86", "locator": "Section I, p.29", "status": "source_checked", "normalization": "Source requires always-correct output and uses worst-case expected queries. R here means modern R0, not bounded-error R.", "alias_kind": "notation" }, { "text": "Las Vegas", "source": "ABBLS17", "locator": "Section 2, Randomized decision tree algorithms", "status": "source_checked", "normalization": "Expected query cost in this source; source error constants differ from fixed-cap R.", "alias_kind": "name" } ], "notes": " ABK16 Section 2.2 uses fixed worst-case cost with failure symbol probability <=1/2; KLLPV26 Definition 16 uses worst-case expected cost and always-correct output. These are equivalent up to universal factors by truncation/restart, not numerically identical. Exact constants require selecting a convention.", "definition_locator": "KLLPV26 Definition 16", "definition_audit": "Definition and normalization inspected; historical alias inventory remains incomplete.", "alias_attribution_history": [ { "text": "Las Vegas query complexity", "source": "ABK16", "status": "seed_alias_exact_occurrence_audit_pending", "normalization": "See definition; verify exact typography during source audit.", "disposition": "replaced_by_source_checked_occurrence", "replacement_source": "ABBLS17" } ], "alias_candidates": [] }, { "id": "R1", "name": "one-sided-error randomized query complexity", "symbol": "R_1", "definition": "Query cap with no false positives and acceptance probability at least 2/3 on positive inputs. Complement orientation must be separate.", "definition_status": "source_checked", "family": "core", "sources": [ "AIK26" ], "aliases": [ { "text": "R_1", "source": "AIK26", "status": "source_checked", "normalization": "same definition", "locator": "Section 2", "alias_kind": "notation" } ], "notes": "Other sources reverse the one-sided orientation; do not merge without complementing f.", "definition_locator": "AIK26 Section 2, preliminaries", "alias_candidates": [] }, { "id": "Q", "name": "bounded-error quantum query complexity", "symbol": "Q", "definition": "Minimum quantum bit-oracle query cap with success at least 2/3 on every promised input.", "definition_status": "source_checked", "family": "core", "sources": [ "ABK16", "KLLPV26", "GIPS21" ], "aliases": [ { "text": "Q", "source": "ABK16", "status": "source_checked", "normalization": "same definition", "locator": "Section 2.2", "alias_kind": "notation" }, { "text": "Q_2", "source": "Aar02", "locator": "Section 1", "status": "source_checked", "normalization": "Equivalent up to constants; inspect the source’s expected-cost and one-sided conventions.", "alias_kind": "notation" }, { "text": "Q_{1/3}", "source": "HS05survey", "locator": "Section 2", "status": "source_checked", "normalization": "Fixed two-sided error 1/3; Q_epsilon is parameterized.", "alias_kind": "notation" }, { "text": "Q^dt", "source": "GIPS21", "locator": "Theorem 4 and Section 2", "status": "source_checked", "normalization": "Search relations; classical/quantum bounded-error depth and pseudo-deterministic depth, as specified by the stable ID." } ], "notes": "", "definition_locator": "KLLPV26 Definition 17", "definition_audit": "Definition and normalization inspected; historical alias inventory remains incomplete.", "alias_attribution_history": [ { "text": "Q_2", "source": "ABK16", "status": "seed_alias_exact_occurrence_audit_pending", "normalization": "See definition; verify exact typography during source audit.", "disposition": "replaced_by_source_checked_occurrence", "replacement_source": "Aar02" }, { "text": "Q_{1/3}", "source": "ABK16", "status": "seed_alias_exact_occurrence_audit_pending", "normalization": "See definition; verify exact typography during source audit.", "disposition": "replaced_by_source_checked_occurrence", "replacement_source": "HS05survey" } ], "alias_candidates": [] }, { "id": "QE", "name": "exact quantum query complexity", "symbol": "Q_E", "definition": "Minimum fixed query cap of a quantum algorithm computing f with certainty.", "definition_status": "source_checked", "family": "core", "sources": [ "Mid04", "KLLPV26" ], "aliases": [ { "text": "Q_E", "source": "ABK16", "locator": "Section 2.2", "status": "source_checked", "normalization": "same definition", "alias_kind": "notation" } ], "notes": "", "definition_locator": "KLLPV26 Definition 19", "definition_audit": "Definition and normalization inspected; historical alias inventory remains incomplete.", "alias_attribution_history": [ { "text": "Q_E", "source": "Mid04", "status": "seed_alias_exact_occurrence_audit_pending", "normalization": "See definition; verify exact typography during source audit.", "disposition": "replaced_by_source_checked_occurrence", "replacement_source": "ABK16" } ], "repository_display_variants": [ { "text": "Q_{\\mathrm E}", "source": "Mid04", "status": "repository_display_not_a_historical_occurrence", "normalization": "See definition; verify exact typography during source audit." } ] }, { "id": "Q0", "name": "zero-error quantum query complexity", "symbol": "Q_0", "definition": "Minimum fixed query cap returning a correct answer with probability at least 1/2 and otherwise inconclusive; never wrong.", "definition_status": "source_checked", "family": "core", "sources": [ "Aar02", "KLLPV26" ], "aliases": [ { "text": "Q_0", "source": "Aar02", "status": "source_checked", "normalization": "Equivalent up to constants; inspect the source’s expected-cost and one-sided conventions.", "locator": "Section 1", "alias_kind": "notation" } ], "notes": "Do not silently identify fixed-cap, expected-time, and exact models when tracking constants.", "definition_locator": "KLLPV26 Definition 18", "definition_audit": "Definition and normalization inspected; historical alias inventory remains incomplete.", "alias_candidates": [] }, { "id": "s", "name": "sensitivity", "symbol": "s", "definition": "max_x |{i : f(x) != f(x xor e_i)}|, with both inputs in the domain when used on a promise.", "definition_status": "source_checked", "family": "core", "sources": [ "BdW02", "KLLPV26", "GNSTW16" ], "aliases": [ { "text": "s", "source": "KLLPV26", "locator": "Definition 31", "status": "source_checked", "normalization": "same definition", "alias_kind": "notation" }, { "text": "critical sensitivity", "source": "Nis89", "locator": "Publisher abstract", "status": "source_checked", "normalization": "Historical name identified in the original introduction/abstract; later sensitivity definitions fix the normalization.", "alias_kind": "name" }, { "text": "s_w(f)", "source": "Nis89", "locator": "Section 2, p.330", "status": "source_checked", "normalization": "Input-local notation, not the global maximum." } ], "notes": "", "definition_locator": "KLLPV26 Definition 31 (output-restricted maxima for s_b)", "alias_attribution_history": [ { "text": "s", "source": "BdW02", "status": "seed_alias_exact_occurrence_audit_pending", "normalization": "See definition; verify exact typography during source audit.", "disposition": "replaced_by_source_checked_occurrence", "replacement_source": "KLLPV26" }, { "text": "critical sensitivity", "source": "BdW02", "status": "seed_alias_exact_occurrence_audit_pending", "normalization": "See definition; verify exact typography during source audit.", "disposition": "replaced_by_source_checked_occurrence", "replacement_source": "Nis89" } ], "alias_candidates": [] }, { "id": "s0", "name": "0-sensitivity", "symbol": "s_0", "definition": "Maximum local sensitivity over promised inputs x with output f(x)=0. This is output-conditioned, not a restriction on the values of the flipped input coordinates.", "definition_status": "source_checked", "family": "one_sided", "sources": [ "ABKBRT21", "KLLPV26" ], "aliases": [ { "text": "s_0", "source": "ABKBRT21", "status": "source_checked", "normalization": "same definition", "locator": "Appendix A, Lemma 31", "alias_kind": "notation" } ], "notes": " Notation clash: CDM23 Section 2 also uses this typography for input-directed flips. Its output-conditioned version is described separately immediately afterwards; do not identify the two definitions.", "definition_locator": "KLLPV26 Definition 31 (output-restricted maxima for s_b)", "alias_candidates": [] }, { "id": "s1", "name": "1-sensitivity", "symbol": "s_1", "definition": "Maximum local sensitivity over promised inputs x with output f(x)=1. This is output-conditioned, not a restriction on the values of the flipped input coordinates.", "definition_status": "source_checked", "family": "one_sided", "sources": [ "ABKBRT21", "KLLPV26" ], "aliases": [ { "text": "s_1", "source": "ABKBRT21", "status": "source_checked", "normalization": "same definition", "locator": "Appendix A, Lemma 31", "alias_kind": "notation" } ], "notes": " Notation clash: CDM23 Section 2 also uses this typography for input-directed flips. Its output-conditioned version is described separately immediately afterwards; do not identify the two definitions.", "definition_locator": "KLLPV26 Definition 31 (output-restricted maxima for s_b)", "alias_candidates": [] }, { "id": "bs", "name": "block sensitivity", "symbol": "bs", "definition": "Maximum over inputs of the largest family of pairwise disjoint sensitive coordinate blocks.", "definition_status": "source_checked", "family": "core", "sources": [ "Nis91", "KLLPV26" ], "aliases": [ { "text": "bs", "source": "NS94", "locator": "Definition 3.6", "status": "source_checked", "normalization": "same definition", "alias_kind": "notation" }, { "text": "bs_w(f)", "source": "Nis89", "locator": "Section 2, p.330", "status": "source_checked", "normalization": "Input-local notation, not the global maximum." } ], "notes": "", "definition_locator": "KLLPV26 Definition 33 (output-restricted maxima for bs_b)", "alias_attribution_history": [ { "text": "bs", "source": "Nis91", "status": "seed_alias_exact_occurrence_audit_pending", "normalization": "See definition; verify exact typography during source audit.", "disposition": "replaced_by_source_checked_occurrence", "replacement_source": "NS94" } ], "alias_candidates": [] }, { "id": "bs0", "name": "0-block sensitivity", "symbol": "bs_0", "definition": "Maximum local block sensitivity over promised inputs x with output f(x)=0. This is output-conditioned, not input-directed.", "definition_status": "source_checked", "family": "one_sided", "sources": [ "ABK16", "KLLPV26" ], "aliases": [ { "text": "bs_0", "source": "GSS13", "locator": "Section 2", "status": "source_checked", "normalization": "same definition", "alias_kind": "notation" } ], "notes": " Notation clash: CDM23 Section 2 also uses this typography for input-directed flips. Its output-conditioned version is described separately immediately afterwards; do not identify the two definitions.", "definition_locator": "KLLPV26 Definition 33 (output-restricted maxima for bs_b)", "alias_attribution_history": [ { "text": "bs_0", "source": "ABK16", "status": "seed_alias_exact_occurrence_audit_pending", "normalization": "See definition; verify exact typography during source audit.", "disposition": "replaced_by_source_checked_occurrence", "replacement_source": "GSS13" } ], "alias_candidates": [] }, { "id": "bs1", "name": "1-block sensitivity", "symbol": "bs_1", "definition": "Maximum local block sensitivity over promised inputs x with output f(x)=1. This is output-conditioned, not input-directed.", "definition_status": "source_checked", "family": "one_sided", "sources": [ "ABK16", "KLLPV26" ], "aliases": [ { "text": "bs_1", "source": "GSS13", "locator": "Section 2", "status": "source_checked", "normalization": "same definition", "alias_kind": "notation" } ], "notes": " Notation clash: CDM23 Section 2 also uses this typography for input-directed flips. Its output-conditioned version is described separately immediately afterwards; do not identify the two definitions.", "definition_locator": "KLLPV26 Definition 33 (output-restricted maxima for bs_b)", "alias_attribution_history": [ { "text": "bs_1", "source": "ABK16", "status": "seed_alias_exact_occurrence_audit_pending", "normalization": "See definition; verify exact typography during source audit.", "disposition": "replaced_by_source_checked_occurrence", "replacement_source": "GSS13" } ], "alias_candidates": [] }, { "id": "bs_l", "name": "bounded-block-size sensitivity", "symbol": "bs_ell", "definition": "Block sensitivity with each sensitive block of size at most ell; ell is an explicit parameter.", "definition_status": "source_checked", "family": "parameterized", "sources": [ "HKP11" ], "aliases": [ { "text": "bs_ell", "source": "HKP11", "status": "source_checked", "normalization": "Same measure; source-specific cost/encoding caveats remain in the definition.", "locator": "Section 3.1", "alias_kind": "notation" }, { "text": "ell-block sensitivity", "source": "HKP11", "status": "source_checked", "normalization": "Same measure; source-specific cost/encoding caveats remain in the definition.", "locator": "Section 3.1", "alias_kind": "name" } ], "notes": "", "definition_locator": "HKP11 Section 3.1 discussion of ell-block sensitivity", "alias_candidates": [] }, { "id": "C", "name": "certificate complexity", "symbol": "C", "definition": "Maximum over x of the minimum number of coordinates fixing f(x) among inputs in the domain.", "definition_status": "source_checked", "family": "core", "sources": [ "BdW02", "KLLPV26", "CDMRS23" ], "aliases": [ { "text": "certificate size", "source": "BdW02", "status": "source_checked", "normalization": "Same measure; source-specific cost/encoding caveats remain in the definition.", "locator": "Section 2.2", "alias_kind": "name" }, { "text": "C", "source": "KLLPV26", "locator": "Definition 22", "status": "source_checked", "normalization": "same definition", "alias_kind": "notation" }, { "text": "N", "source": "SW86", "locator": "Section I, p.31", "status": "source_checked", "normalization": "Nondeterministic complexity means maximum over both output orientations, modern C.", "alias_kind": "notation" }, { "text": "HSC", "source": "CDMRS23", "locator": "Definition 2.4", "status": "source_checked", "normalization": "A set hits every minimal sensitive block iff fixing it certifies the target value." }, { "text": "C_w(f)", "source": "Nis89", "locator": "Section 2, p.330", "status": "source_checked", "normalization": "Input-local notation, not the global maximum." } ], "notes": " Partial-function certificates here are weak: only promised opposite-output inputs must be excluded. Strong certificates that exclude off-promise inputs are a different measure.", "definition_locator": "KLLPV26 Definition 22", "definition_audit": "Definition and normalization inspected; historical alias inventory remains incomplete.", "alias_attribution_history": [ { "text": "C", "source": "BdW02", "status": "seed_alias_exact_occurrence_audit_pending", "normalization": "See definition; verify exact typography during source audit.", "disposition": "replaced_by_source_checked_occurrence", "replacement_source": "KLLPV26" } ], "alias_candidates": [] }, { "id": "C0", "name": "0-certificate complexity", "symbol": "C_0", "definition": "Maximum minimum certificate size over 0-inputs.", "definition_status": "source_checked", "family": "one_sided", "sources": [ "BdW02", "KLLPV26" ], "aliases": [ { "text": "C^0", "source": "KT13", "locator": "Section 1 notation paragraph", "status": "source_checked", "normalization": "same definition" }, { "text": "C_0", "source": "KLLPV26", "locator": "Definition 22", "status": "source_checked", "normalization": "same definition", "alias_kind": "notation" }, { "text": "CNF width", "source": "GPW18", "locator": "Section 1, complement of nondeterministic model", "status": "source_checked", "normalization": "Minimum width over CNFs; follows by complementing the DNF characterization.", "alias_kind": "name" }, { "text": "coNP^dt(f)", "source": "Goos15", "status": "source_checked", "locator": "Sections 1.1–2, pp.2–4", "normalization": "Total Boolean functions; superscript dt denotes query model.", "alias_kind": "symbol" } ], "notes": " Partial-function certificates here are weak: only promised opposite-output inputs must be excluded. Strong certificates that exclude off-promise inputs are a different measure.", "definition_locator": "KLLPV26 Definition 22", "definition_audit": "Definition and normalization inspected; historical alias inventory remains incomplete.", "alias_attribution_history": [ { "text": "C_0", "source": "BdW02", "status": "seed_alias_exact_occurrence_audit_pending", "normalization": "See definition; verify exact typography during source audit.", "disposition": "replaced_by_source_checked_occurrence", "replacement_source": "KLLPV26" } ], "alias_candidates": [] }, { "id": "C1", "name": "1-certificate complexity", "symbol": "C_1", "definition": "Maximum minimum certificate size over 1-inputs.", "definition_status": "source_checked", "family": "one_sided", "sources": [ "BdW02", "KLLPV26" ], "aliases": [ { "text": "C^1", "source": "KT13", "locator": "Section 1 notation paragraph", "status": "source_checked", "normalization": "same definition" }, { "text": "NP^dt", "source": "GPW18", "locator": "Section 1, query notation", "status": "source_checked", "normalization": "same definition", "alias_kind": "notation" }, { "text": "C_1", "source": "KLLPV26", "locator": "Definition 22", "status": "source_checked", "normalization": "same definition", "alias_kind": "notation" }, { "text": "DNF width", "source": "GPW18", "locator": "Section 1", "status": "source_checked", "normalization": "Minimum width over DNFs for a total function.", "alias_kind": "name" }, { "text": "C^(1)(f)", "source": "Nis89", "locator": "Section 4, p.333", "status": "source_checked", "normalization": "As defined in the original." }, { "text": "NP^dt(f)", "source": "Goos15", "status": "source_checked", "locator": "Sections 1.1–2, pp.2–4", "normalization": "Total Boolean functions; superscript dt denotes query model.", "alias_kind": "symbol" } ], "notes": " Partial-function certificates here are weak: only promised opposite-output inputs must be excluded. Strong certificates that exclude off-promise inputs are a different measure.", "definition_locator": "KLLPV26 Definition 22", "definition_audit": "Definition and normalization inspected; historical alias inventory remains incomplete.", "alias_attribution_history": [ { "text": "C_1", "source": "BdW02", "status": "seed_alias_exact_occurrence_audit_pending", "normalization": "See definition; verify exact typography during source audit.", "disposition": "replaced_by_source_checked_occurrence", "replacement_source": "KLLPV26" } ], "alias_candidates": [] }, { "id": "fbs", "name": "fractional block sensitivity", "symbol": "fbs", "definition": "Maximum over x of the LP packing of sensitive blocks: nonnegative weights with total weight through any coordinate at most 1.", "definition_status": "source_checked", "family": "core", "sources": [ "KT13", "KLLPV26" ], "aliases": [ { "text": "fbs", "source": "KT13", "status": "source_checked", "normalization": "same definition", "locator": "Section 2", "alias_kind": "notation" }, { "text": "bs^*", "source": "GSS13", "locator": "Section 2", "status": "source_checked", "normalization": "Fractional block packing LP.", "alias_kind": "notation" } ], "notes": "", "definition_locator": "KLLPV26 Definition 34", "alias_candidates": [] }, { "id": "FC", "name": "fractional certificate complexity", "symbol": "FC", "definition": "Maximum over x of the LP covering of sensitive blocks by nonnegative coordinate weights, each sensitive block receiving weight at least 1.", "definition_status": "source_checked", "family": "core", "sources": [ "KT13", "KLLPV26" ], "aliases": [ { "text": "FC", "source": "KT13", "status": "source_checked", "normalization": "same definition", "locator": "Section 2", "alias_kind": "notation" }, { "text": "C^*", "source": "GSS13", "locator": "Section 2 and Appendix A", "status": "source_checked", "normalization": "Fractional certificate LP; within constants of RC and exactly dual to fbs.", "alias_kind": "notation" }, { "text": "randomized certificate complexity", "source": "FLLV20notes", "locator": "Section 2, p.3, immediately after the certificate LP relaxation", "status": "source_checked", "alias_kind": "name", "normalization": "Here the literal name labels the fractional covering LP, numerically FC. This must not identify its exact value with Aar02 operational expected-query RC; the measures are only constant-factor equivalent under the stored verifier convention." } ], "notes": "LP duality gives FC=fbs under this definition. Keep RC as a distinct, only constant-factor equivalent measure.", "definition_locator": "KLLPV26 Definition 24", "definition_audit": "Definition and normalization inspected; historical alias inventory remains incomplete.", "alias_candidates": [] }, { "id": "RC", "name": "randomized certificate complexity", "symbol": "RC", "definition": "For each fixed x, minimum worst-case expected number of classical queries of a verifier accepting x with probability 1 and rejecting every promised y with f(y)!=f(x) with probability at least 1/2; maximize over x. Its behavior on other same-output inputs is unrestricted.", "definition_status": "source_checked", "family": "core", "sources": [ "Aar02" ], "aliases": [ { "text": "RC", "source": "Aar02", "status": "source_checked", "normalization": "same definition", "locator": "Definition 3" } ], "notes": "", "definition_locator": "Aar02 Definition 3 and Proposition 4", "alias_candidates": [] }, { "id": "QC", "name": "quantum certificate complexity", "symbol": "QC", "definition": "For each fixed x, minimum quantum query cost of a verifier accepting x with certainty and rejecting every promised y with f(y)!=f(x) with probability at least 1/2; maximize over x. Allowing two-sided constant error changes the measure by at most a constant factor.", "definition_status": "source_checked", "family": "core", "sources": [ "Aar02" ], "aliases": [ { "text": "QC", "source": "Aar02", "status": "source_checked", "normalization": "same definition", "locator": "Definition 3" } ], "notes": " Quantum query cost uses a fixed query cap; the original text defines the quantum analogue of its verifier. Do not identify numerical constants across expected-classical and fixed-quantum cost conventions.", "definition_locator": "Aar02 Definition 3 and Proposition 4", "alias_candidates": [] }, { "id": "UC", "name": "unambiguous certificate complexity", "symbol": "UC", "definition": "Maximum of UC_0 and UC_1; UC_b is minimum maximum width of a disjoint monochromatic b-subcube cover.", "definition_status": "source_checked", "family": "core", "sources": [ "AIK26", "KLLPV26" ], "aliases": [ { "text": "UC", "source": "AIK26", "status": "source_checked", "normalization": "same definition", "locator": "Section 2", "alias_kind": "notation" }, { "text": "2UP^dt", "source": "GPW18", "locator": "Section 1, query notation", "status": "source_checked", "normalization": "same definition", "alias_kind": "notation" }, { "text": "subcube partition complexity", "source": "AKK16", "locator": "Introduction, query complexity", "status": "source_checked", "normalization": "Minimum maximum codimension, not logarithm of the number of partition parts.", "alias_kind": "name" } ], "notes": "Do not equate with subcube partition complexity without checking the exact partition objective. Keep as a candidate alias in the notation-completion backlog.", "definition_locator": "KLLPV26 Definition 23", "definition_audit": "Definition and normalization inspected; historical alias inventory remains incomplete.", "alias_candidates": [] }, { "id": "UC0", "name": "0-unambiguous certificate complexity", "symbol": "UC_0", "definition": "Minimum maximum width of an unambiguous DNF for the complement of f.", "definition_status": "source_checked", "family": "one_sided", "sources": [ "AIK26", "KLLPV26" ], "aliases": [ { "text": "UC_0", "source": "AIK26", "status": "source_checked", "normalization": "same definition", "locator": "Section 2", "alias_kind": "notation" }, { "text": "coUP^dt", "source": "GPW18", "locator": "Section 1, query notation", "status": "source_checked", "normalization": "same definition", "alias_kind": "notation" } ], "notes": "", "definition_locator": "KLLPV26 Definition 23", "definition_audit": "Definition and normalization inspected; historical alias inventory remains incomplete.", "alias_candidates": [] }, { "id": "UC1", "name": "1-unambiguous certificate complexity", "symbol": "UC_1", "definition": "Minimum maximum width of an unambiguous DNF for f.", "definition_status": "source_checked", "family": "one_sided", "sources": [ "AIK26", "KLLPV26" ], "aliases": [ { "text": "UC_1", "source": "AIK26", "status": "source_checked", "normalization": "same definition", "locator": "Section 2", "alias_kind": "notation" }, { "text": "UP^dt", "source": "GPW18", "locator": "Section 1, query notation", "status": "source_checked", "normalization": "same definition", "alias_kind": "notation" }, { "text": "one-sided subcube partition complexity", "source": "AKK16", "locator": "Introduction, quantum query complexity", "status": "source_checked", "normalization": "Positive side; complement gives the other orientation.", "alias_kind": "name" }, { "text": "UP^dt(f)", "source": "Goos15", "status": "source_checked", "locator": "Sections 1.1–2, pp.2–4", "normalization": "Total Boolean functions; superscript dt denotes query model.", "alias_kind": "symbol" } ], "notes": "", "definition_locator": "KLLPV26 Definition 23", "definition_audit": "Definition and normalization inspected; historical alias inventory remains incomplete.", "alias_candidates": [] }, { "id": "UCmin", "name": "minimum one-sided unambiguous certificate complexity", "symbol": "UC_min", "definition": "min(UC_0,UC_1).", "definition_status": "source_checked", "family": "one_sided", "sources": [ "AIK26" ], "aliases": [ { "text": "UC_min", "source": "AIK26", "status": "source_checked", "normalization": "same definition", "locator": "Section 2" } ], "notes": "", "definition_locator": "AIK26 Section 2, preliminary definitions", "alias_candidates": [] }, { "id": "deg", "name": "real exact polynomial degree", "symbol": "deg", "definition": "Minimum degree of a real multilinear polynomial equal to f on its domain; for total functions the representing polynomial is unique.", "definition_status": "source_checked", "family": "core", "sources": [ "NS94", "ABK16" ], "aliases": [ { "text": "deg", "source": "NS94", "status": "source_checked", "normalization": "same definition", "locator": "Definition 1.1", "alias_kind": "notation" }, { "text": "Fourier degree", "source": "NS94", "status": "source_checked", "normalization": "Same measure; source-specific cost/encoding caveats remain in the definition.", "locator": "Definition 1.1; representation by real polynomials", "alias_kind": "name" }, { "text": "real degree", "source": "NS94", "status": "definition_equivalence_checked_literal_name_pending", "normalization": "Same measure; source-specific cost/encoding caveats remain in the definition.", "locator": "Definition 1.1; representation by real polynomials", "alias_kind": "descriptive_label" }, { "text": "Fourier-degree", "source": "ABKBRT21", "locator": "Section 2.3", "status": "source_checked", "normalization": "Equal degree under the source’s Fourier encoding.", "alias_kind": "notation" }, { "text": "gdeg", "source": "Sparse25", "locator": "Definition 2.11 and Remark 2.12", "status": "source_checked", "normalization": "Generalized subcube-indicator degree equals ordinary degree by replacing every complement variable with 1-x_i. Approximate version has pointwise error 1/3." } ], "notes": " No condition off the promised domain; keep distinct from cubeBoundedDeg/cubeBoundedAdeg.", "definition_locator": "ABK16 Section 2.2, pp.7–8 (arXiv v2)", "alias_candidates": [] }, { "id": "adeg", "name": "approximate degree", "symbol": "\\widetilde{deg}", "definition": "Minimum real polynomial degree approximating the 0/1 output within 1/3 in absolute value on the domain.", "definition_status": "source_checked", "family": "core", "sources": [ "ABK16" ], "aliases": [ { "text": "\\widetilde{deg}", "source": "ABK16", "status": "source_checked", "normalization": "same definition", "locator": "Section 2.2", "alias_kind": "notation" }, { "text": "approximate polynomial degree", "source": "ABK16", "status": "source_checked", "normalization": "Same measure; source-specific cost/encoding caveats remain in the definition.", "locator": "Section 2.2", "alias_kind": "name" }, { "text": "tilde gdeg", "source": "Sparse25", "locator": "Definition 2.11 and Remark 2.12", "status": "source_checked", "normalization": "Generalized subcube-indicator degree equals ordinary degree by replacing every complement variable with 1-x_i. Approximate version has pointwise error 1/3." } ], "notes": " No condition off the promised domain; keep distinct from cubeBoundedDeg/cubeBoundedAdeg.", "definition_locator": "ABK16 Section 2.2, pp.7–8 (arXiv v2)", "alias_candidates": [] }, { "id": "adeg_eps", "name": "error-parameterized approximate degree", "symbol": "\\widetilde{deg}_epsilon", "definition": "As adeg, with explicit error epsilon in [0,1/2); bounded versus unrestricted polynomial ranges must be recorded.", "definition_status": "source_checked", "family": "parameterized", "sources": [ "ABKBRT21", "ABK16" ], "aliases": [ { "text": "\\widetilde{deg}_epsilon", "source": "BT17", "locator": "Section 2.1", "status": "source_checked", "normalization": "Signed target in this source: epsilon signed error corresponds to epsilon/2 for a 0/1 target." } ], "notes": "Theorem 18 uses bounded signed-range approximation; constants cannot be copied to another normalization. No condition off the promised domain; keep distinct from cubeBoundedDeg/cubeBoundedAdeg.", "definition_locator": "ABK16 Section 2.2, pp.7–8 (arXiv v2)", "alias_candidates": [] }, { "id": "adeg_bounded", "name": "bounded-range approximate degree", "symbol": "\\widetilde{deg}^{bounded}_epsilon", "definition": "Minimum degree of q with 0<=q(x)<=epsilon on 0-inputs and 1-epsilon<=q(x)<=1 on 1-inputs. Equivalently, for signed target 2f-1 use 2q-1 in [-1,-1+2epsilon] and [1-2epsilon,1].", "definition_status": "source_checked", "family": "parameterized", "sources": [ "ABKBRT21" ], "aliases": [ { "text": "\\widetilde{deg}_epsilon", "source": "ABKBRT21", "status": "source_checked", "normalization": "Bounded range on Boolean inputs; source epsilon convention. This occurrence belongs to adeg_bounded, not unrestricted adeg_eps.", "locator": "Section 2.3", "alias_kind": "notation" } ], "notes": "Repository-qualified symbol; source writes a tilde-degree symbol. Keep separate from unrestricted absolute-error degree.", "definition_locator": "ABKBRT21 Section 2.3", "repository_display_variants": [ { "text": "\\widetilde{deg}^{bounded}_epsilon", "source": "ABKBRT21", "status": "repository_display_not_a_historical_occurrence", "normalization": "See definition; verify exact typography during source audit." } ] }, { "id": "lambda", "name": "spectral sensitivity", "symbol": "lambda", "definition": "Spectral norm of the adjacency matrix of the sensitivity graph.", "definition_status": "source_checked", "family": "core", "sources": [ "ABKBRT21", "KLLPV26" ], "aliases": [ { "text": "lambda", "source": "ABKBRT21", "status": "source_checked", "normalization": "same definition", "locator": "Section 3", "alias_kind": "notation" }, { "text": "K", "source": "ABKBRT21", "status": "source_checked", "normalization": "Koutsoupias complexity, with norm unsquared; square gives formula-size lower bound.", "locator": "Appendix A.1, Theorem 28", "alias_kind": "notation" }, { "text": "Adv_1", "source": "ABKBRT21", "locator": "Appendix A.1, Theorem 28", "status": "source_checked", "normalization": "Single-bit positive adversary, equal to lambda.", "alias_kind": "notation" }, { "text": "Adv_1^{pm}", "source": "ABKBRT21", "locator": "Appendix A.1, Theorem 28", "status": "source_checked", "normalization": "Single-bit negative adversary, equal to lambda.", "alias_kind": "notation" }, { "text": "SA_1", "source": "ABKBRT21", "locator": "Appendix A.1, Theorem 28", "status": "source_checked", "normalization": "Single-bit restrictions of the respective adversary formulations; equal to lambda, not to the unrestricted adversary.", "alias_kind": "notation" }, { "text": "SWA_1", "source": "ABKBRT21", "locator": "Appendix A.1, Theorem 28", "status": "source_checked", "normalization": "Single-bit restrictions of the respective adversary formulations; equal to lambda, not to the unrestricted adversary.", "alias_kind": "notation" }, { "text": "MM_1", "source": "ABKBRT21", "locator": "Appendix A.1, Theorem 28", "status": "source_checked", "normalization": "Single-bit restrictions of the respective adversary formulations; equal to lambda, not to the unrestricted adversary.", "alias_kind": "notation" }, { "text": "GSA_1", "source": "ABKBRT21", "locator": "Appendix A.1, Theorem 28", "status": "source_checked", "normalization": "Single-bit restrictions of the respective adversary formulations; equal to lambda, not to the unrestricted adversary.", "alias_kind": "notation" } ], "notes": "Koutsoupias K and the single-bit positive/negative adversary formulations equal lambda in ABKBRT21 Theorem 28. The square of K is the formula-size lower bound; do not confuse the squared normalization with K itself. On a promise domain, take only edges whose endpoints are both promised and have different outputs; a nonconstant promised function can have an empty sensitivity graph.", "definition_locator": "KLLPV26 Definition 32", "alias_candidates": [] }, { "id": "rdeg", "name": "exact rational degree", "symbol": "rdeg", "definition": "Minimum max(deg p,deg q) such that p/q equals f on its domain and q is nonzero there.", "definition_status": "source_checked", "family": "core", "sources": [ "IJKKSWW23", "KLLPV26" ], "aliases": [ { "text": "rational degree", "source": "IJKKSWW23", "status": "source_checked", "normalization": "Same measure; source-specific cost/encoding caveats remain in the definition.", "locator": "Introduction and definitions", "alias_kind": "name" }, { "text": "rdeg", "source": "KLLPV26", "locator": "Definition 37", "status": "source_checked", "normalization": "same definition", "alias_kind": "notation" } ], "notes": "", "definition_locator": "KLLPV26 Definition 37", "definition_audit": "Definition and normalization inspected; historical alias inventory remains incomplete.", "alias_attribution_history": [ { "text": "rdeg", "source": "IJKKSWW23", "status": "seed_alias_exact_occurrence_audit_pending", "normalization": "See definition; verify exact typography during source audit.", "disposition": "replaced_by_source_checked_occurrence", "replacement_source": "KLLPV26" } ], "alias_candidates": [] }, { "id": "ndeg", "name": "nondeterministic polynomial degree", "symbol": "ndeg", "definition": "Minimum degree of a real polynomial nonzero exactly on the 1-inputs.", "definition_status": "source_checked", "family": "polynomial", "sources": [ "IJKKSWW23", "KLLPV26" ], "aliases": [ { "text": "ndeg", "source": "KLLPV26", "locator": "Definition 36", "status": "source_checked", "normalization": "same definition", "alias_kind": "notation" } ], "notes": "Orientation matters. Extract precise original source and quantum nondeterminism characterization.", "definition_locator": "KLLPV26 Definition 36", "definition_audit": "Definition and normalization inspected; historical alias inventory remains incomplete.", "alias_attribution_history": [ { "text": "ndeg", "source": "IJKKSWW23", "status": "seed_alias_exact_occurrence_audit_pending", "normalization": "See definition; verify exact typography during source audit.", "disposition": "replaced_by_source_checked_occurrence", "replacement_source": "KLLPV26" } ], "alias_candidates": [] }, { "id": "AdvPlus", "name": "positive-weight adversary bound", "symbol": "Adv", "definition": "Maximum ||Gamma||/max_i ||Gamma Hadamard D_i|| over nonzero symmetric entrywise nonnegative adversary matrices Gamma supported on pairs with different outputs; D_i[x,y]=1 iff x_i!=y_i. Norm is spectral norm. For a partial f matrices are indexed by its promised domain.", "definition_status": "source_checked", "family": "core", "sources": [ "BK19", "ABKBRT21" ], "aliases": [ { "text": "Adv", "source": "BK19", "status": "source_checked", "normalization": "Same measure; source-specific cost/encoding caveats remain in the definition.", "locator": "Section 1.1", "alias_kind": "notation" }, { "text": "positive adversary", "source": "BK19", "status": "definition_equivalence_checked_literal_name_pending", "normalization": "Same measure; source-specific cost/encoding caveats remain in the definition.", "locator": "Section 1.1", "alias_kind": "descriptive_label" }, { "text": "MM", "source": "BK19", "locator": "Section 1.1 and references to equivalent formulations", "status": "source_checked", "normalization": "same definition" }, { "text": "SA", "source": "ABKBRT21", "locator": "Definition 8", "status": "source_checked", "normalization": "positive spectral adversary" } ], "notes": "Adv without a superscript can mean the general adversary elsewhere; source-specific identification is mandatory.", "definition_locator": "ABKBRT21 Definition 8", "alias_attribution_history": [ { "text": "SA", "source": "BK19", "status": "seed_alias_exact_occurrence_audit_pending", "normalization": "See definition; verify exact typography during source audit.", "disposition": "replaced_by_source_checked_occurrence", "replacement_source": "ABKBRT21" } ], "alias_candidates": [] }, { "id": "AdvGeneral", "name": "general adversary bound", "symbol": "Adv^{\\pm}", "definition": "Maximum ||Gamma||/max_i ||Gamma o D_i|| over real symmetric matrices Gamma indexed by promised inputs, with Gamma_xy=0 whenever f(x)=f(y); D_i(x,y)=1 iff x_i!=y_i. Entries may be negative.", "definition_status": "source_checked", "family": "extension", "sources": [ "ABK16", "HLS07", "LMRSS11" ], "aliases": [ { "text": "Adv^{\\pm}", "source": "ABK16", "locator": "Section 2; general-adversary characterization cited in Section 5", "status": "source_checked", "normalization": "The spectral norm is used; the general adversary differs from the positive-weight bound. Original HLS/LMRSS source reconciliation is in the bibliography queue." }, { "text": "negative-weight adversary bound", "source": "ABK16", "locator": "Section 2; general-adversary characterization cited in Section 5", "status": "source_checked", "normalization": "The spectral norm is used; the general adversary differs from the positive-weight bound. Original HLS/LMRSS source reconciliation is in the bibliography queue." }, { "text": "ADV^{pm}", "source": "HLS07", "locator": "Definition 2", "status": "source_checked", "normalization": "TeX transcription of the printed plus/minus superscript; signed spectral adversary." } ], "notes": "Negative entries are allowed. HLS07 supplies the original generalization and explicit error-dependent lower bound; LMRSS11 gives a tight characterization for finite input/output alphabets.", "definition_locator": "HLS07 Definition 2; LMRSS11 Definition 3.3", "definition_audit_date": "2026-10-17", "alias_candidates": [], "alias_attribution_history": [ { "text": "Adv±", "source": "ABKBRT21", "status": "seed_alias_exact_occurrence_audit_pending", "normalization": "See definition; verify exact typography during source audit.", "disposition": "replaced_by_original_HLS07_occurrence" } ] }, { "id": "QD", "name": "quantum distinguishing complexity", "symbol": "QD", "definition": "Minimum k for a k-query quantum algorithm producing states rho_x with trace distance at least 1/6 for every promised opposite-output pair x,y.", "definition_status": "source_checked", "family": "extension", "sources": [ "BK19" ], "aliases": [ { "text": "QD", "source": "BK19", "locator": "Definition 5, PDF p.7", "status": "source_checked", "normalization": "Trace distance is half the Schatten 1-norm. DQA is attributed to Barnum–Saks–Szegedy 2001 Remark 1 in Section 1.1; retain that historical attribution separately." }, { "text": "DQA", "source": "BK19", "locator": "Definition 5, PDF p.7", "status": "source_checked", "normalization": "Trace distance is half the Schatten 1-norm. DQA is attributed to Barnum–Saks–Szegedy 2001 Remark 1 in Section 1.1; retain that historical attribution separately." } ], "notes": "Trace distance is half the Schatten 1-norm. DQA is attributed to Barnum–Saks–Szegedy 2001 Remark 1 in Section 1.1; retain that historical attribution separately.", "definition_locator": "Definition 5, PDF p.7", "definition_audit_date": "2026-10-17", "alias_candidates": [] }, { "id": "QSZK", "name": "quantum statistical-zero-knowledge query complexity", "symbol": "QSZK", "definition": "Minimum total queries of two state-preparation algorithms producing rho_x,sigma_x with trace distance >=2/3 on 1-inputs and <=1/3 on 0-inputs.", "definition_status": "source_checked", "family": "extension", "sources": [ "BK19" ], "aliases": [ { "text": "QSZK", "source": "BK19", "locator": "Definition 14, PDF p.12", "status": "source_checked", "normalization": "This is the complete-problem-based state-distance definition, not an automatically identical interactive-protocol cost." } ], "notes": "This is the complete-problem-based state-distance definition, not an automatically identical interactive-protocol cost.", "definition_locator": "Definition 14, PDF p.12", "definition_audit_date": "2026-10-17", "alias_candidates": [] }, { "id": "RS", "name": "randomized sabotage complexity", "symbol": "RS", "definition": "R_0(f_sab): a sabotaged input is a partial assignment with blanks all marked * or all marked dagger, consistent with both a 0-input and a 1-input of f; output which marker occurs.", "definition_status": "source_checked", "family": "extension", "sources": [ "CG22" ], "aliases": [ { "text": "RS", "source": "CG22", "locator": "Definition 5.6", "status": "source_checked", "normalization": "Zero-error expected queries. Oracle alphabet {0,1,*,dagger}; underlying f may be partial. Original definition: Ben-David–Kothari randomized query composition and sabotage complexity." } ], "notes": "Zero-error expected queries. Oracle alphabet {0,1,*,dagger}; underlying f may be partial. Original definition: Ben-David–Kothari randomized query composition and sabotage complexity.", "definition_locator": "Definition 5.6", "definition_audit_date": "2026-10-17", "alias_candidates": [] }, { "id": "QS", "name": "weak-input quantum sabotage complexity", "symbol": "QS_{weak}", "definition": "Bounded-error quantum queries to the sabotaged string alone, distinguishing * from dagger in the sabotage promise.", "definition_status": "source_checked", "family": "extension", "sources": [ "CMP24" ], "aliases": [ { "text": "QS_{weak}", "source": "CMP24", "locator": "Section 2 and Definition 1", "status": "source_checked", "normalization": "Do not identify with QD. BK19 suggests the name quantum sabotage for QD, whereas CMP24 studies four distinct sabotage oracle/output variants." }, { "text": "QS", "source": "CMP24", "locator": "Section 1.1, before the four-model split", "status": "source_checked", "normalization": "Initially Q(f_sab), the weak-input distinguishing model. Later unqualified QS may refer schematically to a selected variant; retain the model subscript when making comparisons." } ], "notes": "Do not identify with QD. BK19 suggests the name quantum sabotage for QD, whereas CMP24 studies four distinct sabotage oracle/output variants.", "definition_locator": "Section 2 and Definition 1", "definition_audit_date": "2026-10-17", "alias_candidates": [], "alias_attribution_history": [ { "text": "QS", "source": "KLLPV26", "status": "seed_alias_exact_occurrence_audit_pending", "normalization": "See definition; verify exact typography during source audit.", "disposition": "unqualified_KLLPV26_is_schematic_across_four_models_not_a_weak_model_identity" } ] }, { "id": "CGpub", "name": "public-coin certificate game complexity", "symbol": "CG^{pub}", "definition": "Inverse of the best worst-pair winning probability in the zero-communication game: x is a 0-input, y a 1-input, and players must output the same index i with x_i!=y_i. Strategies use a shared random distribution over deterministic output maps.", "definition_status": "source_checked", "family": "extension", "sources": [ "CG22" ], "aliases": [ { "text": "CG^{pub}", "source": "CG22", "locator": "Definition 2.4", "status": "source_checked", "normalization": "Possibly partial f. This is inverse probability, not its logarithm." } ], "notes": "Possibly partial f. This is inverse probability, not its logarithm.", "definition_locator": "Definition 2.4", "definition_audit_date": "2026-10-17", "alias_candidates": [], "alias_attribution_history": [ { "text": "source-specific", "source": "CG22", "status": "seed_alias_exact_occurrence_audit_pending", "normalization": "See definition; verify exact typography during source audit.", "disposition": "withdrawn_placeholder_not_a_notation" } ] }, { "id": "CGpriv", "name": "private-coin certificate game complexity", "symbol": "CG^{priv}", "definition": "Inverse of the best worst-pair winning probability in the zero-communication game: x is a 0-input, y a 1-input, and players must output the same index i with x_i!=y_i. Strategies use independent randomized output maps.", "definition_status": "source_checked", "family": "extension", "sources": [ "CG22" ], "aliases": [ { "text": "CG^{priv}", "source": "CG22", "locator": "Definition 2.3", "status": "source_checked", "normalization": "Possibly partial f. This is inverse probability, not its logarithm." } ], "notes": "Possibly partial f. This is inverse probability, not its logarithm.", "definition_locator": "Definition 2.3", "definition_audit_date": "2026-10-17", "alias_candidates": [], "alias_attribution_history": [ { "text": "source-specific", "source": "CG22", "status": "seed_alias_exact_occurrence_audit_pending", "normalization": "See definition; verify exact typography during source audit.", "disposition": "withdrawn_placeholder_not_a_notation" } ] }, { "id": "CGent", "name": "entangled certificate game complexity", "symbol": "CG^*", "definition": "Inverse of the best worst-pair winning probability in the zero-communication game: x is a 0-input, y a 1-input, and players must output the same index i with x_i!=y_i. Strategies use a fixed input-independent bipartite state and local measurements.", "definition_status": "source_checked", "family": "extension", "sources": [ "CG22" ], "aliases": [ { "text": "CG^*", "source": "CG22", "locator": "Definition 2.5", "status": "source_checked", "normalization": "Possibly partial f. This is inverse probability, not its logarithm." } ], "notes": "Possibly partial f. This is inverse probability, not its logarithm.", "definition_locator": "Definition 2.5", "definition_audit_date": "2026-10-17", "alias_candidates": [], "alias_attribution_history": [ { "text": "source-specific", "source": "CG22", "status": "seed_alias_exact_occurrence_audit_pending", "normalization": "See definition; verify exact typography during source audit.", "disposition": "withdrawn_placeholder_not_a_notation" } ] }, { "id": "CGns", "name": "nonsignaling certificate game complexity", "symbol": "CG^{ns}", "definition": "Inverse of the best worst-pair winning probability in the zero-communication game: x is a 0-input, y a 1-input, and players must output the same index i with x_i!=y_i. Strategies use conditional output distributions whose marginals do not depend on the other input.", "definition_status": "source_checked", "family": "extension", "sources": [ "CG22" ], "aliases": [ { "text": "CG^{ns}", "source": "CG22", "locator": "Definitions 2.1 and 2.6", "status": "source_checked", "normalization": "Possibly partial f. This is inverse probability, not its logarithm." } ], "notes": "Possibly partial f. This is inverse probability, not its logarithm.", "definition_locator": "Definitions 2.1 and 2.6", "definition_audit_date": "2026-10-17", "alias_candidates": [], "alias_attribution_history": [ { "text": "source-specific", "source": "CG22", "status": "seed_alias_exact_occurrence_audit_pending", "normalization": "See definition; verify exact typography during source audit.", "disposition": "withdrawn_placeholder_not_a_notation" } ] }, { "id": "conical", "name": "two-sided approximate nonnegative degree for decision functions", "symbol": "\\widetilde{deg}^{+2}", "definition": "Minimum maximum conjunction width in nonnegative linear combinations {h_s} of conjunctions of literals, with sum_s h_s<=1 everywhere and sum_{s valid for x} h_s(x)>=2/3 on the promise. Here the valid output set is the singleton {f(x)}.", "definition_status": "source_checked", "family": "extension", "sources": [ "BK24" ], "aliases": [ { "text": "\\widetilde{deg}^{+2}", "source": "BK24", "locator": "Definition 12, decision-function specialization", "status": "source_checked", "normalization": "As in the cited definition." }, { "text": "conical junta degree", "source": "BK24", "locator": "Definition 12, decision-function specialization", "status": "source_checked", "normalization": "As in the cited definition." } ], "notes": "", "definition_locator": "Definition 12, decision-function specialization", "definition_audit_date": "2026-10-17", "alias_candidates": [], "alias_attribution_history": [ { "text": "source-specific", "source": "BK24", "status": "seed_alias_exact_occurrence_audit_pending", "normalization": "See definition; verify exact typography during source audit.", "disposition": "withdrawn_placeholder_not_a_notation" } ] }, { "id": "rel_adeg", "name": "bounded approximate degree for relations", "symbol": "\\widetilde{deg}", "definition": "Minimum maximum degree of real polynomials {p_s}, one per output, with p_s(x)>=0 and sum_s p_s(x)<=1 on the entire cube, and sum_{s valid for x} p_s(x)>=1-epsilon on promised inputs.", "definition_status": "source_checked", "family": "extension", "sources": [ "BK24" ], "aliases": [ { "text": "\\widetilde{deg}", "source": "BK24", "locator": "Definition 10, PDF pp.11–12", "status": "source_checked", "normalization": "Default epsilon=1/3. Positivity and normalization also hold off-promise. Single-output-function degree is a separate specialization." } ], "notes": "Default epsilon=1/3. Positivity and normalization also hold off-promise. Single-output-function degree is a separate specialization.", "definition_locator": "Definition 10, PDF pp.11–12", "definition_audit_date": "2026-10-17", "alias_candidates": [], "alias_attribution_history": [ { "text": "source-specific", "source": "BK24", "status": "seed_alias_exact_occurrence_audit_pending", "normalization": "See definition; verify exact typography during source audit.", "disposition": "withdrawn_placeholder_not_a_notation" } ] }, { "id": "Rexp", "name": "randomized query complexity in expectation", "symbol": "RE", "definition": "Minimum worst-case query cap of a randomized algorithm outputting a nonnegative real variable whose expectation is f(x) on every input.", "definition_status": "source_checked", "family": "extension", "sources": [ "KLW14" ], "aliases": [ { "text": "RE", "source": "KLW14", "locator": "Section 2.3 and Theorem 9", "status": "source_checked", "normalization": "f maps the full cube to nonnegative reals. This is not expected running time." } ], "notes": "f maps the full cube to nonnegative reals. This is not expected running time.", "definition_locator": "Section 2.3 and Theorem 9", "definition_audit_date": "2026-10-17", "alias_candidates": [], "alias_attribution_history": [ { "text": "source-specific", "source": "KLW14", "status": "seed_alias_exact_occurrence_audit_pending", "normalization": "See definition; verify exact typography during source audit.", "disposition": "withdrawn_placeholder_not_a_notation" } ] }, { "id": "Qexp", "name": "quantum query complexity in expectation", "symbol": "QE", "definition": "Quantum analogue of RE: output a nonnegative real variable with expectation f(x), minimizing worst-case oracle calls.", "definition_status": "source_checked", "family": "extension", "sources": [ "KLW14" ], "aliases": [ { "text": "QE", "source": "KLW14", "locator": "Section 2.3 and Theorem 12", "status": "source_checked", "normalization": "Source QE collides with exact quantum query complexity Q_E. Preserve the distinction by stable IDs." } ], "notes": "Source QE collides with exact quantum query complexity Q_E. Preserve the distinction by stable IDs.", "definition_locator": "Section 2.3 and Theorem 12", "definition_audit_date": "2026-10-17", "alias_candidates": [], "alias_attribution_history": [ { "text": "source-specific", "source": "KLW14", "status": "seed_alias_exact_occurrence_audit_pending", "normalization": "See definition; verify exact typography during source audit.", "disposition": "withdrawn_placeholder_not_a_notation" } ] }, { "id": "literaldeg", "name": "nonnegative literal degree", "symbol": "ldeg_+", "definition": "Minimum d such that f is a nonnegative linear combination of conjunctions of at most d literals.", "definition_status": "source_checked", "family": "extension", "sources": [ "KLW14" ], "aliases": [ { "text": "ldeg_+", "source": "KLW14", "locator": "Section 2.3 and Theorem 9", "status": "source_checked", "normalization": "As in the cited definition." } ], "notes": "", "definition_locator": "Section 2.3 and Theorem 9", "definition_audit_date": "2026-10-17", "alias_candidates": [], "alias_attribution_history": [ { "text": "source-specific", "source": "KLW14", "status": "seed_alias_exact_occurrence_audit_pending", "normalization": "See definition; verify exact typography during source audit.", "disposition": "withdrawn_placeholder_not_a_notation" } ] }, { "id": "sosdeg", "name": "sum-of-squares degree", "symbol": "deg_sos", "definition": "Minimum d such that f(x)=sum_j p_j(x)^2 on the cube with every real p_j of degree <=d.", "definition_status": "source_checked", "family": "extension", "sources": [ "KLW14" ], "aliases": [ { "text": "deg_sos", "source": "KLW14", "locator": "Section 2.3 and Theorem 12", "status": "source_checked", "normalization": "The roots have degree d; the sum of squares has formal degree at most 2d. Other SOS conventions count 2d." } ], "notes": "The roots have degree d; the sum of squares has formal degree at most 2d. Other SOS conventions count 2d.", "definition_locator": "Section 2.3 and Theorem 12", "definition_audit_date": "2026-10-17", "alias_candidates": [], "alias_attribution_history": [ { "text": "source-specific", "source": "KLW14", "status": "seed_alias_exact_occurrence_audit_pending", "normalization": "See definition; verify exact typography during source audit.", "disposition": "withdrawn_placeholder_not_a_notation" } ] }, { "id": "psR", "name": "pseudo-deterministic randomized classical query complexity", "symbol": "psR", "definition": "Minimum worst-case query cap of a randomized classical algorithm for a relation S, for which there exists a fixed valid selector g with Pr[A(x)=g(x)]>=2/3 for every promised input.", "definition_status": "source_checked", "family": "extension", "sources": [ "AGL26", "CDM23full", "GIPS21" ], "aliases": [ { "text": "psR", "source": "AGL26", "locator": "Definitions 2.1–2.2", "status": "source_checked", "normalization": "The canonical selector may depend on the algorithm, but cannot depend on its random coins or measurement outcomes." }, { "text": "psD^dt", "source": "CDM23full", "locator": "Section 2: decision-tree and pseudo-deterministic definitions", "status": "source_checked", "normalization": "Depth is worst case; psD^dt is randomized pseudo-deterministic depth despite the letter D. The notation also applies to search relations." }, { "text": "psP^dt", "source": "GIPS21", "locator": "Theorem 4 and Section 2", "status": "source_checked", "normalization": "Search relations; classical/quantum bounded-error depth and pseudo-deterministic depth, as specified by the stable ID." } ], "notes": "The canonical selector may depend on the algorithm, but cannot depend on its random coins or measurement outcomes.", "definition_locator": "Definitions 2.1–2.2", "definition_audit_date": "2026-10-17", "alias_candidates": [] }, { "id": "psQ", "name": "pseudo-deterministic quantum query complexity", "symbol": "psQ", "definition": "Minimum worst-case query cap of a quantum algorithm for a relation S, for which there exists a fixed valid selector g with Pr[A(x)=g(x)]>=2/3 for every promised input.", "definition_status": "source_checked", "family": "extension", "sources": [ "AGL26", "GIPS21" ], "aliases": [ { "text": "psQ", "source": "AGL26", "locator": "Definitions 2.1–2.2", "status": "source_checked", "normalization": "The canonical selector may depend on the algorithm, but cannot depend on its random coins or measurement outcomes." }, { "text": "psQ^dt", "source": "GIPS21", "locator": "Theorem 4 and Section 2", "status": "source_checked", "normalization": "Search relations; classical/quantum bounded-error depth and pseudo-deterministic depth, as specified by the stable ID." } ], "notes": "The canonical selector may depend on the algorithm, but cannot depend on its random coins or measurement outcomes.", "definition_locator": "Definitions 2.1–2.2", "definition_audit_date": "2026-10-17", "alias_candidates": [] }, { "id": "blocknumber", "name": "block number", "symbol": "N_bs", "definition": "Maximum over x of the number of inclusion-minimal sensitive blocks of f at x.", "definition_status": "source_checked", "family": "extension", "sources": [ "CG26", "CDMRS23" ], "aliases": [ { "text": "N_b", "source": "CG26", "locator": "Definitions 3–4", "status": "source_checked", "normalization": "KT13 Section 5 uses nbs. The size-ell version counts blocks of size at most ell in CG26, unlike exact-size nbs_k in KT13." }, { "text": "nbs", "source": "CG26", "locator": "Definitions 3–4", "status": "source_checked", "normalization": "KT13 Section 5 uses nbs. The size-ell version counts blocks of size at most ell in CG26, unlike exact-size nbs_k in KT13." }, { "text": "#bs", "source": "CDMRS23", "locator": "Definition 2.4", "status": "source_checked", "normalization": "Number of minimal sensitive blocks, not a maximum disjoint packing." } ], "notes": "KT13 Section 5 uses nbs. The size-ell version counts blocks of size at most ell in CG26, unlike exact-size nbs_k in KT13.", "definition_locator": "Definitions 3–4", "definition_audit_date": "2026-10-17", "alias_candidates": [], "alias_attribution_history": [ { "text": "source-specific", "source": "CG26", "status": "seed_alias_exact_occurrence_audit_pending", "normalization": "See definition; verify exact typography during source audit.", "disposition": "withdrawn_placeholder_not_a_notation" } ] }, { "id": "bricknumber", "name": "brick number", "symbol": "N_{br}", "definition": "Maximum over x of the number of nonempty blocks B such that f(x^B)!=f(x^(B minus {i})) for every i in B.", "definition_status": "source_checked", "family": "extension", "sources": [ "CG26" ], "aliases": [ { "text": "N_{br}", "source": "CG26", "locator": "Definitions 7–8", "status": "source_checked", "normalization": "A brick need not itself be a sensitive block when |B|>1. Do not replace the defining condition by minimal sensitivity." } ], "notes": "A brick need not itself be a sensitive block when |B|>1. Do not replace the defining condition by minimal sensitivity.", "definition_locator": "Definitions 7–8", "definition_audit_date": "2026-10-17", "alias_candidates": [], "alias_attribution_history": [ { "text": "source-specific", "source": "CG26", "status": "seed_alias_exact_occurrence_audit_pending", "normalization": "See definition; verify exact typography during source audit.", "disposition": "withdrawn_placeholder_not_a_notation" } ] }, { "id": "rel_conical", "name": "two-sided approximate nonnegative degree for relations", "symbol": "\\widetilde{deg}^{+2}", "definition": "Minimum maximum conjunction width in nonnegative linear combinations {h_s} of conjunctions of literals, with sum_s h_s<=1 everywhere and sum_{s valid for x} h_s(x)>=2/3 on the promise.", "definition_status": "source_checked", "family": "extension", "sources": [ "BK24" ], "aliases": [ { "text": "\\widetilde{deg}^{+2}", "source": "BK24", "locator": "Definition 12", "status": "source_checked", "normalization": "As in the cited definition." }, { "text": "conical junta degree", "source": "BK24", "locator": "Definition 12", "status": "source_checked", "normalization": "As in the cited definition." }, { "text": "two-sided conical junta degree", "source": "BK24", "locator": "Definition 12", "status": "source_checked", "normalization": "As in the cited definition." } ], "notes": "", "definition_locator": "Definition 12", "definition_audit_date": "2026-10-17", "alias_candidates": [] }, { "id": "mbs", "name": "monotone block sensitivity", "symbol": "mbs", "definition": "For each x, fix every coordinate in supp(x) to 1 and let f_x be the function of the remaining coordinates. Take the block sensitivity of f_x at its all-zero input, then maximize over x.", "definition_status": "source_checked", "family": "extension", "sources": [ "BJKM24", "CDM23full", "CDM23", "KLMY21" ], "aliases": [ { "text": "mbs", "source": "BJKM24", "locator": "Definitions 5–8", "status": "source_checked", "normalization": "Defined for arbitrary total f. This is not merely ordinary complexity restricted to monotone f." }, { "text": "bs_0", "source": "CDM23full", "locator": "Section 2, lines 235–245", "status": "source_checked", "normalization": "Flip only input-zero coordinates; NOT the maximum over zero-output inputs. The source expressly identifies these with monotone sensitivity / monotone block sensitivity on total Boolean functions." }, { "text": "bs_0", "source": "CDM23", "locator": "Section 2: directed sensitivity definitions", "status": "source_checked", "normalization": "Subscript is the value of the input bits being flipped, not an output restriction. Final Theorem 5.2 uses this convention." }, { "text": "MBS", "source": "KLMY21", "locator": "Section 2.1", "status": "source_checked", "normalization": "Monotone block sensitivity on the upward restriction at each input." }, { "text": "bs^{0→1}", "source": "CDM25", "locator": "Section 2 and Theorem 6.1", "status": "source_checked", "normalization": "Direction of input-coordinate flips, not output value." } ], "notes": "Defined for arbitrary total f. This is not merely ordinary complexity restricted to monotone f. CDM23 also defines this input-directed quantity for promised and multi-output functions, requiring the flipped input to remain promised.", "definition_locator": "Definitions 5–8", "definition_audit_date": "2026-10-17", "alias_candidates": [] }, { "id": "fmbs", "name": "fractional monotone block sensitivity", "symbol": "fmbs", "definition": "For each x, fix every coordinate in supp(x) to 1 and let f_x be the function of the remaining coordinates. Take the fractional block sensitivity of f_x at its all-zero input, then maximize over x.", "definition_status": "source_checked", "family": "extension", "sources": [ "BJKM24", "CDMRS23", "KLMY21" ], "aliases": [ { "text": "fmbs", "source": "BJKM24", "locator": "Definitions 5–8", "status": "source_checked", "normalization": "Defined for arbitrary total f. This is not merely ordinary complexity restricted to monotone f." }, { "text": "FHSC_0", "source": "CDMRS23", "locator": "Definition 2.6", "status": "source_checked", "normalization": "Fractional hitting-set LP dual to the fractional packing of sensitive input-zero blocks. Not output-zero fractional block sensitivity." }, { "text": "FHSC", "source": "KLMY21", "locator": "Definition 2.13", "status": "source_checked", "normalization": "Fractional input-directed hitting-set LP, equal to FMBS. Interpret the printed smallest-p wording using the explicitly stated LP and largest feasible p." }, { "text": "FMBS", "source": "KLMY21", "locator": "Section 2.1", "status": "source_checked", "normalization": "Fractional monotone block sensitivity, equal by LP duality to FHSC." } ], "notes": "Defined for arbitrary total f. This is not merely ordinary complexity restricted to monotone f.", "definition_locator": "Definitions 5–8", "definition_audit_date": "2026-10-17", "alias_candidates": [] }, { "id": "MCC", "name": "monotone certificate complexity", "symbol": "MCC", "definition": "For each x, fix every coordinate in supp(x) to 1 and let f_x be the function of the remaining coordinates. Take the certificate complexity of f_x at its all-zero input, then maximize over x.", "definition_status": "source_checked", "family": "extension", "sources": [ "BJKM24", "CDMRS23", "KLMY21" ], "aliases": [ { "text": "MCC", "source": "BJKM24", "locator": "Definitions 5–8", "status": "source_checked", "normalization": "Defined for arbitrary total f. This is not merely ordinary complexity restricted to monotone f." }, { "text": "hitting set complexity", "source": "BJKM24", "locator": "Definitions 5–8", "status": "source_checked", "normalization": "Defined for arbitrary total f. This is not merely ordinary complexity restricted to monotone f." }, { "text": "HSC_0", "source": "CDMRS23", "locator": "Definition 2.5", "status": "source_checked", "normalization": "Hit every sensitive block consisting of input-zero positions, at every input. This matches monotone certificate complexity, not output-zero C0." }, { "text": "HSC", "source": "KLMY21", "locator": "Definition 2.10", "status": "source_checked", "normalization": "Input-zero/upward sensitive blocks. CDMRS23 uses bare HSC for ordinary C and HSC_0 for this quantity." } ], "notes": "Defined for arbitrary total f. This is not merely ordinary complexity restricted to monotone f.", "definition_locator": "Definitions 5–8", "definition_audit_date": "2026-10-17", "alias_candidates": [] }, { "id": "ms", "name": "monotone sensitivity", "symbol": "ms", "definition": "For each x, fix every coordinate in supp(x) to 1 and let f_x be the function of the remaining coordinates. Take the sensitivity of f_x at its all-zero input, then maximize over x.", "definition_status": "source_checked", "family": "extension", "sources": [ "BJKM24", "CDM23full", "CDM23" ], "aliases": [ { "text": "ms", "source": "BJKM24", "locator": "Definitions 5–8", "status": "source_checked", "normalization": "Defined for arbitrary total f. This is not merely ordinary complexity restricted to monotone f." }, { "text": "s_0", "source": "CDM23full", "locator": "Section 2, lines 235–245", "status": "source_checked", "normalization": "Flip only input-zero coordinates; NOT the maximum over zero-output inputs. The source expressly identifies these with monotone sensitivity / monotone block sensitivity on total Boolean functions." }, { "text": "s_0", "source": "CDM23", "locator": "Section 2: directed sensitivity definitions", "status": "source_checked", "normalization": "Subscript is the value of the input bits being flipped, not an output restriction. Final Theorem 5.2 uses this convention." } ], "notes": "Defined for arbitrary total f. This is not merely ordinary complexity restricted to monotone f. CDM23 also defines this input-directed quantity for promised and multi-output functions, requiring the flipped input to remain promised.", "definition_locator": "Definitions 5–8", "definition_audit_date": "2026-10-17", "alias_candidates": [] }, { "id": "DTSize", "name": "decision tree leaf size", "symbol": "DTSize", "definition": "Minimum number of leaves of a deterministic bit-query tree computing f.", "definition_status": "source_checked", "family": "extension", "sources": [ "DM23", "CDM23full" ], "aliases": [ { "text": "DTSize", "source": "DM23", "status": "source_checked", "normalization": "same definition", "locator": "Section 2" }, { "text": "DSize^dt", "source": "CDM23full", "locator": "Section 2: decision-tree and pseudo-deterministic definitions", "status": "source_checked", "normalization": "Depth is worst case; psD^dt is randomized pseudo-deterministic depth despite the letter D. The notation also applies to search relations." } ], "notes": "", "definition_locator": "DM23 Section 2, Decision trees", "alias_candidates": [] }, { "id": "DTRank", "name": "decision tree rank", "symbol": "Rank", "definition": "Minimum Strahler rank: leaves have rank 0; unequal child ranks give their maximum, equal child ranks give that rank plus 1.", "definition_status": "source_checked", "family": "extension", "sources": [ "DM23", "CDMRS23", "CMP22" ], "aliases": [ { "text": "Rank", "source": "DM23", "status": "source_checked", "normalization": "same definition", "locator": "Section 2", "alias_kind": "notation" }, { "text": "Strahler number", "source": "DM23", "locator": "Section 1", "status": "source_checked", "normalization": "First a rooted-tree statistic; function measure minimizes over computing trees." }, { "text": "Horton-Strahler number", "source": "DM23", "locator": "Section 1", "status": "source_checked", "normalization": "First a rooted-tree statistic; function measure minimizes over computing trees." }, { "text": "tree rank", "source": "DM23", "locator": "Section 1", "status": "source_checked", "normalization": "First a rooted-tree statistic; function measure minimizes over computing trees." }, { "text": "rank", "source": "CDMRS23", "locator": "Definition 2.10", "status": "source_checked", "normalization": "Minimum Strahler rank of a correct deterministic tree." }, { "text": "rank", "source": "CMP22", "locator": "Definition 3", "status": "source_checked", "normalization": "Strahler rank; randomized rank takes the maximum rank in the support." }, { "text": "r(f)", "source": "EH89", "locator": "Definition p.241", "status": "source_checked", "normalization": "As defined in the original." } ], "notes": "", "definition_locator": "DM23 Section 2, Decision Tree Rank, PDF p.7", "alias_candidates": [] }, { "id": "sparsity", "name": "Fourier sparsity", "symbol": "spar", "definition": "Number of nonzero coefficients in f=sum_S fhat(S) (-1)^(sum_{i in S} x_i), including the empty-set coefficient.", "definition_status": "source_checked", "family": "extension", "sources": [ "DM23" ], "aliases": [ { "text": "spar", "source": "DM23", "locator": "Section 2, Fourier representation, PDF p.6", "status": "source_checked", "normalization": "As in the cited definition." } ], "notes": "", "definition_locator": "Section 2, Fourier representation, PDF p.6", "definition_audit_date": "2026-10-17", "alias_candidates": [] }, { "id": "ANDD", "name": "conjunctive decision tree depth", "symbol": "Conj", "definition": "Minimum worst-case number of queries in a decision tree whose internal nodes query conjunctions of literals, allowing both positive and negated variables.", "definition_status": "source_checked", "family": "extension", "sources": [ "DM23" ], "aliases": [ { "text": "D_wedge", "source": "DM23", "locator": "Section 7, AND-decision tree model", "status": "source_checked", "normalization": "This source model uses monotone conjunctions; do not silently allow arbitrary negated literals." } ], "notes": "The original seed incorrectly restricted queries to monotone conjunctions. Corrected from the full paper.", "definition_locator": "DM23 Section 7.2", "definition_audit_date": "2026-10-17", "alias_candidates": [], "alias_attribution_history": [ { "text": "source-specific", "source": "DM23", "status": "seed_alias_exact_occurrence_audit_pending", "normalization": "Source-specific convention.", "disposition": "withdrawn_placeholder_not_a_notation" } ] }, { "id": "IC", "name": "query instance complexity", "symbol": "InstC", "definition": "Minimum over deterministic decision trees T computing nonconstant f of max_x T(x)/C(f,x).", "definition_status": "source_checked", "family": "extension", "sources": [ "LM26" ], "aliases": [ { "text": "InstC", "source": "LM26", "locator": "Definition 2.3", "status": "source_checked", "normalization": "Competitive ratio. Constant functions require a separate zero-denominator convention; source results involving them are stated separately." } ], "notes": "Competitive ratio. Constant functions require a separate zero-denominator convention; source results involving them are stated separately.", "definition_locator": "Definition 2.3", "definition_audit_date": "2026-10-17", "alias_candidates": [], "alias_attribution_history": [ { "text": "source-specific", "source": "LM26", "status": "seed_alias_exact_occurrence_audit_pending", "normalization": "Source-specific convention.", "disposition": "withdrawn_placeholder_not_a_notation" } ] }, { "id": "hazardD", "name": "deterministic u-query complexity", "symbol": "D_u", "definition": "Minimum depth of a ternary decision tree computing the hazard-free extension. For a Boolean f, its extension maps z in {0,u,1}^n to b if every Boolean resolution of z has value b, and to u otherwise.", "definition_status": "source_checked", "family": "extension", "sources": [ "BKMNSS25" ], "aliases": [ { "text": "D_u", "source": "BKMNSS25", "locator": "Definition 2.1", "status": "source_checked", "normalization": "Input and output alphabets are ternary. The symbol u is a queried value, distinct from an unrevealed coordinate." } ], "notes": "Input and output alphabets are ternary. The symbol u is a queried value, distinct from an unrevealed coordinate.", "definition_locator": "Definition 2.1", "definition_audit_date": "2026-10-17", "alias_candidates": [], "alias_attribution_history": [ { "text": "source-specific", "source": "BKMNSS25", "status": "seed_alias_exact_occurrence_audit_pending", "normalization": "Source-specific convention.", "disposition": "withdrawn_placeholder_not_a_notation" } ] }, { "id": "hazardR", "name": "randomized u-query complexity", "symbol": "R_u", "definition": "Minimum worst-case query cap with error <=1/3 for the hazard-free extension. For a Boolean f, its extension maps z in {0,u,1}^n to b if every Boolean resolution of z has value b, and to u otherwise.", "definition_status": "source_checked", "family": "extension", "sources": [ "BKMNSS25" ], "aliases": [ { "text": "R_u", "source": "BKMNSS25", "locator": "Definition 2.5", "status": "source_checked", "normalization": "Input and output alphabets are ternary. The symbol u is a queried value, distinct from an unrevealed coordinate." } ], "notes": "Input and output alphabets are ternary. The symbol u is a queried value, distinct from an unrevealed coordinate.", "definition_locator": "Definition 2.5", "definition_audit_date": "2026-10-17", "alias_candidates": [], "alias_attribution_history": [ { "text": "source-specific", "source": "BKMNSS25", "status": "seed_alias_exact_occurrence_audit_pending", "normalization": "Source-specific convention.", "disposition": "withdrawn_placeholder_not_a_notation" } ] }, { "id": "hazardQ", "name": "quantum u-query complexity", "symbol": "Q_u", "definition": "Minimum bounded-error queries to |i,b,w> -> |i,b+x_i mod 3,w>, encoding u as 2, for the hazard-free extension. For a Boolean f, its extension maps z in {0,u,1}^n to b if every Boolean resolution of z has value b, and to u otherwise.", "definition_status": "source_checked", "family": "extension", "sources": [ "BKMNSS25" ], "aliases": [ { "text": "Q_u", "source": "BKMNSS25", "locator": "Definition 2.6", "status": "source_checked", "normalization": "Input and output alphabets are ternary. The symbol u is a queried value, distinct from an unrevealed coordinate." } ], "notes": "Input and output alphabets are ternary. The symbol u is a queried value, distinct from an unrevealed coordinate.", "definition_locator": "Definition 2.6", "definition_audit_date": "2026-10-17", "alias_candidates": [], "alias_attribution_history": [ { "text": "source-specific", "source": "BKMNSS25", "status": "seed_alias_exact_occurrence_audit_pending", "normalization": "Source-specific convention.", "disposition": "withdrawn_placeholder_not_a_notation" } ] }, { "id": "rdeg_eps", "name": "approximate rational degree", "symbol": "rdeg_epsilon", "definition": "Minimum max(deg p,deg q) with q nonzero on the domain and |p(x)/q(x)-f(x)|<=epsilon throughout the domain.", "definition_status": "source_checked", "family": "extension", "sources": [ "IJKKSWW23" ], "aliases": [ { "text": "rdeg_epsilon", "source": "IJKKSWW23", "locator": "Introduction and Section 2", "status": "source_checked", "normalization": "Explicit error and denominator nonvanishing condition; no off-promise bound is imposed here." } ], "notes": "Explicit error and denominator nonvanishing condition; no off-promise bound is imposed here.", "definition_locator": "Introduction and Section 2", "definition_audit_date": "2026-10-17", "alias_candidates": [] }, { "id": "PostQ", "name": "postselected bounded-error quantum query complexity", "symbol": "PostQ_epsilon", "definition": "Minimum quantum queries allowing a postselection event of nonzero probability on every promised input, with conditional output error at most epsilon.", "definition_status": "source_checked", "family": "extension", "sources": [ "IJKKSWW23" ], "aliases": [ { "text": "PostQ_epsilon", "source": "IJKKSWW23", "locator": "Section 2, postselection definitions", "status": "source_checked", "normalization": "The postselection success probability need not be inverse polynomial." } ], "notes": "The postselection success probability need not be inverse polynomial.", "definition_locator": "Section 2, postselection definitions", "definition_audit_date": "2026-10-17", "alias_candidates": [] }, { "id": "PostQE", "name": "exact postselected quantum query complexity", "symbol": "PostQ_E", "definition": "Postselected quantum query complexity with zero conditional error and positive postselection probability on every promised input.", "definition_status": "source_checked", "family": "extension", "sources": [ "IJKKSWW23" ], "aliases": [ { "text": "PostQ_E", "source": "IJKKSWW23", "locator": "Section 2 and Appendix A", "status": "source_checked", "normalization": "Exact conditioned output; not zero-error quantum query complexity Q_0." }, { "text": "PostEQP query complexity", "source": "IJKKSWW23", "locator": "Section 2 and Appendix A", "status": "source_checked", "normalization": "Exact conditioned output; not zero-error quantum query complexity Q_0." } ], "notes": "Exact conditioned output; not zero-error quantum query complexity Q_0.", "definition_locator": "Section 2 and Appendix A", "definition_audit_date": "2026-10-17", "alias_candidates": [] }, { "id": "QSweakIndex", "family": "extension", "name": "quantum sabotage complexity (QSweakIndex)", "symbol": "QS^{ind}_{weak}", "definition": "Weak input, but output both the marker type and an index containing it.", "definition_status": "source_checked", "sources": [ "CMP24" ], "definition_locator": "Section 2; four-variant definitions", "definition_audit_date": "2026-10-17", "notes": "All component input queries count; source access conventions retained.", "alias_candidates": [], "aliases": [ { "text": "QS^{ind}_{weak}", "source": "CMP24", "locator": "Section 2; four-variant definitions", "status": "source_checked", "normalization": "All component input queries count; source access conventions retained." } ] }, { "id": "QSstrong", "family": "extension", "name": "quantum sabotage complexity (QSstrong)", "symbol": "QS_{str}", "definition": "Strong input: also query the underlying opposite-output pair x,y; only determine the marker type.", "definition_status": "source_checked", "sources": [ "CMP24" ], "definition_locator": "Section 2; four-variant definitions", "definition_audit_date": "2026-10-17", "notes": "All component input queries count; source access conventions retained.", "alias_candidates": [], "aliases": [ { "text": "QS_{str}", "source": "CMP24", "locator": "Section 2; four-variant definitions", "status": "source_checked", "normalization": "All component input queries count; source access conventions retained." } ] }, { "id": "QSstrongIndex", "family": "extension", "name": "quantum sabotage complexity (QSstrongIndex)", "symbol": "QS^{ind}_{str}", "definition": "Strong input: also query x,y and output the marker type and a differing index.", "definition_status": "source_checked", "sources": [ "CMP24" ], "definition_locator": "Section 2; four-variant definitions", "definition_audit_date": "2026-10-17", "notes": "All component input queries count; source access conventions retained.", "alias_candidates": [], "aliases": [ { "text": "QS^{ind}_{str}", "source": "CMP24", "locator": "Section 2; four-variant definitions", "status": "source_checked", "normalization": "All component input queries count; source access conventions retained." } ] }, { "id": "CGpubSingle", "family": "extension", "name": "single-bit public-coin certificate game complexity", "symbol": "CG^{pub}_{[1]}", "definition": "Inverse of the best worst-pair winning probability in the zero-communication game: x is a 0-input, y a 1-input, and players must output the same index i with x_i!=y_i. Strategies use a shared random distribution over deterministic output maps. Only pairs with Hamming distance one are allowed.", "definition_status": "source_checked", "sources": [ "CG22" ], "definition_locator": "Section 8, Theorems 8.4 and 8.8", "definition_audit_date": "2026-10-17", "notes": "", "alias_candidates": [], "aliases": [ { "text": "CG^{pub}_{[1]}", "source": "CG22", "locator": "Section 8, Theorems 8.4 and 8.8", "status": "source_checked", "normalization": "As in the cited definition." } ] }, { "id": "CGprivSingle", "family": "extension", "name": "single-bit private-coin certificate game complexity", "symbol": "CG^{priv}_{[1]}", "definition": "Inverse of the best worst-pair winning probability in the zero-communication game: x is a 0-input, y a 1-input, and players must output the same index i with x_i!=y_i. Strategies use independent randomized output maps. Only pairs with Hamming distance one are allowed.", "definition_status": "source_checked", "sources": [ "CG22" ], "definition_locator": "Section 8, Theorems 8.4 and 8.8", "definition_audit_date": "2026-10-17", "notes": "", "alias_candidates": [], "aliases": [ { "text": "CG^{priv}_{[1]}", "source": "CG22", "locator": "Section 8, Theorems 8.4 and 8.8", "status": "source_checked", "normalization": "As in the cited definition." } ] }, { "id": "CAdv", "family": "extension", "name": "classical adversary bound", "symbol": "CAdv", "definition": "Minimum max_x sum_i m_x(i) over nonnegative weights with sum_{i:x_i!=y_i} min(m_x(i),m_y(i))>=1 for every opposite-output promised pair.", "definition_status": "source_checked", "sources": [ "BK24" ], "definition_locator": "Definition 14", "definition_audit_date": "2026-10-17", "notes": "CG22 Definition 4.2 uses probability-normalized CMM; its equivalent normalization is recorded. Total-function CAdv=FC does not extend to all partial functions.", "alias_candidates": [], "aliases": [ { "text": "CAdv", "source": "BK24", "locator": "Definition 14", "status": "source_checked", "normalization": "CG22 Definition 4.2 uses probability-normalized CMM; its equivalent normalization is recorded. Total-function CAdv=FC does not extend to all partial functions." }, { "text": "CMM", "source": "BK24", "locator": "Definition 14", "status": "source_checked", "normalization": "CG22 Definition 4.2 uses probability-normalized CMM; its equivalent normalization is recorded. Total-function CAdv=FC does not extend to all partial functions." } ] }, { "id": "EC", "family": "extension", "name": "expectational certificate complexity", "symbol": "EC", "definition": "Minimum max_x sum_i w_xi, with 0<=w_xi<=1 and sum_{i:x_i!=y_i} w_xi w_yi>=1 for every opposite-output pair.", "definition_status": "source_checked", "sources": [ "CG22" ], "definition_locator": "Definition 4.5", "definition_audit_date": "2026-10-17", "notes": "The source extends this definition to partial functions. Distinct from randomized query complexity in expectation.", "alias_candidates": [], "aliases": [ { "text": "EC", "source": "CG22", "locator": "Definition 4.5", "status": "source_checked", "normalization": "The source extends this definition to partial functions. Distinct from randomized query complexity in expectation." } ] }, { "id": "rel_udeg", "family": "extension", "name": "promise-only approximate degree for relations", "symbol": "\\widetilde{udeg}", "definition": "Minimum maximum degree of real polynomials {p_s}, one per output, with p_s(x)>=0 and sum_s p_s(x)<=1 on the promised domain, and sum_{s valid for x} p_s(x)>=1-epsilon on promised inputs.", "definition_status": "source_checked", "sources": [ "BK24" ], "definition_locator": "Definition 11", "definition_audit_date": "2026-10-17", "notes": "No off-promise boundedness. Do not conflate with bounded relation degree.", "alias_candidates": [], "aliases": [ { "text": "\\widetilde{udeg}", "source": "BK24", "locator": "Definition 11", "status": "source_checked", "normalization": "No off-promise boundedness. Do not conflate with bounded relation degree." } ] }, { "id": "rel_deg", "family": "extension", "name": "exact bounded polynomial degree for relations", "symbol": "deg", "definition": "Minimum maximum degree of real polynomials {p_s}, one per output, with p_s(x)>=0 and sum_s p_s(x)<=1 on the entire cube, and sum_{s valid for x} p_s(x)>=1 on promised inputs.", "definition_status": "source_checked", "sources": [ "BK24" ], "definition_locator": "Definition 10, epsilon=0", "definition_audit_date": "2026-10-17", "notes": "", "alias_candidates": [], "aliases": [ { "text": "deg", "source": "BK24", "locator": "Definition 10, epsilon=0", "status": "source_checked", "normalization": "As in the cited definition." } ] }, { "id": "rel_Adv", "family": "extension", "name": "relation positive adversary bound", "symbol": "Adv", "definition": "Minimum max_x sum_i m_x(i), with nonnegative m and sum_{i:x_i!=y_i} sqrt(m_x(i)m_y(i))>=1 whenever the valid-output sets for x,y are disjoint.", "definition_status": "source_checked", "sources": [ "BK24" ], "definition_locator": "Definition 13", "definition_audit_date": "2026-10-17", "notes": "", "alias_candidates": [], "aliases": [ { "text": "Adv", "source": "BK24", "locator": "Definition 13", "status": "source_checked", "normalization": "As in the cited definition." } ] }, { "id": "rel_CAdv", "family": "extension", "name": "relation classical adversary bound", "symbol": "CAdv", "definition": "Minimum max_x sum_i m_x(i), with nonnegative m and sum_{i:x_i!=y_i} min(m_x(i),m_y(i))>=1 whenever the valid-output sets for x,y are disjoint.", "definition_status": "source_checked", "sources": [ "BK24" ], "definition_locator": "Definition 14", "definition_audit_date": "2026-10-17", "notes": "", "alias_candidates": [], "aliases": [ { "text": "CAdv", "source": "BK24", "locator": "Definition 14", "status": "source_checked", "normalization": "As in the cited definition." } ] }, { "id": "rel_bs", "family": "extension", "name": "relation block sensitivity", "symbol": "bs", "definition": "Maximum number of pairwise disjoint blocks B at an input x such that f(x) and f(x^B) have disjoint valid-output sets. Maximize over promised x.", "definition_status": "source_checked", "sources": [ "BK24" ], "definition_locator": "Definition 15", "definition_audit_date": "2026-10-17", "notes": "", "alias_candidates": [], "aliases": [ { "text": "bs", "source": "BK24", "locator": "Definition 15", "status": "source_checked", "normalization": "As in the cited definition." } ] }, { "id": "rel_cbs", "family": "extension", "name": "critical block sensitivity", "symbol": "cbs", "definition": "Minimum over single-valued completions g(x) in f(x) of max bs_x(g) over critical inputs x with exactly one valid output.", "definition_status": "source_checked", "sources": [ "BK24" ], "definition_locator": "Definition 16", "definition_audit_date": "2026-10-17", "notes": " The maximum over an empty critical-input set is taken to be zero; a relation with no singleton valid-output set can therefore have critical measure zero.", "alias_candidates": [], "aliases": [ { "text": "cbs", "source": "BK24", "locator": "Definition 16", "status": "source_checked", "normalization": "As in the cited definition." } ] }, { "id": "rel_fbs", "family": "extension", "name": "relation fractional block sensitivity", "symbol": "fbs", "definition": "Fractional packing of the sensitive blocks from relation block sensitivity, with total weight through each coordinate at most one; maximize over x.", "definition_status": "source_checked", "sources": [ "BK24" ], "definition_locator": "Definition 17", "definition_audit_date": "2026-10-17", "notes": "", "alias_candidates": [], "aliases": [ { "text": "fbs", "source": "BK24", "locator": "Definition 17", "status": "source_checked", "normalization": "As in the cited definition." } ] }, { "id": "rel_fcbs", "family": "extension", "name": "fractional critical block sensitivity", "symbol": "fcbs", "definition": "Minimum over single-valued completions g(x) in f(x) of max fbs_x(g) over critical inputs x with exactly one valid output.", "definition_status": "source_checked", "sources": [ "BK24" ], "definition_locator": "Definition 17", "definition_audit_date": "2026-10-17", "notes": " The maximum over an empty critical-input set is taken to be zero; a relation with no singleton valid-output set can therefore have critical measure zero.", "alias_candidates": [], "aliases": [ { "text": "fcbs", "source": "BK24", "locator": "Definition 17", "status": "source_checked", "normalization": "As in the cited definition." } ] }, { "id": "blocknumber_l", "family": "extension", "name": "bounded-size block number", "symbol": "N_bs^ell", "definition": "Maximum over x of the number of inclusion-minimal sensitive blocks of cardinality at most ell.", "definition_status": "source_checked", "sources": [ "CG26" ], "definition_locator": "Definition 4", "definition_audit_date": "2026-10-17", "notes": "", "alias_candidates": [], "aliases": [ { "text": "N_b^{ell}", "source": "CG26", "locator": "Definition 4", "status": "source_checked", "normalization": "As in the cited definition." } ] }, { "id": "bricknumber_l", "family": "extension", "name": "bounded-size brick number", "symbol": "N_{br}^{ell}", "definition": "Brick number restricted to blocks of size at most ell.", "definition_status": "source_checked", "sources": [ "CG26" ], "definition_locator": "Definition 8", "definition_audit_date": "2026-10-17", "notes": "", "alias_candidates": [], "aliases": [ { "text": "N_{br}^{ell}", "source": "CG26", "locator": "Definition 8", "status": "source_checked", "normalization": "As in the cited definition." } ] }, { "id": "sparsityNonempty", "family": "extension", "name": "nonconstant Fourier sparsity", "symbol": "\\widetilde{spar}", "definition": "Number of nonzero Fourier coefficients indexed by nonempty S.", "definition_status": "source_checked", "sources": [ "DM23" ], "definition_locator": "Section 2, Fourier representation, PDF p.6", "definition_audit_date": "2026-10-17", "notes": "", "alias_candidates": [], "aliases": [ { "text": "\\widetilde{spar}", "source": "DM23", "locator": "Section 2, Fourier representation, PDF p.6", "status": "source_checked", "normalization": "As in the cited definition." } ] }, { "id": "Cmin", "family": "extension", "name": "minimum certificate complexity", "symbol": "C_min", "definition": "min_x C(f,x), equivalently minimum codimension of a monochromatic subcube.", "definition_status": "source_checked", "sources": [ "LM26" ], "definition_locator": "Definition 2.2", "definition_audit_date": "2026-10-17", "notes": "K in DM23 is unrelated to the Koutsoupias spectral-sensitivity alias K.", "alias_candidates": [], "aliases": [ { "text": "C_min", "source": "LM26", "locator": "Definition 2.2", "status": "source_checked", "normalization": "K in DM23 is unrelated to the Koutsoupias spectral-sensitivity alias K." }, { "text": "subcube kill number", "source": "LM26", "locator": "Definition 2.2", "status": "source_checked", "normalization": "K in DM23 is unrelated to the Koutsoupias spectral-sensitivity alias K." }, { "text": "K", "source": "LM26", "locator": "Definition 2.2", "status": "source_checked", "normalization": "K in DM23 is unrelated to the Koutsoupias spectral-sensitivity alias K." } ] }, { "id": "hazardS", "family": "extension", "name": "u-sensitivity", "symbol": "s_u", "definition": "Maximum over ternary inputs of the number of coordinates admitting a replacement that changes the hazard-free output. For a Boolean f, its extension maps z in {0,u,1}^n to b if every Boolean resolution of z has value b, and to u otherwise.", "definition_status": "source_checked", "sources": [ "BKMNSS25" ], "definition_locator": "Definition 2.2", "definition_audit_date": "2026-10-17", "notes": "Input and output alphabets are ternary. The symbol u is a queried value, distinct from an unrevealed coordinate.", "alias_candidates": [], "aliases": [ { "text": "s_u", "source": "BKMNSS25", "locator": "Definition 2.2", "status": "source_checked", "normalization": "Input and output alphabets are ternary. The symbol u is a queried value, distinct from an unrevealed coordinate." } ] }, { "id": "hazardBS", "family": "extension", "name": "u-block sensitivity", "symbol": "bs_u", "definition": "Maximum number of disjoint coordinate sets, each admitting a replacement that changes the hazard-free output. For a Boolean f, its extension maps z in {0,u,1}^n to b if every Boolean resolution of z has value b, and to u otherwise.", "definition_status": "source_checked", "sources": [ "BKMNSS25" ], "definition_locator": "Definition 2.3", "definition_audit_date": "2026-10-17", "notes": "Input and output alphabets are ternary. The symbol u is a queried value, distinct from an unrevealed coordinate.", "alias_candidates": [], "aliases": [ { "text": "bs_u", "source": "BKMNSS25", "locator": "Definition 2.3", "status": "source_checked", "normalization": "Input and output alphabets are ternary. The symbol u is a queried value, distinct from an unrevealed coordinate." } ] }, { "id": "hazardC", "family": "extension", "name": "u-certificate complexity", "symbol": "cc_u", "definition": "Maximum minimum number of revealed ternary coordinates fixing the hazard-free output among all consistent ternary completions. For a Boolean f, its extension maps z in {0,u,1}^n to b if every Boolean resolution of z has value b, and to u otherwise.", "definition_status": "source_checked", "sources": [ "BKMNSS25" ], "definition_locator": "Definition 2.4", "definition_audit_date": "2026-10-17", "notes": "Input and output alphabets are ternary. The symbol u is a queried value, distinct from an unrevealed coordinate.", "alias_candidates": [], "aliases": [ { "text": "cc_u", "source": "BKMNSS25", "locator": "Definition 2.4", "status": "source_checked", "normalization": "Input and output alphabets are ternary. The symbol u is a queried value, distinct from an unrevealed coordinate." } ] }, { "id": "Qnegligent", "family": "extension", "name": "noisy quantum query complexity (Qnegligent)", "symbol": "Q^{neg}_{p,epsilon}", "definition": "Minimum queries to the channel (1-p) O_x rho O_x^dagger + p rho, succeeding with probability >=1-epsilon.", "definition_status": "source_checked", "sources": [ "ARS26" ], "definition_locator": "Section 1, equations (1.4) and (1.6)", "definition_audit_date": "2026-10-17", "notes": "Fix p and epsilon explicitly; query-index dephasing differs from oracle negligence.", "alias_candidates": [], "aliases": [ { "text": "Q^{neg}_{p,epsilon}", "source": "ARS26", "locator": "Section 1, equations (1.4) and (1.6)", "status": "source_checked", "normalization": "Fix p and epsilon explicitly; query-index dephasing differs from oracle negligence." } ] }, { "id": "Qdephasing", "family": "extension", "name": "noisy quantum query complexity (Qdephasing)", "symbol": "Q^{deph}_{p,epsilon}", "definition": "Minimum queries to the bit oracle followed by complete dephasing of its index register independently with probability p, succeeding with probability >=1-epsilon.", "definition_status": "source_checked", "sources": [ "ARS26" ], "definition_locator": "Section 1, equations (1.4) and (1.6)", "definition_audit_date": "2026-10-17", "notes": "Fix p and epsilon explicitly; query-index dephasing differs from oracle negligence.", "alias_candidates": [], "aliases": [ { "text": "Q^{deph}_{p,epsilon}", "source": "ARS26", "locator": "Section 1, equations (1.4) and (1.6)", "status": "source_checked", "normalization": "Fix p and epsilon explicitly; query-index dephasing differs from oracle negligence." } ] }, { "id": "cubeBoundedDeg", "name": "globally bounded exact polynomial degree", "symbol": "deg", "definition": "Minimum real multilinear degree of p agreeing with f on its domain and satisfying 0<=p<=1 on the entire Boolean cube.", "definition_status": "source_checked", "family": "extension", "sources": [ "KLLPV26", "AB23" ], "definition_locator": "Definition 7", "aliases": [ { "text": "deg", "source": "KLLPV26", "locator": "Definition 7", "status": "source_checked", "normalization": "For partial f this differs from unrestricted exact degree. The source uses the same symbol deg; the stable ID distinguishes them." }, { "text": "deg", "source": "AB23", "locator": "Section 2.1", "status": "source_checked", "normalization": "Exact representing polynomial is in [0,1] on the entire cube, including off-promise inputs; this is not unrestricted promise interpolation degree." } ], "notes": "For partial f this differs from unrestricted exact degree. The source uses the same symbol deg; the stable ID distinguishes them.", "alias_candidates": [] }, { "id": "cubeBoundedAdeg", "name": "globally bounded approximate polynomial degree", "symbol": "\\widetilde{deg}", "definition": "Minimum degree of p in [0,1] on the whole cube with |p-f|<=1/3 on the promised domain.", "definition_status": "source_checked", "family": "extension", "sources": [ "KLLPV26", "AB23" ], "definition_locator": "Definition 7", "aliases": [ { "text": "\\widetilde{deg}", "source": "KLLPV26", "locator": "Definition 7", "status": "source_checked", "normalization": "same definition" }, { "text": "tilde deg", "source": "AB23", "locator": "Section 2.1", "status": "definition_convention_checked_not_literal_alias", "normalization": "AB23 explicitly says approximate degree can be defined analogously but does not require it. Its detailed representing-polynomial definition is for exact degree; do not count this as a verified literal approximate-degree alias." } ], "notes": "", "alias_candidates": [] }, { "id": "boundedPolyDelta", "name": "asymmetric bounded polynomial degree", "symbol": "Delta_{epsilon0,epsilon1,beta}", "definition": "Minimum multilinear degree of p in [0,beta] on the entire cube, approximating 0-inputs within epsilon0 and 1-inputs within epsilon1.", "definition_status": "source_checked", "family": "extension", "sources": [ "KLLPV26" ], "definition_locator": "Definition 7", "aliases": [ { "text": "Delta_{epsilon0,epsilon1,beta}", "source": "KLLPV26", "locator": "Definition 7", "status": "source_checked", "normalization": "All three parameters are explicit." } ], "notes": "All three parameters are explicit.", "alias_candidates": [] }, { "id": "osdeg0", "name": "one-sided bounded polynomial degree (osdeg0)", "symbol": "\\widehat{osdeg}^0", "definition": "Delta_{1/3,0,1}(f): globally [0,1]-bounded polynomial with error only on the indicated output side.", "definition_status": "source_checked", "family": "extension", "sources": [ "KLLPV26" ], "definition_locator": "Definition 7", "aliases": [ { "text": "\\widehat{osdeg}^0", "source": "KLLPV26", "locator": "Definition 7", "status": "source_checked", "normalization": "same definition" } ], "notes": "", "alias_candidates": [] }, { "id": "osdeg1", "name": "one-sided bounded polynomial degree (osdeg1)", "symbol": "\\widehat{osdeg}^1", "definition": "Delta_{0,1/3,1}(f): globally [0,1]-bounded polynomial with error only on the indicated output side.", "definition_status": "source_checked", "family": "extension", "sources": [ "KLLPV26" ], "definition_locator": "Definition 7", "aliases": [ { "text": "\\widehat{osdeg}^1", "source": "KLLPV26", "locator": "Definition 7", "status": "source_checked", "normalization": "same definition" } ], "notes": "", "alias_candidates": [] }, { "id": "osdeg", "name": "two-orientation one-sided bounded polynomial degree", "symbol": "\\widehat{osdeg}", "definition": "max(osdeg0,osdeg1).", "definition_status": "source_checked", "family": "extension", "sources": [ "KLLPV26" ], "definition_locator": "Definition 7", "aliases": [ { "text": "\\widehat{osdeg}", "source": "KLLPV26", "locator": "Definition 7", "status": "source_checked", "normalization": "Distinct from one-sided approximation notions allowing unbounded values on the other side." } ], "notes": "Distinct from one-sided approximation notions allowing unbounded values on the other side.", "alias_candidates": [] }, { "id": "bmAdeg", "name": "block-multilinear bounded approximate degree", "symbol": "\\widehat{bm-deg}", "definition": "Minimum d for a degree-d block-multilinear real polynomial P on d independent blocks of n+1 signed variables, each monomial containing exactly one variable from each block, such that |P(z)|<=1 on all independent signed block inputs; on promised diagonal inputs x, 0<=P((1,x),...,(1,x))<=1 and |P((1,x),...,(1,x))-f(x)|<=1/3. The target f is 0/1-valued.", "definition_status": "source_checked", "family": "extension", "sources": [ "KLLPV26" ], "definition_locator": "Definition 38", "aliases": [ { "text": "\\widehat{bm-deg}", "source": "KLLPV26", "locator": "Definition 38", "status": "source_checked", "normalization": "Source error/encoding follows Definition 38. Each monomial uses exactly one variable from each block." } ], "notes": "Source error/encoding follows Definition 38. Each monomial uses exactly one variable from each block.", "alias_candidates": [] }, { "id": "D_cert", "name": "certification of deterministic query complexity", "symbol": "D^{cert}", "definition": "max_z deterministic query complexity of the partial decision problem accepting z and rejecting all x with f(x)!=f(z).", "definition_status": "source_checked", "family": "extension", "sources": [ "KLLPV26" ], "definition_locator": "Definitions 1–2 and corresponding Theorems 3–12", "aliases": [ { "text": "D^{cert}", "source": "KLLPV26", "locator": "Definitions 1–2 and corresponding Theorems 3–12", "status": "source_checked", "normalization": "The certification problem always has a promise, even when the parent f is total. Preserve the base degree’s off-promise boundedness convention." } ], "notes": "The certification problem always has a promise, even when the parent f is total. Preserve the base degree’s off-promise boundedness convention.", "base_measure": "D", "alias_candidates": [] }, { "id": "R_cert", "name": "certification of bounded-error randomized query complexity", "symbol": "R^{cert}", "definition": "max_z bounded-error randomized query complexity of the partial decision problem accepting z and rejecting all x with f(x)!=f(z).", "definition_status": "source_checked", "family": "extension", "sources": [ "KLLPV26" ], "definition_locator": "Definitions 1–2 and corresponding Theorems 3–12", "aliases": [ { "text": "R^{cert}", "source": "KLLPV26", "locator": "Definitions 1–2 and corresponding Theorems 3–12", "status": "source_checked", "normalization": "The certification problem always has a promise, even when the parent f is total. Preserve the base degree’s off-promise boundedness convention." } ], "notes": "The certification problem always has a promise, even when the parent f is total. Preserve the base degree’s off-promise boundedness convention.", "base_measure": "R", "alias_candidates": [] }, { "id": "R0_cert", "name": "certification of zero-error randomized query complexity", "symbol": "R_0^{cert}", "definition": "max_z zero-error randomized query complexity of the partial decision problem accepting z and rejecting all x with f(x)!=f(z).", "definition_status": "source_checked", "family": "extension", "sources": [ "KLLPV26" ], "definition_locator": "Definitions 1–2 and corresponding Theorems 3–12", "aliases": [ { "text": "R_0^{cert}", "source": "KLLPV26", "locator": "Definitions 1–2 and corresponding Theorems 3–12", "status": "source_checked", "normalization": "The certification problem always has a promise, even when the parent f is total. Preserve the base degree’s off-promise boundedness convention." } ], "notes": "The certification problem always has a promise, even when the parent f is total. Preserve the base degree’s off-promise boundedness convention.", "base_measure": "R0", "alias_candidates": [] }, { "id": "QE_cert", "name": "certification of exact quantum query complexity", "symbol": "Q_E^{cert}", "definition": "max_z exact quantum query complexity of the partial decision problem accepting z and rejecting all x with f(x)!=f(z).", "definition_status": "source_checked", "family": "extension", "sources": [ "KLLPV26" ], "definition_locator": "Definitions 1–2 and corresponding Theorems 3–12", "aliases": [ { "text": "Q_E^{cert}", "source": "KLLPV26", "locator": "Definitions 1–2 and corresponding Theorems 3–12", "status": "source_checked", "normalization": "The certification problem always has a promise, even when the parent f is total. Preserve the base degree’s off-promise boundedness convention." } ], "notes": "The certification problem always has a promise, even when the parent f is total. Preserve the base degree’s off-promise boundedness convention.", "base_measure": "QE", "alias_candidates": [] }, { "id": "Q0_cert", "name": "certification of zero-error quantum query complexity", "symbol": "Q_0^{cert}", "definition": "max_z zero-error quantum query complexity of the partial decision problem accepting z and rejecting all x with f(x)!=f(z).", "definition_status": "source_checked", "family": "extension", "sources": [ "KLLPV26" ], "definition_locator": "Definitions 1–2 and corresponding Theorems 3–12", "aliases": [ { "text": "Q_0^{cert}", "source": "KLLPV26", "locator": "Definitions 1–2 and corresponding Theorems 3–12", "status": "source_checked", "normalization": "The certification problem always has a promise, even when the parent f is total. Preserve the base degree’s off-promise boundedness convention." } ], "notes": "The certification problem always has a promise, even when the parent f is total. Preserve the base degree’s off-promise boundedness convention.", "base_measure": "Q0", "alias_candidates": [] }, { "id": "Q_cert", "name": "certification of bounded-error quantum query complexity", "symbol": "Q^{cert}", "definition": "max_z bounded-error quantum query complexity of the partial decision problem accepting z and rejecting all x with f(x)!=f(z).", "definition_status": "source_checked", "family": "extension", "sources": [ "KLLPV26" ], "definition_locator": "Definitions 1–2 and corresponding Theorems 3–12", "aliases": [ { "text": "Q^{cert}", "source": "KLLPV26", "locator": "Definitions 1–2 and corresponding Theorems 3–12", "status": "source_checked", "normalization": "The certification problem always has a promise, even when the parent f is total. Preserve the base degree’s off-promise boundedness convention." } ], "notes": "The certification problem always has a promise, even when the parent f is total. Preserve the base degree’s off-promise boundedness convention.", "base_measure": "Q", "alias_candidates": [] }, { "id": "QD_cert", "name": "certification of quantum distinguishing complexity", "symbol": "QD^{cert}", "definition": "max_z quantum distinguishing complexity of the partial decision problem accepting z and rejecting all x with f(x)!=f(z).", "definition_status": "source_checked", "family": "extension", "sources": [ "KLLPV26" ], "definition_locator": "Definitions 1–2 and corresponding Theorems 3–12", "aliases": [ { "text": "QD^{cert}", "source": "KLLPV26", "locator": "Definitions 1–2 and corresponding Theorems 3–12", "status": "source_checked", "normalization": "The certification problem always has a promise, even when the parent f is total. Preserve the base degree’s off-promise boundedness convention." } ], "notes": "The certification problem always has a promise, even when the parent f is total. Preserve the base degree’s off-promise boundedness convention.", "base_measure": "QD", "alias_candidates": [] }, { "id": "AdvPlus_cert", "name": "certification of positive-weight adversary bound", "symbol": "Adv^{cert}", "definition": "max_z positive-weight adversary bound of the partial decision problem accepting z and rejecting all x with f(x)!=f(z).", "definition_status": "source_checked", "family": "extension", "sources": [ "KLLPV26" ], "definition_locator": "Definitions 1–2 and corresponding Theorems 3–12", "aliases": [ { "text": "Adv^{cert}", "source": "KLLPV26", "locator": "Definitions 1–2 and corresponding Theorems 3–12", "status": "source_checked", "normalization": "The certification problem always has a promise, even when the parent f is total. Preserve the base degree’s off-promise boundedness convention." } ], "notes": "The certification problem always has a promise, even when the parent f is total. Preserve the base degree’s off-promise boundedness convention.", "base_measure": "AdvPlus", "alias_candidates": [] }, { "id": "RS_cert", "name": "certification of randomized sabotage complexity", "symbol": "RS^{cert}", "definition": "max_z randomized sabotage complexity of the partial decision problem accepting z and rejecting all x with f(x)!=f(z).", "definition_status": "source_checked", "family": "extension", "sources": [ "KLLPV26" ], "definition_locator": "Definitions 1–2 and corresponding Theorems 3–12", "aliases": [ { "text": "RS^{cert}", "source": "KLLPV26", "locator": "Definitions 1–2 and corresponding Theorems 3–12", "status": "source_checked", "normalization": "The certification problem always has a promise, even when the parent f is total. Preserve the base degree’s off-promise boundedness convention." } ], "notes": "The certification problem always has a promise, even when the parent f is total. Preserve the base degree’s off-promise boundedness convention.", "base_measure": "RS", "alias_candidates": [] }, { "id": "QS_cert", "name": "certification of weak-input quantum sabotage complexity", "symbol": "QS_{weak}^{cert}", "definition": "max_z weak-input quantum sabotage complexity of the partial decision problem accepting z and rejecting all x with f(x)!=f(z).", "definition_status": "source_checked", "family": "extension", "sources": [ "KLLPV26" ], "definition_locator": "Definitions 1–2 and corresponding Theorems 3–12", "aliases": [ { "text": "QS_{weak}^{cert}", "source": "KLLPV26", "locator": "Definitions 1–2 and corresponding Theorems 3–12", "status": "source_checked", "normalization": "The certification problem always has a promise, even when the parent f is total. Preserve the base degree’s off-promise boundedness convention." } ], "notes": "The certification problem always has a promise, even when the parent f is total. Preserve the base degree’s off-promise boundedness convention.", "base_measure": "QS", "alias_candidates": [] }, { "id": "EC_cert", "name": "certification of expectational certificate complexity", "symbol": "EC^{cert}", "definition": "max_z expectational certificate complexity of the partial decision problem accepting z and rejecting all x with f(x)!=f(z).", "definition_status": "source_checked", "family": "extension", "sources": [ "KLLPV26" ], "definition_locator": "Definitions 1–2 and corresponding Theorems 3–12", "aliases": [ { "text": "EC^{cert}", "source": "KLLPV26", "locator": "Definitions 1–2 and corresponding Theorems 3–12", "status": "source_checked", "normalization": "The certification problem always has a promise, even when the parent f is total. Preserve the base degree’s off-promise boundedness convention." } ], "notes": "The certification problem always has a promise, even when the parent f is total. Preserve the base degree’s off-promise boundedness convention.", "base_measure": "EC", "alias_candidates": [] }, { "id": "UC_cert", "name": "certification of unambiguous certificate complexity", "symbol": "UC^{cert}", "definition": "max_z unambiguous certificate complexity of the partial decision problem accepting z and rejecting all x with f(x)!=f(z).", "definition_status": "source_checked", "family": "extension", "sources": [ "KLLPV26" ], "definition_locator": "Definitions 1–2 and corresponding Theorems 3–12", "aliases": [ { "text": "UC^{cert}", "source": "KLLPV26", "locator": "Definitions 1–2 and corresponding Theorems 3–12", "status": "source_checked", "normalization": "The certification problem always has a promise, even when the parent f is total. Preserve the base degree’s off-promise boundedness convention." } ], "notes": "The certification problem always has a promise, even when the parent f is total. Preserve the base degree’s off-promise boundedness convention.", "base_measure": "UC", "alias_candidates": [] }, { "id": "CAdv_cert", "name": "certification of classical adversary bound", "symbol": "CAdv^{cert}", "definition": "max_z classical adversary bound of the partial decision problem accepting z and rejecting all x with f(x)!=f(z).", "definition_status": "source_checked", "family": "extension", "sources": [ "KLLPV26" ], "definition_locator": "Definitions 1–2 and corresponding Theorems 3–12", "aliases": [ { "text": "CAdv^{cert}", "source": "KLLPV26", "locator": "Definitions 1–2 and corresponding Theorems 3–12", "status": "source_checked", "normalization": "The certification problem always has a promise, even when the parent f is total. Preserve the base degree’s off-promise boundedness convention." } ], "notes": "The certification problem always has a promise, even when the parent f is total. Preserve the base degree’s off-promise boundedness convention.", "base_measure": "CAdv", "alias_candidates": [] }, { "id": "cubeBoundedDeg_cert", "name": "certification of globally bounded exact polynomial degree", "symbol": "deg^{cert}", "definition": "max_z globally bounded exact polynomial degree of the partial decision problem accepting z and rejecting all x with f(x)!=f(z).", "definition_status": "source_checked", "family": "extension", "sources": [ "KLLPV26" ], "definition_locator": "Definitions 1–2 and corresponding Theorems 3–12", "aliases": [ { "text": "deg^{cert}", "source": "KLLPV26", "locator": "Definitions 1–2 and corresponding Theorems 3–12", "status": "source_checked", "normalization": "The certification problem always has a promise, even when the parent f is total. Preserve the base degree’s off-promise boundedness convention." } ], "notes": "The certification problem always has a promise, even when the parent f is total. Preserve the base degree’s off-promise boundedness convention.", "base_measure": "cubeBoundedDeg", "alias_candidates": [] }, { "id": "cubeBoundedAdeg_cert", "name": "certification of globally bounded approximate polynomial degree", "symbol": "\\widetilde{deg}^{cert}", "definition": "max_z globally bounded approximate polynomial degree of the partial decision problem accepting z and rejecting all x with f(x)!=f(z).", "definition_status": "source_checked", "family": "extension", "sources": [ "KLLPV26" ], "definition_locator": "Definitions 1–2 and corresponding Theorems 3–12", "aliases": [ { "text": "\\widetilde{deg}^{cert}", "source": "KLLPV26", "locator": "Definitions 1–2 and corresponding Theorems 3–12", "status": "source_checked", "normalization": "The certification problem always has a promise, even when the parent f is total. Preserve the base degree’s off-promise boundedness convention." } ], "notes": "The certification problem always has a promise, even when the parent f is total. Preserve the base degree’s off-promise boundedness convention.", "base_measure": "cubeBoundedAdeg", "alias_candidates": [] }, { "id": "osdeg_cert", "name": "certification of two-orientation one-sided bounded polynomial degree", "symbol": "\\widehat{osdeg}^{cert}", "definition": "max_z two-orientation one-sided bounded polynomial degree of the partial decision problem accepting z and rejecting all x with f(x)!=f(z).", "definition_status": "source_checked", "family": "extension", "sources": [ "KLLPV26" ], "definition_locator": "Definitions 1–2 and corresponding Theorems 3–12", "aliases": [ { "text": "\\widehat{osdeg}^{cert}", "source": "KLLPV26", "locator": "Definitions 1–2 and corresponding Theorems 3–12", "status": "source_checked", "normalization": "The certification problem always has a promise, even when the parent f is total. Preserve the base degree’s off-promise boundedness convention." } ], "notes": "The certification problem always has a promise, even when the parent f is total. Preserve the base degree’s off-promise boundedness convention.", "base_measure": "osdeg", "alias_candidates": [] }, { "id": "lambda_cert", "name": "certification of spectral sensitivity", "symbol": "lambda^{cert}", "definition": "max_z spectral sensitivity of the partial decision problem accepting z and rejecting all x with f(x)!=f(z).", "definition_status": "source_checked", "family": "extension", "sources": [ "KLLPV26" ], "definition_locator": "Definitions 1–2 and corresponding Theorems 3–12", "aliases": [ { "text": "lambda^{cert}", "source": "KLLPV26", "locator": "Definitions 1–2 and corresponding Theorems 3–12", "status": "source_checked", "normalization": "The certification problem always has a promise, even when the parent f is total. Preserve the base degree’s off-promise boundedness convention." } ], "notes": "The certification problem always has a promise, even when the parent f is total. Preserve the base degree’s off-promise boundedness convention.", "base_measure": "lambda", "alias_candidates": [] }, { "id": "rdeg_cert", "name": "certification of exact rational degree", "symbol": "rdeg^{cert}", "definition": "max_z exact rational degree of the partial decision problem accepting z and rejecting all x with f(x)!=f(z).", "definition_status": "source_checked", "family": "extension", "sources": [ "KLLPV26" ], "definition_locator": "Definitions 1–2 and corresponding Theorems 3–12", "aliases": [ { "text": "rdeg^{cert}", "source": "KLLPV26", "locator": "Definitions 1–2 and corresponding Theorems 3–12", "status": "source_checked", "normalization": "The certification problem always has a promise, even when the parent f is total. Preserve the base degree’s off-promise boundedness convention." } ], "notes": "The certification problem always has a promise, even when the parent f is total. Preserve the base degree’s off-promise boundedness convention.", "base_measure": "rdeg", "alias_candidates": [] }, { "id": "signDeg", "name": "sign / threshold degree", "symbol": "deg_{\\pm}", "definition": "Minimum real multilinear degree of p with p(x)<0 exactly on 1-inputs and p(x)>=0 on 0-inputs; equivalent to strict sign representation after an arbitrarily small shift on a finite cube.", "definition_status": "source_checked", "family": "extension", "sources": [ "KKWY26" ], "definition_locator": "Definition 3", "aliases": [ { "text": "deg_{\\pm}", "source": "KKWY26", "locator": "Definition 3", "status": "source_checked", "normalization": "Source conventions use sign (-1)^f. Nonconstant Boolean inputs are the intended setting." } ], "notes": "Source conventions use sign (-1)^f. Nonconstant Boolean inputs are the intended setting.", "alias_candidates": [] }, { "id": "nRelevant", "name": "number of relevant variables", "symbol": "n(f)", "definition": "Number of coordinates i for which flipping i changes f somewhere.", "definition_status": "source_checked", "family": "extension", "sources": [ "Wel22", "CHS20", "GNSTW16" ], "definition_locator": "Section 2", "aliases": [ { "text": "n(f)", "source": "Wel22", "locator": "Section 2", "status": "source_checked", "normalization": "Does not count dummy input variables; distinguish from ambient input length n." }, { "text": "R(f)", "source": "CHS20", "locator": "Introduction, p.1", "status": "source_checked", "normalization": "Number of relevant variables, not randomized query complexity." } ], "notes": "Does not count dummy input variables; distinguish from ambient input length n.", "alias_candidates": [] }, { "id": "Dweighted", "name": "weighted deterministic query complexity", "symbol": "Depth_w", "definition": "Minimum over decision trees of the maximum sum of prescribed nonnegative coordinate-query weights along an input path.", "definition_status": "source_checked", "family": "extension", "sources": [ "DM23" ], "definition_locator": "Section 2, weighted decision trees", "aliases": [ { "text": "Depth_w", "source": "DM23", "locator": "Section 2, weighted decision trees", "status": "source_checked", "normalization": "Coordinate weights are part of the measure’s parameters." } ], "notes": "Coordinate weights are part of the measure’s parameters.", "alias_candidates": [] }, { "id": "RCmin", "name": "minimum randomized certificate complexity", "symbol": "RC_min", "definition": "min_x RC_x(f).", "definition_status": "source_checked", "family": "extension", "sources": [ "KKWY26" ], "definition_locator": "Definition 10", "aliases": [ { "text": "RC_min", "source": "KKWY26", "locator": "Definition 10", "status": "source_checked", "normalization": "same definition" } ], "notes": "", "alias_candidates": [] }, { "id": "fbsmin", "name": "minimum fractional block sensitivity", "symbol": "fbs_min", "definition": "min_x fbs_x(f).", "definition_status": "source_checked", "family": "extension", "sources": [ "KKWY26" ], "definition_locator": "Definition 10", "aliases": [ { "text": "fbs_min", "source": "KKWY26", "locator": "Definition 10", "status": "source_checked", "normalization": "same definition" } ], "notes": "", "alias_candidates": [] }, { "id": "RCminDown", "name": "downward-closed minimum randomized certificate complexity", "symbol": "RC_min^downarrow", "definition": "Maximum of RC_min(f restricted by rho) over coordinate restrictions rho.", "definition_status": "source_checked", "family": "extension", "sources": [ "KKWY26" ], "definition_locator": "Definition 11", "aliases": [ { "text": "RC_min^downarrow", "source": "KKWY26", "locator": "Definition 11", "status": "source_checked", "normalization": "same definition" } ], "notes": "", "alias_candidates": [] }, { "id": "fbsminDown", "name": "downward-closed minimum fractional block sensitivity", "symbol": "fbs_min^downarrow", "definition": "Maximum of fbs_min(f restricted by rho) over coordinate restrictions rho.", "definition_status": "source_checked", "family": "extension", "sources": [ "KKWY26" ], "definition_locator": "Definition 11", "aliases": [ { "text": "fbs_min^downarrow", "source": "KKWY26", "locator": "Definition 11", "status": "source_checked", "normalization": "same definition" } ], "notes": "", "alias_candidates": [] }, { "id": "fbs_at0", "name": "fractional block sensitivity at the all-zero input", "symbol": "fbs(f,0^n)", "definition": "The fractional packing of sensitive blocks at 0^n, provided that 0^n lies in the domain.", "definition_status": "source_checked", "family": "extension", "sources": [ "ARS26" ], "definition_locator": "Theorem 1.1 and Section 2", "aliases": [ { "text": "fbs(f,0^n)", "source": "ARS26", "locator": "Theorem 1.1 and Section 2", "status": "source_checked", "normalization": "same definition" } ], "notes": "", "alias_candidates": [] }, { "id": "fbs0", "name": "0-input fractional block sensitivity", "symbol": "fbs^0", "definition": "Maximum local fbs value over inputs with f(x)=0.", "definition_status": "source_checked", "family": "extension", "sources": [ "KT13" ], "definition_locator": "Section 1 notation and Section 2 definitions", "aliases": [ { "text": "fbs^0", "source": "KT13", "locator": "Section 1 notation and Section 2 definitions", "status": "source_checked", "normalization": "same definition" } ], "notes": "", "base_measure": "fbs", "output_side": 0, "alias_candidates": [] }, { "id": "fbs1", "name": "1-input fractional block sensitivity", "symbol": "fbs^1", "definition": "Maximum local fbs value over inputs with f(x)=1.", "definition_status": "source_checked", "family": "extension", "sources": [ "KT13" ], "definition_locator": "Section 1 notation and Section 2 definitions", "aliases": [ { "text": "fbs^1", "source": "KT13", "locator": "Section 1 notation and Section 2 definitions", "status": "source_checked", "normalization": "same definition" } ], "notes": "", "base_measure": "fbs", "output_side": 1, "alias_candidates": [] }, { "id": "FC0", "name": "0-input fractional certificate complexity", "symbol": "FC^0", "definition": "Maximum local FC value over inputs with f(x)=0.", "definition_status": "source_checked", "family": "extension", "sources": [ "KT13" ], "definition_locator": "Section 1 notation and Section 2 definitions", "aliases": [ { "text": "FC^0", "source": "KT13", "locator": "Section 1 notation and Section 2 definitions", "status": "source_checked", "normalization": "same definition" } ], "notes": "", "base_measure": "FC", "output_side": 0, "alias_candidates": [] }, { "id": "FC1", "name": "1-input fractional certificate complexity", "symbol": "FC^1", "definition": "Maximum local FC value over inputs with f(x)=1.", "definition_status": "source_checked", "family": "extension", "sources": [ "KT13" ], "definition_locator": "Section 1 notation and Section 2 definitions", "aliases": [ { "text": "FC^1", "source": "KT13", "locator": "Section 1 notation and Section 2 definitions", "status": "source_checked", "normalization": "same definition" } ], "notes": "", "base_measure": "FC", "output_side": 1, "alias_candidates": [] }, { "id": "RC0", "name": "0-input randomized certificate complexity", "symbol": "RC^0", "definition": "Maximum local RC value over inputs with f(x)=0.", "definition_status": "source_checked", "family": "extension", "sources": [ "KT13" ], "definition_locator": "Section 1 notation and Section 2 definitions", "aliases": [ { "text": "RC^0", "source": "KT13", "locator": "Section 1 notation and Section 2 definitions", "status": "source_checked", "normalization": "same definition" } ], "notes": "", "base_measure": "RC", "output_side": 0, "alias_candidates": [] }, { "id": "RC1", "name": "1-input randomized certificate complexity", "symbol": "RC^1", "definition": "Maximum local RC value over inputs with f(x)=1.", "definition_status": "source_checked", "family": "extension", "sources": [ "KT13" ], "definition_locator": "Section 1 notation and Section 2 definitions", "aliases": [ { "text": "RC^1", "source": "KT13", "locator": "Section 1 notation and Section 2 definitions", "status": "source_checked", "normalization": "same definition" } ], "notes": "", "base_measure": "RC", "output_side": 1, "alias_candidates": [] }, { "id": "QC0", "name": "0-input quantum certificate complexity", "symbol": "QC^0", "definition": "Maximum local QC value over inputs with f(x)=0.", "definition_status": "source_checked", "family": "extension", "sources": [ "KT13" ], "definition_locator": "Section 1 notation and Section 2 definitions", "aliases": [ { "text": "QC^0", "source": "KT13", "locator": "Section 1 notation and Section 2 definitions", "status": "source_checked", "normalization": "same definition" } ], "notes": "", "base_measure": "QC", "output_side": 0, "alias_candidates": [] }, { "id": "QC1", "name": "1-input quantum certificate complexity", "symbol": "QC^1", "definition": "Maximum local QC value over inputs with f(x)=1.", "definition_status": "source_checked", "family": "extension", "sources": [ "KT13" ], "definition_locator": "Section 1 notation and Section 2 definitions", "aliases": [ { "text": "QC^1", "source": "KT13", "locator": "Section 1 notation and Section 2 definitions", "status": "source_checked", "normalization": "same definition" } ], "notes": "", "base_measure": "QC", "output_side": 1, "alias_candidates": [] }, { "id": "cfbs_boolean", "name": "critical fractional block sensitivity of a partial Boolean function", "symbol": "cfbs", "family": "certificate", "definition": "Minimum over total Boolean completions g of f of max over x in Dom(f) of fbs(g,x). The maximum ranges over the original promise, rather than all inputs of g.", "definition_status": "source_checked", "definition_locator": "KLLPV26 Definition 35", "sources": [ "KLLPV26" ], "aliases": [ { "text": "cfbs", "source": "KLLPV26", "locator": "Definition 35", "status": "source_checked" } ], "notes": "Keep separate from critical fractional block sensitivity of general search relations; the two definitions require a model translation.", "alias_candidates": [] }, { "id": "Cset", "name": "output-set certificate complexity", "symbol": "C_Sigma", "family": "certificate", "definition": "For f:{0,1}^n->{0,1,*} and Sigma a subset of {0,1,*}, C_Sigma(f,x) is the fewest fixed coordinates of a subcube containing x on which every output belongs to Sigma. Maximize over x with f(x) in Sigma for C_Sigma(f).", "definition_status": "source_checked", "definition_locator": "Bal21 Section 1; BBGJK21 Puzzle II", "sources": [ "Bal21", "BBGJK21" ], "aliases": [ { "text": "C_Sigma", "source": "Bal21", "locator": "Section 1", "status": "source_checked", "normalization": "Output sets 0,1,bar0={1,*},bar1={0,*}." } ], "notes": "Here * is an output denoting undefined, not an unqueried input coordinate. C_0/C_1 in this strong convention exclude undefined completions; do not combine with weak partial certificates.", "alias_candidates": [] }, { "id": "Cbar0", "name": "certificate complexity for excluding output zero", "symbol": "C_bar0", "family": "certificate", "definition": "C_Sigma(f) with Sigma={1,*}, maximizing the output-set certificate complexity over all x with f(x) in Sigma.", "definition_status": "source_checked", "definition_locator": "Bal21 Section 1; BBGJK21 Puzzle II", "sources": [ "Bal21", "BBGJK21" ], "aliases": [ { "text": "C_{\\bar{0}}", "source": "Bal21", "locator": "Section 1", "status": "source_checked", "normalization": "Repository-qualified names; the source uses the displayed notation for ternary output functions. Not weak promise certificates." } ], "notes": "Undefined output * is included explicitly. Local quantities C_Sigma(f,x) remain parameterized; this ID denotes their indicated maximum." }, { "id": "Cbar1", "name": "certificate complexity for excluding output one", "symbol": "C_bar1", "family": "certificate", "definition": "C_Sigma(f) with Sigma={0,*}, maximizing the output-set certificate complexity over all x with f(x) in Sigma.", "definition_status": "source_checked", "definition_locator": "Bal21 Section 1; BBGJK21 Puzzle II", "sources": [ "Bal21", "BBGJK21" ], "aliases": [ { "text": "C_{\\bar{1}}", "source": "Bal21", "locator": "Section 1", "status": "source_checked", "normalization": "Repository-qualified names; the source uses the displayed notation for ternary output functions. Not weak promise certificates." } ], "notes": "Undefined output * is included explicitly. Local quantities C_Sigma(f,x) remain parameterized; this ID denotes their indicated maximum." }, { "id": "Cstrong", "name": "strong certificate complexity of a partial function", "symbol": "C", "family": "certificate", "definition": "max(C_{Sigma={0}}(f),C_{Sigma={1}}(f)). Certificates must force the declared output on all cube completions, excluding undefined outputs as well as the opposite Boolean output.", "definition_status": "source_checked", "definition_locator": "Bal21 Section 1; BBGJK21 Puzzle II", "sources": [ "Bal21", "BBGJK21" ], "aliases": [ { "text": "C", "source": "Bal21", "locator": "Section 1", "status": "source_checked", "normalization": "Repository-qualified names; the source uses the displayed notation for ternary output functions. Not weak promise certificates." } ], "notes": "Undefined output * is included explicitly. Local quantities C_Sigma(f,x) remain parameterized; this ID denotes their indicated maximum." }, { "id": "Q_eps", "name": "error-parameter quantum query complexity", "symbol": "Q_epsilon", "family": "parameterized", "definition": "Minimum worst-case quantum oracle queries giving the correct Boolean output with probability at least 1-epsilon on every promised input. Epsilon is explicit; fixed 0{0,1}; each query returns a symbol in [n].", "definition_status": "source_checked", "family": "extension", "sources": [ "DFLLV21", "DFLLV26" ], "definition_locator": "Introduction and Section 2.2", "aliases": [ { "text": "D", "source": "DFLLV21", "locator": "Introduction and Section 2.2", "status": "source_checked" } ], "notes": "", "journal_definition_check": "Final journal Section 2.2; forward/inverse query family, packing sensitivity and detectable failure 2/3 preserved." }, { "id": "permDegree", "name": "permutation polynomial degree", "symbol": "deg", "definition": "Minimum degree of a real polynomial in permutation-matrix entries agreeing with f on S_n. Equivalently, span of functions depending on at most d values pi(i).", "definition_status": "source_checked", "family": "extension", "sources": [ "DFLLV21", "DFLLV26" ], "definition_locator": "Introduction and Section 2.2", "aliases": [ { "text": "deg", "source": "DFLLV21", "locator": "Introduction and Section 2.2", "status": "source_checked" } ], "notes": "", "journal_definition_check": "Final journal Section 2.2; forward/inverse query family, packing sensitivity and detectable failure 2/3 preserved." }, { "id": "permAdeg", "name": "permutation approximate degree", "symbol": "tilde deg", "definition": "Minimum degree of a real polynomial in permutation-matrix entries approximating f within 1/3 on S_n.", "definition_status": "source_checked", "family": "extension", "sources": [ "DFLLV21", "DFLLV26" ], "definition_locator": "Introduction and Section 2.2", "aliases": [ { "text": "tilde deg", "source": "DFLLV21", "locator": "Introduction and Section 2.2", "status": "source_checked" } ], "notes": "", "journal_definition_check": "Final journal Section 2.2; forward/inverse query family, packing sensitivity and detectable failure 2/3 preserved." }, { "id": "permS", "name": "disjoint-transposition sensitivity", "symbol": "s", "definition": "Maximum over pi of the largest family of pairwise disjoint transpositions tau with f(tau*pi)!=f(pi).", "definition_status": "source_checked", "family": "extension", "sources": [ "DFLLV21", "DFLLV26" ], "definition_locator": "Introduction and Section 2.2", "aliases": [ { "text": "s", "source": "DFLLV21", "locator": "Introduction and Section 2.2", "status": "source_checked" } ], "notes": "", "journal_definition_check": "Final journal Section 2.2; forward/inverse query family, packing sensitivity and detectable failure 2/3 preserved." }, { "id": "permEdgeS", "name": "transposition edge sensitivity", "symbol": "t", "definition": "Maximum over pi of the total number of transpositions tau with f(pi*tau)!=f(pi); disjointness is not required.", "definition_status": "source_checked", "family": "extension", "sources": [ "DFLLV21", "DFLLV26" ], "definition_locator": "Introduction and Section 2.2", "aliases": [ { "text": "t", "source": "DFLLV21", "locator": "Introduction and Section 2.2", "status": "source_checked" } ], "notes": "", "journal_definition_check": "Final journal Section 2.2; forward/inverse query family, packing sensitivity and detectable failure 2/3 preserved." }, { "id": "permBS", "name": "permutation block sensitivity", "symbol": "bs", "definition": "Maximum over pi of the largest family of permutations tau_j with pairwise disjoint moved-point supports and f(tau_j*pi)!=f(pi).", "definition_status": "source_checked", "family": "extension", "sources": [ "DFLLV21", "DFLLV26" ], "definition_locator": "Introduction and Section 2.2", "aliases": [ { "text": "bs", "source": "DFLLV21", "locator": "Introduction and Section 2.2", "status": "source_checked" } ], "notes": "", "journal_definition_check": "Final journal Section 2.2; forward/inverse query family, packing sensitivity and detectable failure 2/3 preserved." }, { "id": "completion_D", "name": "minimum-completion deterministic query complexity", "symbol": "overline D", "definition": "Minimum of D(F) over all total Boolean completions F on the same cube with F(x)=f(x) for every promised x. On a total domain this equals D(f).", "definition_status": "source_checked", "family": "extension", "sources": [ "HKMP26" ], "definition_locator": "Definition 45, p.22", "aliases": [ { "text": "overline M", "source": "HKMP26", "locator": "Definition 45, p.22", "status": "source_checked", "normalization": "Generic source template, explicitly instantiated here with M=D. The instantiated display symbol is repository notation, not a separate literal occurrence." } ], "notes": "The source defines an arbitrary-measure template; this record is an explicit instantiation. Each minimum may use a different completion. Overline notation visually checked in the original PDF.", "completion_of": "D", "repository_display_variants": [ { "text": "overline D", "source": "HKMP26", "status": "repository_display_not_a_historical_occurrence", "normalization": "Instantiation of the source overline-M template." } ] }, { "id": "completion_R0", "name": "minimum-completion zero-error randomized query complexity", "symbol": "overline R_0", "definition": "Minimum of R0(F) over all total Boolean completions F on the same cube with F(x)=f(x) for every promised x. On a total domain this equals R0(f).", "definition_status": "source_checked", "family": "extension", "sources": [ "HKMP26" ], "definition_locator": "Definition 45, p.22", "aliases": [ { "text": "overline M", "source": "HKMP26", "locator": "Definition 45, p.22", "status": "source_checked", "normalization": "Generic source template, explicitly instantiated here with M=R0. The instantiated display symbol is repository notation, not a separate literal occurrence." } ], "notes": "The source defines an arbitrary-measure template; this record is an explicit instantiation. Each minimum may use a different completion. Overline notation visually checked in the original PDF.", "completion_of": "R0", "repository_display_variants": [ { "text": "overline R_0", "source": "HKMP26", "status": "repository_display_not_a_historical_occurrence", "normalization": "Instantiation of the source overline-M template." } ] }, { "id": "completion_R", "name": "minimum-completion bounded-error randomized query complexity", "symbol": "overline R", "definition": "Minimum of R(F) over all total Boolean completions F on the same cube with F(x)=f(x) for every promised x. On a total domain this equals R(f).", "definition_status": "source_checked", "family": "extension", "sources": [ "HKMP26" ], "definition_locator": "Definition 45, p.22", "aliases": [ { "text": "overline M", "source": "HKMP26", "locator": "Definition 45, p.22", "status": "source_checked", "normalization": "Generic source template, explicitly instantiated here with M=R. The instantiated display symbol is repository notation, not a separate literal occurrence." } ], "notes": "The source defines an arbitrary-measure template; this record is an explicit instantiation. Each minimum may use a different completion. Overline notation visually checked in the original PDF.", "completion_of": "R", "repository_display_variants": [ { "text": "overline R", "source": "HKMP26", "status": "repository_display_not_a_historical_occurrence", "normalization": "Instantiation of the source overline-M template." } ] }, { "id": "completion_C", "name": "minimum-completion certificate complexity", "symbol": "overline C", "definition": "Minimum of C(F) over all total Boolean completions F on the same cube with F(x)=f(x) for every promised x. On a total domain this equals C(f).", "definition_status": "source_checked", "family": "extension", "sources": [ "HKMP26" ], "definition_locator": "Definition 45, p.22", "aliases": [ { "text": "overline M", "source": "HKMP26", "locator": "Definition 45, p.22", "status": "source_checked", "normalization": "Generic source template, explicitly instantiated here with M=C. The instantiated display symbol is repository notation, not a separate literal occurrence." } ], "notes": "The source defines an arbitrary-measure template; this record is an explicit instantiation. Each minimum may use a different completion. Overline notation visually checked in the original PDF.", "completion_of": "C", "repository_display_variants": [ { "text": "overline C", "source": "HKMP26", "status": "repository_display_not_a_historical_occurrence", "normalization": "Instantiation of the source overline-M template." } ] }, { "id": "completion_RC", "name": "minimum-completion randomized certificate complexity", "symbol": "overline RC", "definition": "Minimum of RC(F) over all total Boolean completions F on the same cube with F(x)=f(x) for every promised x. On a total domain this equals RC(f).", "definition_status": "source_checked", "family": "extension", "sources": [ "HKMP26" ], "definition_locator": "Definition 45, p.22", "aliases": [ { "text": "overline M", "source": "HKMP26", "locator": "Definition 45, p.22", "status": "source_checked", "normalization": "Generic source template, explicitly instantiated here with M=RC. The instantiated display symbol is repository notation, not a separate literal occurrence." } ], "notes": "The source defines an arbitrary-measure template; this record is an explicit instantiation. Each minimum may use a different completion. Overline notation visually checked in the original PDF.", "completion_of": "RC", "repository_display_variants": [ { "text": "overline RC", "source": "HKMP26", "status": "repository_display_not_a_historical_occurrence", "normalization": "Instantiation of the source overline-M template." } ] }, { "id": "completion_bs", "name": "minimum-completion block sensitivity", "symbol": "overline bs", "definition": "Minimum of bs(F) over all total Boolean completions F on the same cube with F(x)=f(x) for every promised x. On a total domain this equals bs(f).", "definition_status": "source_checked", "family": "extension", "sources": [ "HKMP26" ], "definition_locator": "Definition 45, p.22", "aliases": [ { "text": "overline M", "source": "HKMP26", "locator": "Definition 45, p.22", "status": "source_checked", "normalization": "Generic source template, explicitly instantiated here with M=bs. The instantiated display symbol is repository notation, not a separate literal occurrence." } ], "notes": "The source defines an arbitrary-measure template; this record is an explicit instantiation. Each minimum may use a different completion. Overline notation visually checked in the original PDF.", "completion_of": "bs", "repository_display_variants": [ { "text": "overline bs", "source": "HKMP26", "status": "repository_display_not_a_historical_occurrence", "normalization": "Instantiation of the source overline-M template." } ] }, { "id": "completion_s", "name": "minimum-completion sensitivity", "symbol": "overline s", "definition": "Minimum of s(F) over all total Boolean completions F on the same cube with F(x)=f(x) for every promised x. On a total domain this equals s(f).", "definition_status": "source_checked", "family": "extension", "sources": [ "HKMP26" ], "definition_locator": "Definition 45, p.22", "aliases": [ { "text": "overline M", "source": "HKMP26", "locator": "Definition 45, p.22", "status": "source_checked", "normalization": "Generic source template, explicitly instantiated here with M=s. The instantiated display symbol is repository notation, not a separate literal occurrence." } ], "notes": "The source defines an arbitrary-measure template; this record is an explicit instantiation. Each minimum may use a different completion. Overline notation visually checked in the original PDF.", "completion_of": "s", "repository_display_variants": [ { "text": "overline s", "source": "HKMP26", "status": "repository_display_not_a_historical_occurrence", "normalization": "Instantiation of the source overline-M template." } ] }, { "id": "completion_lambda", "name": "minimum-completion spectral sensitivity", "symbol": "overline lambda", "definition": "Minimum of lambda(F) over all total Boolean completions F on the same cube with F(x)=f(x) for every promised x. On a total domain this equals lambda(f).", "definition_status": "source_checked", "family": "extension", "sources": [ "HKMP26" ], "definition_locator": "Definition 45, p.22", "aliases": [ { "text": "overline M", "source": "HKMP26", "locator": "Definition 45, p.22", "status": "source_checked", "normalization": "Generic source template, explicitly instantiated here with M=lambda. The instantiated display symbol is repository notation, not a separate literal occurrence." } ], "notes": "The source defines an arbitrary-measure template; this record is an explicit instantiation. Each minimum may use a different completion. Overline notation visually checked in the original PDF.", "completion_of": "lambda", "repository_display_variants": [ { "text": "overline lambda", "source": "HKMP26", "status": "repository_display_not_a_historical_occurrence", "normalization": "Instantiation of the source overline-M template." } ] }, { "id": "completion_QE", "name": "minimum-completion exact quantum query complexity", "symbol": "overline Q_E", "definition": "Minimum of QE(F) over all total Boolean completions F on the same cube with F(x)=f(x) for every promised x. On a total domain this equals QE(f).", "definition_status": "source_checked", "family": "extension", "sources": [ "HKMP26" ], "definition_locator": "Definition 45, p.22", "aliases": [ { "text": "overline M", "source": "HKMP26", "locator": "Definition 45, p.22", "status": "source_checked", "normalization": "Generic source template, explicitly instantiated here with M=QE. The instantiated display symbol is repository notation, not a separate literal occurrence." } ], "notes": "The source defines an arbitrary-measure template; this record is an explicit instantiation. Each minimum may use a different completion. Overline notation visually checked in the original PDF.", "completion_of": "QE", "repository_display_variants": [ { "text": "overline Q_E", "source": "HKMP26", "status": "repository_display_not_a_historical_occurrence", "normalization": "Instantiation of the source overline-M template." } ] }, { "id": "completion_deg", "name": "minimum-completion real exact polynomial degree", "symbol": "overline deg", "definition": "Minimum of deg(F) over all total Boolean completions F on the same cube with F(x)=f(x) for every promised x. On a total domain this equals deg(f).", "definition_status": "source_checked", "family": "extension", "sources": [ "HKMP26" ], "definition_locator": "Definition 45, p.22", "aliases": [ { "text": "overline M", "source": "HKMP26", "locator": "Definition 45, p.22", "status": "source_checked", "normalization": "Generic source template, explicitly instantiated here with M=deg. The instantiated display symbol is repository notation, not a separate literal occurrence." } ], "notes": "The source defines an arbitrary-measure template; this record is an explicit instantiation. Each minimum may use a different completion. Overline notation visually checked in the original PDF.", "completion_of": "deg", "repository_display_variants": [ { "text": "overline deg", "source": "HKMP26", "status": "repository_display_not_a_historical_occurrence", "normalization": "Instantiation of the source overline-M template." } ] }, { "id": "completion_Q", "name": "minimum-completion bounded-error quantum query complexity", "symbol": "overline Q", "definition": "Minimum of Q(F) over all total Boolean completions F on the same cube with F(x)=f(x) for every promised x. On a total domain this equals Q(f).", "definition_status": "source_checked", "family": "extension", "sources": [ "HKMP26" ], "definition_locator": "Definition 45, p.22", "aliases": [ { "text": "overline M", "source": "HKMP26", "locator": "Definition 45, p.22", "status": "source_checked", "normalization": "Generic source template, explicitly instantiated here with M=Q. The instantiated display symbol is repository notation, not a separate literal occurrence." } ], "notes": "The source defines an arbitrary-measure template; this record is an explicit instantiation. Each minimum may use a different completion. Overline notation visually checked in the original PDF.", "completion_of": "Q", "repository_display_variants": [ { "text": "overline Q", "source": "HKMP26", "status": "repository_display_not_a_historical_occurrence", "normalization": "Instantiation of the source overline-M template." } ] }, { "id": "completion_adeg", "name": "minimum-completion approximate degree", "symbol": "overline \\widetilde{deg}", "definition": "Minimum of adeg(F) over all total Boolean completions F on the same cube with F(x)=f(x) for every promised x. On a total domain this equals adeg(f).", "definition_status": "source_checked", "family": "extension", "sources": [ "HKMP26" ], "definition_locator": "Definition 45, p.22", "aliases": [ { "text": "overline M", "source": "HKMP26", "locator": "Definition 45, p.22", "status": "source_checked", "normalization": "Generic source template, explicitly instantiated here with M=adeg. The instantiated display symbol is repository notation, not a separate literal occurrence." } ], "notes": "The source defines an arbitrary-measure template; this record is an explicit instantiation. Each minimum may use a different completion. Overline notation visually checked in the original PDF.", "completion_of": "adeg", "repository_display_variants": [ { "text": "overline \\widetilde{deg}", "source": "HKMP26", "status": "repository_display_not_a_historical_occurrence", "normalization": "Instantiation of the source overline-M template." } ] }, { "id": "promiseS", "name": "exit-sensitive sensitivity", "symbol": "s prime", "definition": "Maximum over promised x of the number of single-bit flips yielding the opposite output OR leaving the promise.", "definition_status": "source_checked", "family": "extension", "sources": [ "HKMP26" ], "definition_locator": "Definition 15", "aliases": [ { "text": "s prime", "source": "HKMP26", "locator": "Definition 15", "status": "source_checked" } ], "notes": "" }, { "id": "promiseBS", "name": "exit-sensitive block sensitivity", "symbol": "bs prime", "definition": "Maximum over promised x of the number of pairwise disjoint blocks whose flip yields the opposite output OR leaves the promise.", "definition_status": "source_checked", "family": "extension", "sources": [ "HKMP26" ], "definition_locator": "Definition 15", "aliases": [ { "text": "bs prime", "source": "HKMP26", "locator": "Definition 15", "status": "source_checked" } ], "notes": "" }, { "id": "criticalBScompletion", "name": "completion-critical block sensitivity", "symbol": "cbs", "definition": "Minimum over total Boolean completions F of max_{x in Dom(f)} bs(F,x). Off-promise inputs are excluded from the outer maximum, but blocks may lead to them.", "definition_status": "source_checked", "family": "extension", "sources": [ "HKMP26" ], "definition_locator": "Definition 18", "aliases": [ { "text": "cbs", "source": "HKMP26", "locator": "Definition 18", "status": "source_checked" } ], "notes": "" }, { "id": "criticalCertificate", "name": "block-optimal-completion critical certificate complexity", "symbol": "C_cri", "definition": "Among total completions F attaining completion-critical block sensitivity, minimize max_{x in Dom(f)} C(F,x). The minimization is constrained to block-optimal completions.", "definition_status": "source_checked", "family": "extension", "sources": [ "HKMP26" ], "definition_locator": "Definition 19", "aliases": [ { "text": "C_cri", "source": "HKMP26", "locator": "Definition 19", "status": "source_checked" } ], "notes": "" }, { "id": "permC", "name": "permutation certificate complexity", "symbol": "C", "definition": "Maximum over pi of the minimum number of specified permutation entries whose presence forces f(pi) on S_n.", "definition_status": "source_checked", "family": "extension", "sources": [ "DFLLV21", "DFLLV26" ], "definition_locator": "Section 2.2", "aliases": [ { "text": "C", "source": "DFLLV21", "locator": "Section 2.2", "status": "source_checked" } ], "notes": "Not the Boolean-cube measure. In particular R_0 here is a capped detectable-failure model with failure 2/3, not expected Las Vegas cost.", "journal_definition_check": "Final journal Section 2.2; forward/inverse query family, packing sensitivity and detectable failure 2/3 preserved." }, { "id": "permUC", "name": "permutation unambiguous certificate complexity", "symbol": "U", "definition": "Minimum maximum size of permutation-entry certificates forming a partition: each permutation extends exactly one certificate.", "definition_status": "source_checked", "family": "extension", "sources": [ "DFLLV21", "DFLLV26" ], "definition_locator": "Section 2.2", "aliases": [ { "text": "U", "source": "DFLLV21", "locator": "Section 2.2", "status": "source_checked" } ], "notes": "Not the Boolean-cube measure. In particular R_0 here is a capped detectable-failure model with failure 2/3, not expected Las Vegas cost.", "journal_definition_check": "Final journal Section 2.2; forward/inverse query family, packing sensitivity and detectable failure 2/3 preserved." }, { "id": "permFBS", "name": "permutation fractional block sensitivity", "symbol": "fbs", "definition": "At pi, maximize the total nonnegative weight on opposite-output permutations, with total weight at most one for each removed entry of pi. Maximize over pi.", "definition_status": "source_checked", "family": "extension", "sources": [ "DFLLV21", "DFLLV26" ], "definition_locator": "Section 2.2", "aliases": [ { "text": "fbs", "source": "DFLLV21", "locator": "Section 2.2", "status": "source_checked" } ], "notes": "Not the Boolean-cube measure. In particular R_0 here is a capped detectable-failure model with failure 2/3, not expected Las Vegas cost.", "journal_definition_check": "Final journal Section 2.2; forward/inverse query family, packing sensitivity and detectable failure 2/3 preserved." }, { "id": "permR", "name": "permutation randomized query depth", "symbol": "R", "definition": "Minimum worst-case depth cap for forward/inverse symbol queries, with error at most 1/3 on every permutation.", "definition_status": "source_checked", "family": "extension", "sources": [ "DFLLV21", "DFLLV26" ], "definition_locator": "Section 2.2", "aliases": [ { "text": "R", "source": "DFLLV21", "locator": "Section 2.2", "status": "source_checked" } ], "notes": "Not the Boolean-cube measure. In particular R_0 here is a capped detectable-failure model with failure 2/3, not expected Las Vegas cost.", "journal_definition_check": "Final journal Section 2.2; forward/inverse query family, packing sensitivity and detectable failure 2/3 preserved." }, { "id": "permRfail23", "name": "permutation detectable-failure query depth", "symbol": "R_0", "definition": "Minimum depth cap returning f(pi) or a detectable star, never wrong, and star probability at most 2/3 on each permutation.", "definition_status": "source_checked", "family": "extension", "sources": [ "DFLLV21", "DFLLV26" ], "definition_locator": "Section 2.2", "aliases": [ { "text": "R_0", "source": "DFLLV21", "locator": "Section 2.2", "status": "source_checked" } ], "notes": "Not the Boolean-cube measure. In particular R_0 here is a capped detectable-failure model with failure 2/3, not expected Las Vegas cost.", "journal_definition_check": "Final journal Section 2.2; forward/inverse query family, packing sensitivity and detectable failure 2/3 preserved." }, { "id": "permQ", "name": "permutation bounded-error quantum query complexity", "symbol": "Q", "definition": "Minimum number of coherent forward/inverse symbol-oracle queries computing f on S_n with error at most 1/3.", "definition_status": "source_checked", "family": "extension", "sources": [ "DFLLV21", "DFLLV26" ], "definition_locator": "Section 2.2", "aliases": [ { "text": "Q", "source": "DFLLV21", "locator": "Section 2.2", "status": "source_checked" } ], "notes": "Not the Boolean-cube measure. In particular R_0 here is a capped detectable-failure model with failure 2/3, not expected Las Vegas cost.", "journal_definition_check": "Final journal Section 2.2; forward/inverse query family, packing sensitivity and detectable failure 2/3 preserved." }, { "id": "permQE", "name": "permutation exact quantum query complexity", "symbol": "Q_E", "definition": "Minimum fixed number of coherent forward/inverse symbol-oracle queries computing f on S_n with certainty.", "definition_status": "source_checked", "family": "extension", "sources": [ "DFLLV21", "DFLLV26" ], "definition_locator": "Section 2.2", "aliases": [ { "text": "Q_E", "source": "DFLLV21", "locator": "Section 2.2", "status": "source_checked" } ], "notes": "Not the Boolean-cube measure. In particular R_0 here is a capped detectable-failure model with failure 2/3, not expected Las Vegas cost.", "journal_definition_check": "Final journal Section 2.2; forward/inverse query family, packing sensitivity and detectable failure 2/3 preserved." }, { "id": "completion_nRelevant", "name": "minimum relevant-variable count of a total completion", "symbol": "overline nRelevant", "definition": "Minimum nRelevant(F) over total Boolean functions F agreeing with f on its promised domain.", "definition_status": "source_checked", "family": "extension", "sources": [ "HKMP26" ], "definition_locator": "Definition 45, p.22", "aliases": [ { "text": "overline M", "source": "HKMP26", "locator": "Definition 45, p.22", "status": "source_checked", "normalization": "Generic overline-M template instantiated at relevant-variable count." } ], "notes": "Generic completion-template instantiation. Equals deterministic nonadaptive bit-query complexity of a partial decision function; not polynomially equivalent to D even on total functions.", "completion_of": "nRelevant", "repository_display_variants": [ { "text": "overline nRelevant", "source": "HKMP26", "status": "repository_display_not_a_historical_occurrence", "normalization": "Instantiation of the source overline-M template." } ] }, { "id": "completion_C0", "name": "minimum-completion 0-certificate complexity", "symbol": "overline C_0", "definition": "Minimum C0(F) over total Boolean functions F agreeing with f on every promised input.", "definition_status": "source_checked", "family": "extension", "sources": [ "HKMP26" ], "definition_locator": "Definition 45, p.22", "aliases": [ { "text": "overline M", "source": "HKMP26", "locator": "Definition 45, p.22", "status": "source_checked", "normalization": "Generic overline-M template instantiated at M=C0." } ], "notes": "Explicit instance of the generic source template, not a separately named source measure. One-sided measures need not be polynomially equivalent to D on total functions; no automatic equivalence is asserted.", "completion_of": "C0", "repository_display_variants": [ { "text": "overline C_0", "source": "HKMP26", "status": "repository_display_not_a_historical_occurrence", "normalization": "Instantiation of the source overline-M template." } ] }, { "id": "completion_C1", "name": "minimum-completion 1-certificate complexity", "symbol": "overline C_1", "definition": "Minimum C1(F) over total Boolean functions F agreeing with f on every promised input.", "definition_status": "source_checked", "family": "extension", "sources": [ "HKMP26" ], "definition_locator": "Definition 45, p.22", "aliases": [ { "text": "overline M", "source": "HKMP26", "locator": "Definition 45, p.22", "status": "source_checked", "normalization": "Generic overline-M template instantiated at M=C1." } ], "notes": "Explicit instance of the generic source template, not a separately named source measure. One-sided measures need not be polynomially equivalent to D on total functions; no automatic equivalence is asserted.", "completion_of": "C1", "repository_display_variants": [ { "text": "overline C_1", "source": "HKMP26", "status": "repository_display_not_a_historical_occurrence", "normalization": "Instantiation of the source overline-M template." } ] }, { "id": "completion_UC", "name": "minimum-completion unambiguous certificate complexity", "symbol": "overline UC", "definition": "Minimum UC(F) over total Boolean functions F agreeing with f on every promised input.", "definition_status": "source_checked", "family": "extension", "sources": [ "HKMP26" ], "definition_locator": "Definition 45, p.22", "aliases": [ { "text": "overline M", "source": "HKMP26", "locator": "Definition 45, p.22", "status": "source_checked", "normalization": "Generic overline-M template instantiated at M=UC." } ], "notes": "Explicit instance of the generic source template, not a separately named source measure. One-sided measures need not be polynomially equivalent to D on total functions; no automatic equivalence is asserted.", "completion_of": "UC", "repository_display_variants": [ { "text": "overline UC", "source": "HKMP26", "status": "repository_display_not_a_historical_occurrence", "normalization": "Instantiation of the source overline-M template." } ] }, { "id": "completion_UC0", "name": "minimum-completion 0-unambiguous certificate complexity", "symbol": "overline UC_0", "definition": "Minimum UC0(F) over total Boolean functions F agreeing with f on every promised input.", "definition_status": "source_checked", "family": "extension", "sources": [ "HKMP26" ], "definition_locator": "Definition 45, p.22", "aliases": [ { "text": "overline M", "source": "HKMP26", "locator": "Definition 45, p.22", "status": "source_checked", "normalization": "Generic overline-M template instantiated at M=UC0." } ], "notes": "Explicit instance of the generic source template, not a separately named source measure. One-sided measures need not be polynomially equivalent to D on total functions; no automatic equivalence is asserted.", "completion_of": "UC0", "repository_display_variants": [ { "text": "overline UC_0", "source": "HKMP26", "status": "repository_display_not_a_historical_occurrence", "normalization": "Instantiation of the source overline-M template." } ] }, { "id": "completion_UC1", "name": "minimum-completion 1-unambiguous certificate complexity", "symbol": "overline UC_1", "definition": "Minimum UC1(F) over total Boolean functions F agreeing with f on every promised input.", "definition_status": "source_checked", "family": "extension", "sources": [ "HKMP26" ], "definition_locator": "Definition 45, p.22", "aliases": [ { "text": "overline M", "source": "HKMP26", "locator": "Definition 45, p.22", "status": "source_checked", "normalization": "Generic overline-M template instantiated at M=UC1." } ], "notes": "Explicit instance of the generic source template, not a separately named source measure. One-sided measures need not be polynomially equivalent to D on total functions; no automatic equivalence is asserted.", "completion_of": "UC1", "repository_display_variants": [ { "text": "overline UC_1", "source": "HKMP26", "status": "repository_display_not_a_historical_occurrence", "normalization": "Instantiation of the source overline-M template." } ] }, { "id": "completion_fbs", "name": "minimum-completion fractional block sensitivity", "symbol": "overline fbs", "definition": "Minimum fbs(F) over total Boolean functions F agreeing with f on every promised input.", "definition_status": "source_checked", "family": "extension", "sources": [ "HKMP26" ], "definition_locator": "Definition 45, p.22", "aliases": [ { "text": "overline M", "source": "HKMP26", "locator": "Definition 45, p.22", "status": "source_checked", "normalization": "Generic overline-M template instantiated at M=fbs." } ], "notes": "Explicit instance of the generic source template, not a separately named source measure. One-sided measures need not be polynomially equivalent to D on total functions; no automatic equivalence is asserted.", "completion_of": "fbs", "repository_display_variants": [ { "text": "overline fbs", "source": "HKMP26", "status": "repository_display_not_a_historical_occurrence", "normalization": "Instantiation of the source overline-M template." } ] }, { "id": "completion_FC", "name": "minimum-completion fractional certificate complexity", "symbol": "overline FC", "definition": "Minimum FC(F) over total Boolean functions F agreeing with f on every promised input.", "definition_status": "source_checked", "family": "extension", "sources": [ "HKMP26" ], "definition_locator": "Definition 45, p.22", "aliases": [ { "text": "overline M", "source": "HKMP26", "locator": "Definition 45, p.22", "status": "source_checked", "normalization": "Generic overline-M template instantiated at M=FC." } ], "notes": "Explicit instance of the generic source template, not a separately named source measure. One-sided measures need not be polynomially equivalent to D on total functions; no automatic equivalence is asserted.", "completion_of": "FC", "repository_display_variants": [ { "text": "overline FC", "source": "HKMP26", "status": "repository_display_not_a_historical_occurrence", "normalization": "Instantiation of the source overline-M template." } ] }, { "id": "completion_QC", "name": "minimum-completion quantum certificate complexity", "symbol": "overline QC", "definition": "Minimum QC(F) over total Boolean functions F agreeing with f on every promised input.", "definition_status": "source_checked", "family": "extension", "sources": [ "HKMP26" ], "definition_locator": "Definition 45, p.22", "aliases": [ { "text": "overline M", "source": "HKMP26", "locator": "Definition 45, p.22", "status": "source_checked", "normalization": "Generic overline-M template instantiated at M=QC." } ], "notes": "Explicit instance of the generic source template, not a separately named source measure. One-sided measures need not be polynomially equivalent to D on total functions; no automatic equivalence is asserted.", "completion_of": "QC", "repository_display_variants": [ { "text": "overline QC", "source": "HKMP26", "status": "repository_display_not_a_historical_occurrence", "normalization": "Instantiation of the source overline-M template." } ] }, { "id": "completion_Q0", "name": "minimum-completion zero-error quantum query complexity", "symbol": "overline Q_0", "definition": "Minimum Q0(F) over total Boolean functions F agreeing with f on every promised input.", "definition_status": "source_checked", "family": "extension", "sources": [ "HKMP26" ], "definition_locator": "Definition 45, p.22", "aliases": [ { "text": "overline M", "source": "HKMP26", "locator": "Definition 45, p.22", "status": "source_checked", "normalization": "Generic overline-M template instantiated at M=Q0." } ], "notes": "Explicit instance of the generic source template, not a separately named source measure. One-sided measures need not be polynomially equivalent to D on total functions; no automatic equivalence is asserted.", "completion_of": "Q0", "repository_display_variants": [ { "text": "overline Q_0", "source": "HKMP26", "status": "repository_display_not_a_historical_occurrence", "normalization": "Instantiation of the source overline-M template." } ] }, { "id": "completion_R1", "name": "minimum-completion one-sided-error randomized query complexity", "symbol": "overline R_1", "definition": "Minimum R1(F) over total Boolean functions F agreeing with f on every promised input.", "definition_status": "source_checked", "family": "extension", "sources": [ "HKMP26" ], "definition_locator": "Definition 45, p.22", "aliases": [ { "text": "overline M", "source": "HKMP26", "locator": "Definition 45, p.22", "status": "source_checked", "normalization": "Generic overline-M template instantiated at M=R1." } ], "notes": "Explicit instance of the generic source template, not a separately named source measure. One-sided measures need not be polynomially equivalent to D on total functions; no automatic equivalence is asserted.", "completion_of": "R1", "repository_display_variants": [ { "text": "overline R_1", "source": "HKMP26", "status": "repository_display_not_a_historical_occurrence", "normalization": "Instantiation of the source overline-M template." } ] }, { "id": "domainD", "name": "query-domain deterministic depth", "symbol": "D", "definition": "Minimum depth of a decision tree whose query Q returns the unique element of x intersect Q, for x in X.", "definition_status": "source_checked", "family": "extension", "sources": [ "DFLLV22author", "DFLLV26" ], "definition_locator": "Sections 2.1–2.2", "aliases": [ { "text": "D", "source": "DFLLV22author", "locator": "Sections 2.1–2.2", "status": "source_checked" } ], "notes": "Domain X consists of n-element subsets of U. Each allowed query intersects every input in exactly one element, and queries cover U. Use domain-qualified scopes; this is not an additional Boolean-cube definition.", "journal_definition_check": "Final journal Section 2.2; forward/inverse query family, packing sensitivity and detectable failure 2/3 preserved." }, { "id": "domainR", "name": "query-domain bounded-error randomized depth", "symbol": "R", "definition": "Minimum common depth cap of a distribution over domain decision trees, with success at least 2/3 on each x in X.", "definition_status": "source_checked", "family": "extension", "sources": [ "DFLLV22author", "DFLLV26" ], "definition_locator": "Sections 2.1–2.2", "aliases": [ { "text": "R", "source": "DFLLV22author", "locator": "Sections 2.1–2.2", "status": "source_checked" } ], "notes": "Domain X consists of n-element subsets of U. Each allowed query intersects every input in exactly one element, and queries cover U. Use domain-qualified scopes; this is not an additional Boolean-cube definition.", "journal_definition_check": "Final journal Section 2.2; forward/inverse query family, packing sensitivity and detectable failure 2/3 preserved." }, { "id": "domainRfail23", "name": "query-domain detectable-failure depth", "symbol": "R_0", "definition": "Minimum common depth cap of zero-error trees with outputs 0,1,star and failure probability at most 2/3 on every x.", "definition_status": "source_checked", "family": "extension", "sources": [ "DFLLV22author", "DFLLV26" ], "definition_locator": "Sections 2.1–2.2", "aliases": [ { "text": "R_0", "source": "DFLLV22author", "locator": "Sections 2.1–2.2", "status": "source_checked" } ], "notes": "Domain X consists of n-element subsets of U. Each allowed query intersects every input in exactly one element, and queries cover U. Use domain-qualified scopes; this is not an additional Boolean-cube definition.", "journal_definition_check": "Final journal Section 2.2; forward/inverse query family, packing sensitivity and detectable failure 2/3 preserved." }, { "id": "domainQ", "name": "query-domain bounded-error quantum queries", "symbol": "Q", "definition": "Minimum coherent domain-symbol oracle queries computing the Boolean output with error at most 1/3.", "definition_status": "source_checked", "family": "extension", "sources": [ "DFLLV22author", "DFLLV26" ], "definition_locator": "Sections 2.1–2.2", "aliases": [ { "text": "Q", "source": "DFLLV22author", "locator": "Sections 2.1–2.2", "status": "source_checked" } ], "notes": "Domain X consists of n-element subsets of U. Each allowed query intersects every input in exactly one element, and queries cover U. Use domain-qualified scopes; this is not an additional Boolean-cube definition.", "journal_definition_check": "Final journal Section 2.2; forward/inverse query family, packing sensitivity and detectable failure 2/3 preserved." }, { "id": "domainQE", "name": "query-domain exact quantum queries", "symbol": "Q_E", "definition": "Minimum fixed coherent domain-symbol oracle query count computing the output with certainty.", "definition_status": "source_checked", "family": "extension", "sources": [ "DFLLV22author", "DFLLV26" ], "definition_locator": "Sections 2.1–2.2", "aliases": [ { "text": "Q_E", "source": "DFLLV22author", "locator": "Sections 2.1–2.2", "status": "source_checked" } ], "notes": "Domain X consists of n-element subsets of U. Each allowed query intersects every input in exactly one element, and queries cover U. Use domain-qualified scopes; this is not an additional Boolean-cube definition.", "journal_definition_check": "Final journal Section 2.2; forward/inverse query family, packing sensitivity and detectable failure 2/3 preserved." }, { "id": "domainDegree", "name": "query-domain spatial degree", "symbol": "deg", "definition": "Minimum maximum |A| in a linear combination of indicators [A subseteq x] representing f on X; equivalently degree in membership indicators.", "definition_status": "source_checked", "family": "extension", "sources": [ "DFLLV22author", "DFLLV26" ], "definition_locator": "Sections 2.1–2.2", "aliases": [ { "text": "deg", "source": "DFLLV22author", "locator": "Sections 2.1–2.2", "status": "source_checked" } ], "notes": "Domain X consists of n-element subsets of U. Each allowed query intersects every input in exactly one element, and queries cover U. Use domain-qualified scopes; this is not an additional Boolean-cube definition.", "journal_definition_check": "Final journal Section 2.2; forward/inverse query family, packing sensitivity and detectable failure 2/3 preserved." }, { "id": "domainAdeg", "name": "query-domain spatial approximate degree", "symbol": "tilde deg", "definition": "Minimum maximum |A| in a real linear combination of indicators [A subseteq x] approximating f within 1/3 on X. No bound outside X is imposed.", "definition_status": "source_checked", "family": "extension", "sources": [ "DFLLV22author", "DFLLV26" ], "definition_locator": "Sections 2.1–2.2", "aliases": [ { "text": "tilde deg", "source": "DFLLV22author", "locator": "Sections 2.1–2.2", "status": "source_checked" } ], "notes": "Domain X consists of n-element subsets of U. Each allowed query intersects every input in exactly one element, and queries cover U. Use domain-qualified scopes; this is not an additional Boolean-cube definition.", "journal_definition_check": "Final journal Section 2.2; forward/inverse query family, packing sensitivity and detectable failure 2/3 preserved." }, { "id": "domainC", "name": "query-domain certificate complexity", "symbol": "C", "definition": "Maximum over x in X of the minimum |a| for a subset a of x such that every y in X extending a has f(y)=f(x).", "definition_status": "source_checked", "family": "extension", "sources": [ "DFLLV22author", "DFLLV26" ], "definition_locator": "Sections 2.1–2.2", "aliases": [ { "text": "C", "source": "DFLLV22author", "locator": "Sections 2.1–2.2", "status": "source_checked" } ], "notes": "Domain X consists of n-element subsets of U. Each allowed query intersects every input in exactly one element, and queries cover U. Use domain-qualified scopes; this is not an additional Boolean-cube definition.", "journal_definition_check": "Final journal Section 2.2; forward/inverse query family, packing sensitivity and detectable failure 2/3 preserved." }, { "id": "domainUC", "name": "query-domain unambiguous certificate complexity", "symbol": "U", "definition": "Minimum largest certificate size in a collection of subsets of the universe such that every x in X extends exactly one certificate.", "definition_status": "source_checked", "family": "extension", "sources": [ "DFLLV22author", "DFLLV26" ], "definition_locator": "Sections 2.1–2.2", "aliases": [ { "text": "U", "source": "DFLLV22author", "locator": "Sections 2.1–2.2", "status": "source_checked" } ], "notes": "Domain X consists of n-element subsets of U. Each allowed query intersects every input in exactly one element, and queries cover U. Use domain-qualified scopes; this is not an additional Boolean-cube definition.", "journal_definition_check": "Final journal Section 2.2; forward/inverse query family, packing sensitivity and detectable failure 2/3 preserved." }, { "id": "domainBS", "name": "query-domain block sensitivity", "symbol": "bs", "definition": "Maximum over x of the largest family of opposite-output y_j in X for which the removed-entry sets x minus y_j are pairwise disjoint.", "definition_status": "source_checked", "family": "extension", "sources": [ "DFLLV22author", "DFLLV26" ], "definition_locator": "Sections 2.1–2.2", "aliases": [ { "text": "bs", "source": "DFLLV22author", "locator": "Sections 2.1–2.2", "status": "source_checked" } ], "notes": "Domain X consists of n-element subsets of U. Each allowed query intersects every input in exactly one element, and queries cover U. Use domain-qualified scopes; this is not an additional Boolean-cube definition.", "journal_definition_check": "Final journal Section 2.2; forward/inverse query family, packing sensitivity and detectable failure 2/3 preserved." }, { "id": "domainFBS", "name": "query-domain fractional block sensitivity", "symbol": "fbs", "definition": "Maximum over x of the nonnegative weighted packing of opposite-output y, subject to total weight at most one through each removed element of x.", "definition_status": "source_checked", "family": "extension", "sources": [ "DFLLV22author", "DFLLV26" ], "definition_locator": "Sections 2.1–2.2", "aliases": [ { "text": "fbs", "source": "DFLLV22author", "locator": "Sections 2.1–2.2", "status": "source_checked" } ], "notes": "Domain X consists of n-element subsets of U. Each allowed query intersects every input in exactly one element, and queries cover U. Use domain-qualified scopes; this is not an additional Boolean-cube definition.", "journal_definition_check": "Final journal Section 2.2; forward/inverse query family, packing sensitivity and detectable failure 2/3 preserved." }, { "id": "domainS", "name": "query-domain chunk sensitivity", "symbol": "s", "definition": "Maximum over x of the largest family of opposite-output y_j with pairwise disjoint x minus y_j and each |x minus y_j| equal to the domain chunk size c.", "definition_status": "source_checked", "family": "extension", "sources": [ "DFLLV22author", "DFLLV26", "BD16" ], "definition_locator": "Sections 2.1–2.2", "aliases": [ { "text": "s", "source": "DFLLV22author", "locator": "Sections 2.1–2.2", "status": "source_checked" }, { "text": "s_2(f)", "source": "BD16", "locator": "Definition 7", "status": "source_checked", "normalization": "Packing sensitivity for disjoint sensitive two-coordinate blocks on a single multislice; equals domain chunk sensitivity there." } ], "notes": "Domain X consists of n-element subsets of U. Each allowed query intersects every input in exactly one element, and queries cover U. Use domain-qualified scopes; this is not an additional Boolean-cube definition.", "journal_definition_check": "Final journal Section 2.2; forward/inverse query family, packing sensitivity and detectable failure 2/3 preserved." }, { "id": "domainCoverRadius", "name": "covering radius of a promise domain", "symbol": "r_C", "definition": "For nonempty X=Dom(f) subseteq {+1,-1}^n, max over cube points y of min_{x in X} HammingDistance(x,y).", "definition_status": "source_checked", "family": "extension", "sources": [ "HKMP26" ], "definition_locator": "Definition 61", "aliases": [ { "text": "r_C", "source": "HKMP26", "locator": "Definition 61", "status": "source_checked" } ], "notes": "These are domain/polynomial parameters, not canonical Fourier statistics of an incompletely defined Boolean function.", "parameters": [] }, { "id": "polyCoordSparsity", "name": "coordinate Fourier sparsity of a fixed polynomial", "symbol": "spar_i", "definition": "Number of nonzero Walsh Fourier coefficients of p whose index sets contain the specified coordinate i.", "definition_status": "source_checked", "family": "extension", "sources": [ "HKMP26" ], "definition_locator": "Definition 28 and Lemma 63", "aliases": [ { "text": "spar_i", "source": "HKMP26", "locator": "Definition 28", "status": "source_checked" } ], "notes": "These are domain/polynomial parameters, not canonical Fourier statistics of an incompletely defined Boolean function.", "parameters": [ "p", "i" ] }, { "id": "polyCoordInfluence", "name": "coordinate Fourier influence of a fixed polynomial", "symbol": "Inf_i", "definition": "E[(D_i p)^2] under the uniform full cube, where D_i p(x)=(p(x)-p(x^i))/2; equals sum_{S containing i} p_hat(S)^2.", "definition_status": "source_checked", "family": "extension", "sources": [ "HKMP26" ], "definition_locator": "Definitions 21–22", "aliases": [ { "text": "Inf_i", "source": "HKMP26", "locator": "Definitions 21–22", "status": "source_checked" } ], "notes": "These are domain/polynomial parameters, not canonical Fourier statistics of an incompletely defined Boolean function.", "parameters": [ "p", "i" ] }, { "id": "polyMaxDerivative", "name": "maximum coordinate derivative sup norm", "symbol": "max_i ||D_i p||_infinity", "definition": "Maximum over i and full-cube x of |p(x)-p(x^i)|/2 for a fixed polynomial p.", "definition_status": "source_checked", "family": "extension", "sources": [ "HKMP26" ], "definition_locator": "Corollary 62", "aliases": [ { "text": "max_i ||D_i p||_infinity", "source": "HKMP26", "locator": "Corollary 62", "status": "definition_equivalence_checked_literal_name_pending" } ], "notes": "These are domain/polynomial parameters, not canonical Fourier statistics of an incompletely defined Boolean function. The maximum is an equivalent repository aggregation of the source’s bounds for every coordinate; its literal displayed name is not verified.", "parameters": [ "p" ] }, { "id": "distFlipInfluence", "name": "distributional bit-flip squared influence", "symbol": "Inf_i^(lambda)", "definition": "E_{X drawn from lambda}[(p(X)-p(X flipped at i))^2] for a function p on the entire Boolean cube. The off-support value is used even if lambda is a slice distribution.", "definition_status": "source_checked", "family": "extension", "sources": [ "GMNR26", "AA14" ], "definition_locator": "Definition 2.5", "aliases": [ { "text": "Inf_i^(lambda)", "source": "GMNR26", "locator": "Definition 2.5", "status": "source_checked" }, { "text": "Inf_i[p]", "source": "AA14", "locator": "Before Conjecture 1.7, p.139", "status": "source_checked", "normalization": "Uniform cube; unnormalized squared flip difference. Equals four times the HKMP26 derivative-normalized influence for the same full-cube function." } ], "parameters": [ "lambda", "i" ], "notes": "Distribution/coordinate parameters stay explicit. This is an averaged squared difference, not maximum disjoint-transposition sensitivity." }, { "id": "distTotalInfluence", "name": "distributional total bit-flip squared influence", "symbol": "Inf^(lambda)", "definition": "Sum over coordinates i of E_lambda[(p(X)-p(X flipped at i))^2] for a full-cube function p.", "definition_status": "source_checked", "family": "extension", "sources": [ "GMNR26" ], "definition_locator": "Definition 2.5", "aliases": [ { "text": "Inf^(lambda)", "source": "GMNR26", "locator": "Definition 2.5", "status": "source_checked" } ], "parameters": [ "lambda" ], "notes": "Distribution/coordinate parameters stay explicit. This is an averaged squared difference, not maximum disjoint-transposition sensitivity." }, { "id": "distVariance", "name": "distributional variance", "symbol": "Var^(lambda)", "definition": "E_lambda[(p(X)-E_lambda p)^2] for a full-cube function p; only its restriction to the support is needed for variance.", "definition_status": "source_checked", "family": "extension", "sources": [ "GMNR26" ], "definition_locator": "Definition 2.5", "aliases": [ { "text": "Var^(lambda)", "source": "GMNR26", "locator": "Definition 2.5", "status": "source_checked" } ], "parameters": [ "lambda" ], "notes": "Distribution/coordinate parameters stay explicit. This is an averaged squared difference, not maximum disjoint-transposition sensitivity." }, { "id": "sliceSwapInfluence", "name": "slice transposition squared influence", "symbol": "swInf_(i,j)", "definition": "E over the uniform k-slice of (p(X)-p(X with coordinates i,j swapped))^2. This depends only on values on the slice.", "definition_status": "source_checked", "family": "extension", "sources": [ "GMNR26" ], "definition_locator": "Lemma 2.6", "aliases": [ { "text": "swInf_(i,j)", "source": "GMNR26", "locator": "Lemma 2.6", "status": "source_checked" } ], "parameters": [ "k", "i", "j" ], "notes": "Distribution/coordinate parameters stay explicit. This is an averaged squared difference, not maximum disjoint-transposition sensitivity." }, { "id": "degreeOptimalFourierSparsity", "name": "degree-optimal bounded exact Fourier sparsity of a partial function", "symbol": "spar", "definition": "Minimum number of nonzero Walsh coefficients among full-cube [-1,1]-bounded polynomials representing the ±1-valued partial function exactly on its promise, constrained to have minimum possible globally bounded exact degree.", "definition_status": "source_checked", "family": "extension", "sources": [ "HKMP26" ], "definition_locator": "Definition 29", "aliases": [ { "text": "spar", "source": "HKMP26", "locator": "Definition 29", "status": "source_checked" } ], "notes": "Keep degree optimality, the basis, output encoding and bias convention explicit. These are not ordinary unconstrained approximate monomial sparsity." }, { "id": "adegOptimalFourierSparsity", "name": "degree-optimal bounded approximate Fourier sparsity of a partial function", "symbol": "tilde spar", "definition": "Minimum number of nonzero Walsh coefficients among full-cube [-1,1]-bounded polynomials approximating the ±1-valued partial function within 1/3 on its promise, constrained to have minimum possible bounded approximate degree at that error.", "definition_status": "source_checked", "family": "extension", "sources": [ "HKMP26" ], "definition_locator": "Definition 29", "aliases": [ { "text": "tilde spar", "source": "HKMP26", "locator": "Definition 29", "status": "source_checked" } ], "notes": "Keep degree optimality, the basis, output encoding and bias convention explicit. These are not ordinary unconstrained approximate monomial sparsity." }, { "id": "biasedCoordInfluence", "name": "biased Fourier coefficient influence", "symbol": "Inf_i", "definition": "Sum of squared p-biased orthonormal Fourier coefficients indexed by sets containing i. HKMP26 uses p=Pr[x_i=-1], mean mu=1-2p and sigma=2*sqrt(p*(1-p)). This equals sigma^2*E_p[(D_i f)^2].", "definition_status": "source_checked", "family": "extension", "sources": [ "HKMP26" ], "definition_locator": "Definitions 22–25 and Lemma 66", "aliases": [ { "text": "Inf_i", "source": "HKMP26", "locator": "Definitions 22–25 and Lemma 66", "status": "source_checked" } ], "notes": "Keep degree optimality, the basis, output encoding and bias convention explicit. These are not ordinary unconstrained approximate monomial sparsity.", "parameters": [ "bias_p", "i" ] } ], "claims": [ { "id": "s-bs", "kind": "upper_bound", "A": "s", "B": "bs", "exponent": 1, "statement": "s(f) <= bs(f)", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "definition_proof", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Singleton blocks are disjoint sensitive blocks.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "bs-C", "kind": "upper_bound", "A": "bs", "B": "C", "exponent": 1, "statement": "bs(f) <= C(f)", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "definition_proof", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Every certificate intersects every sensitive block.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "C-D", "kind": "upper_bound", "A": "C", "B": "D", "exponent": 1, "statement": "C(f) <= D(f)", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "definition_proof", "source": "definitions", "locator": "Repository derivation", "proof_summary": "A deterministic transcript fixes the output.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "adeg-deg", "kind": "upper_bound", "A": "adeg", "B": "deg", "exponent": 1, "statement": "adeg(f) <= deg(f)", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "definition_proof", "source": "definitions", "locator": "Repository derivation", "proof_summary": "An exact polynomial is an approximant.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "R-D", "kind": "upper_bound", "A": "R", "B": "D", "exponent": 1, "statement": "R(f) <= D(f)", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "definition_proof", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Simulate a deterministic algorithm.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "Q-R", "kind": "upper_bound", "A": "Q", "B": "R", "exponent": 1, "statement": "Q(f) <= R(f)", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "definition_proof", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Simulate randomized queries coherently or with mixed states.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "QE-D", "kind": "upper_bound", "A": "QE", "B": "D", "exponent": 1, "statement": "QE(f) <= D(f)", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "definition_proof", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Simulate deterministic queries exactly.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "Q-QE", "kind": "upper_bound", "A": "Q", "B": "QE", "exponent": 1, "statement": "Q(f) <= QE(f)", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "definition_proof", "source": "definitions", "locator": "Repository derivation", "proof_summary": "An exact answer meets the error requirement.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "C-UC", "kind": "upper_bound", "A": "C", "B": "UC", "exponent": 1, "statement": "C(f) <= UC(f)", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "definition_proof", "source": "definitions", "locator": "Repository derivation", "proof_summary": "An unambiguous certificate is a certificate.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "bs-fbs", "kind": "upper_bound", "A": "bs", "B": "fbs", "exponent": 1, "statement": "bs(f) <= fbs(f)", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "definition_proof", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Put unit weight on disjoint blocks.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "FC-C", "kind": "upper_bound", "A": "FC", "B": "C", "exponent": 1, "statement": "FC(f) <= C(f)", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "definition_proof", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Use an integral certificate as an LP cover.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "FC-fbs", "kind": "identity", "A": "FC", "B": "fbs", "exponent": 1, "statement": "FC(f) = fbs(f)", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "definition_proof", "source": "definitions", "locator": "LP primal/dual", "proof_summary": "Finite LP duality.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "C-C0C1", "kind": "multivariate_upper_bound", "A": "D", "B": null, "exponent": null, "statement": "D(f) <= C_0(f) C_1(f)", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "survey_checked", "source": "BdW02", "locator": "Certificate/decision-tree relations", "proof_summary": "Query compatible certificates; each failed round reduces the opposite certificate budget.", "last_checked": "2026-10-17", "factors": [ { "measure": "C0", "power": 1 }, { "measure": "C1", "power": 1 } ], "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "C-bs2", "kind": "upper_bound", "A": "C", "B": "bs", "exponent": 2, "statement": "C(f) <= bs(f)^2", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "source": "BBCMW01", "locator": "Lemma 5.2 (Nisan), PDF pp.7–8", "proof_summary": "At x, take a maximal disjoint family of minimal sensitive blocks. Its union is a certificate; each block has size at most bs at the flipped input. The original proof is reproduced here.", "last_checked": "2026-10-17", "original_source": "Nis91", "best_known_certified": false, "proof_audit": "proof_route_checked_not_independent_full_proof", "proof_sources": [ "Nis89" ], "original_locator": "Lemma 2.4, p.331" }, { "id": "D-Cbs", "kind": "multivariate_upper_bound", "A": "D", "B": null, "exponent": null, "statement": "D(f) <= C(f) bs(f)", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "source": "BBCMW01", "locator": "Lemma 5.3, PDF p.8", "proof_summary": "Query a consistent 1-certificate per round. If more than bs rounds were needed, the failures would yield too many disjoint sensitive blocks; each round costs at most C_1.", "last_checked": "2026-10-17", "factors": [ { "measure": "C", "power": 1 }, { "measure": "bs", "power": 1 } ], "original_source": "Nis91", "best_known_certified": false, "proof_audit": "proof_route_checked_not_independent_full_proof" }, { "id": "D-deg-bs", "kind": "multivariate_upper_bound", "A": "D", "B": null, "exponent": null, "statement": "D(f) <= deg(f) bs(f)", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "source_checked", "source": "Mid04", "locator": "Proof of Theorem 4, final paragraph", "proof_summary": "Query a maxonomial repeatedly; each round supplies a disjoint sensitive block.", "last_checked": "2026-10-17", "factors": [ { "measure": "deg", "power": 1 }, { "measure": "bs", "power": 1 } ], "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "D-deg3", "kind": "upper_bound", "A": "D", "B": "deg", "exponent": 3, "statement": "D(f) <= 2 deg(f)^3", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "source_checked", "source": "Mid04", "locator": "Theorem 4", "proof_summary": "Combine the maxonomial algorithm with bs <= 2 deg^2.", "last_checked": "2026-10-17", "constant": { "value": 2, "normalization": "real 0/1 exact degree" }, "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "deg-QE", "kind": "upper_bound", "A": "deg", "B": "QE", "exponent": 1, "statement": "deg(f) <= 2 Q_E(f)", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "source_checked", "source": "BBCMW01", "locator": "Polynomial-method lemma", "proof_summary": "T quantum queries give amplitudes of degree T and probabilities of degree at most 2T.", "last_checked": "2026-10-17", "constant": { "value": 2, "normalization": "standard bit oracle" }, "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "adeg-Q", "kind": "upper_bound", "A": "adeg", "B": "Q", "exponent": 1, "statement": "adeg(f) <= 2 Q(f)", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "source_checked", "source": "BBCMW01", "locator": "Polynomial-method lemma", "proof_summary": "The acceptance polynomial approximates f.", "last_checked": "2026-10-17", "constant": { "value": 2, "normalization": "0/1 absolute error 1/3" }, "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "D-Q6-history", "kind": "upper_bound", "A": "D", "B": "Q", "exponent": 6, "statement": "D(f) = O(Q(f)^6)", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "source_checked", "source": "BBCMW01", "locator": "Main classical simulation bound", "proof_summary": "Polynomial method plus classical degree and block-sensitivity inequalities.", "last_checked": "2026-10-17", "historical": true, "superseded_by": [ "D-Q4" ], "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "deg-s2", "kind": "upper_bound", "A": "deg", "B": "s", "exponent": 2, "statement": "deg(f) <= s(f)^2", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "source_checked", "source": "Hua19", "locator": "Sensitivity-conjecture consequence", "proof_summary": "Signed hypercube adjacency and eigenvalue interlacing.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "deg-lambda2", "kind": "upper_bound", "A": "deg", "B": "lambda", "exponent": 2, "statement": "deg(f) <= lambda(f)^2", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "source_checked", "source": "ABKBRT21", "locator": "Theorem 1", "proof_summary": "Huang spectral argument.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "lambda-s", "kind": "upper_bound", "A": "lambda", "B": "s", "exponent": 1, "statement": "lambda(f) <= s(f)", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "definition_proof", "source": "definitions", "locator": "Adjacency norm", "proof_summary": "Spectral radius is at most maximum degree.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "s-lambda2", "kind": "upper_bound", "A": "s", "B": "lambda", "exponent": 2, "statement": "s(f) <= lambda(f)^2", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "definition_proof", "source": "definitions", "locator": "Star subgraph", "proof_summary": "A vertex of degree s induces a star with norm sqrt(s).", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "lambda-deg", "kind": "upper_bound", "A": "lambda", "B": "deg", "exponent": 1, "statement": "lambda(f) <= deg(f)", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "source_checked", "source": "ABKBRT21", "locator": "Theorem 10", "proof_summary": "Fourier matrix norm estimate.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "lambda-adeg", "kind": "upper_bound", "A": "lambda", "B": "adeg", "exponent": 1, "statement": "lambda(f) = O(adeg(f))", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "source_checked", "source": "ABKBRT21", "locator": "Theorem 17", "proof_summary": "Composition and tensor powers.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "lambda-adeg-constant", "kind": "upper_bound", "A": "lambda", "B": "adeg_bounded", "exponent": 1, "statement": "lambda(f) <= adeg_bounded_epsilon(f)/(1-2epsilon)", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "source_checked", "source": "ABKBRT21", "locator": "Theorem 18", "proof_summary": "Polynomial difference matrix.", "last_checked": "2026-10-17", "constant": { "value": "1/(1-2epsilon)", "normalization": "bounded signed-range approximation, epsilon < 1/2" }, "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "deg-adeg2", "kind": "upper_bound", "A": "deg", "B": "adeg", "exponent": 2, "statement": "deg(f) = O(adeg(f)^2)", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "source_checked", "source": "ABKBRT21", "locator": "Theorem 4", "proof_summary": "Huang’s signed-hypercube argument gives deg<=lambda^2. A spectral bound proved using approximating polynomials gives lambda=O(adeg); composing them yields the quadratic relation.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "deg-Q2", "kind": "upper_bound", "A": "deg", "B": "Q", "exponent": 2, "statement": "deg(f) = O(Q(f)^2)", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "source_checked", "source": "ABKBRT21", "locator": "Theorem 3", "proof_summary": "Huang plus spectral adversary.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "D-Q4", "kind": "upper_bound", "A": "D", "B": "Q", "exponent": 4, "statement": "D(f) = O(Q(f)^4)", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "source_checked", "source": "ABKBRT21", "locator": "Theorem 2", "proof_summary": "Combine Midrijanis D<=deg*bs with the spectral-sensitivity bound deg=O(Q^2) and the polynomial-method bound bs=O(Q^2). Both factors are essential to the quartic exponent.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "RC-adeg2", "kind": "upper_bound", "A": "RC", "B": "adeg", "exponent": 2, "statement": "RC(f) = O(adeg(f)^2)", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "source": "KT13", "locator": "Theorem 3.4 and Corollary 1.2, PDF pp.7–8", "proof_summary": "Amplify a hypothetical counterexample by repeated composition; composition limits make integral and fractional block sensitivity comparable, while approximate degree grows multiplicatively up to constants.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "proof_route_checked_not_independent_full_proof" }, { "id": "R0-R2", "kind": "upper_bound", "A": "R0", "B": "R", "exponent": 2, "statement": "R_0(f) = O(R(f)^2 log(2+R(f)))", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "explicit_log", "status": "primary_statement_and_proof_route_checked", "source": "KT13", "locator": "Theorem 5.3, PDF p.10", "proof_summary": "Bound the number of minimal flipping blocks by 2 fbs(f)^s(f), substitute into Midrijanis R_0=O(R log nbs), and use s<=fbs=O(R).", "last_checked": "2026-10-17", "log_factor": "log(2+R)", "best_known_certified": false, "proof_audit": "proof_route_checked_not_independent_full_proof" }, { "id": "QC-RC", "kind": "theta_bound", "A": "QC", "B": "RC", "exponent": "1/2", "statement": "QC(f) = Theta(sqrt(RC(f)))", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "source_checked", "source": "Aar02", "locator": "Main characterization", "proof_summary": "An optimal fractional query distribution gives a quantum verifier by amplitude amplification, costing the square root of classical verification. A positive-weight adversary argument supplies the matching lower bound.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "fbs-RC", "kind": "theta_bound", "A": "fbs", "B": "RC", "exponent": 1, "statement": "fbs(f) = Theta(RC(f))", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "source": "KT13", "locator": "Section 2.1, Definition 2.2, and Section 1 related work", "proof_summary": "A fractional cover gives a nonadaptive verifier by sampling coordinates proportional to their weights; a verifier induces cover weights via its coordinate-query probabilities.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "proof_route_checked_not_independent_full_proof" }, { "id": "C-R2", "kind": "upper_bound", "A": "C", "B": "R", "exponent": 2, "statement": "C(f) = O(R(f)^2)", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "derived", "source": "definitions", "locator": "BBCMW01 Lemma 5.2 plus coupling", "proof_summary": "C<=bs^2 and bs<=3R imply C<=9R^2. This verifies the exponent without relying on the later paper’s attribution.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "depends_on": [ "C-bs2", "bs-R" ] }, { "id": "C-Q4", "kind": "upper_bound", "A": "C", "B": "Q", "exponent": 4, "statement": "C(f) = O(Q(f)^4)", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "derived", "source": "BK26", "locator": "Introduction, paragraph following Theorem 1", "proof_summary": "Use C <= D and the quartic simulation.", "last_checked": "2026-10-17", "depends_on": [ "C-D", "D-Q4" ], "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "Q-C2-sep", "kind": "separation", "A": "Q", "B": "C", "exponent": 2, "statement": "There is a total family with Q(f) = Omega_tilde(C(f)^2)", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "polylog", "status": "source_checked", "source": "ABK16", "locator": "Theorem 3", "proof_summary": "BKK construction and cheat sheets.", "last_checked": "2026-10-17", "witness": "BKK/cheat-sheet construction", "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "Q-adeg4-sep", "kind": "separation", "A": "Q", "B": "adeg", "exponent": 4, "statement": "There is a total family with Q(f) >= adeg(f)^(4-o(1))", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "subpolynomial", "status": "source_checked", "source": "ABK16", "locator": "Theorem 2", "proof_summary": "Recursive BKK with cheat sheets.", "last_checked": "2026-10-17", "witness": "RecBKK cheat-sheet family", "endpoint_attained": false, "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "D-Q4-sep", "kind": "separation", "A": "D", "B": "Q", "exponent": 4, "statement": "There is a total family with D(f) = Omega_tilde(Q(f)^4)", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "polylog", "status": "primary_statement_and_proof_route_checked", "source": "ABBLS17", "locator": "Corollary 7, from Theorems 2, 4, 5", "proof_summary": "A unique all-one column carries pointers certifying zeros elsewhere. An adversary forces deterministic nm queries, while quantum search finds and verifies the candidate in O_tilde(sqrt(n)+sqrt(m)); take n=2m.", "last_checked": "2026-10-17", "witness": "pointer-function family", "best_known_certified": false, "proof_audit": "proof_route_checked_not_independent_full_proof" }, { "id": "C-R2-sep-BK", "kind": "separation", "A": "C", "B": "R", "exponent": 2, "statement": "C(f_n)=Omega(n^2), R(f_n)=O(n log n)", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "polylog", "status": "source_checked", "source": "BK26", "locator": "Theorem 1; Lemmas 3 and 4", "proof_summary": "Sample race winners, eliminate candidates, verify; untouched cards force large 0-certificates.", "last_checked": "2026-10-17", "witness": "championship function", "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "C-Q4-sep-BK", "kind": "separation", "A": "C", "B": "Q", "exponent": 4, "statement": "C(f_n)=Omega(n^2), Q(f_n)=O_tilde(sqrt(n))", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "polylog", "status": "source_checked", "source": "BK26", "locator": "Theorem 1; Lemma 5", "proof_summary": "Virtual candidates, quantum tournament sink finding, and Grover verification.", "last_checked": "2026-10-17", "witness": "championship function", "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "C-Q4-sep-AIK", "kind": "separation", "A": "C", "B": "Q", "exponent": 4, "statement": "C(F_n)>=n(ceil(n/2)+1), Q(F_n)=O(sqrt(n) log(n)^3)", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "explicit_log", "status": "source_checked", "source": "AIK26", "locator": "v3 Theorem 2.1", "proof_summary": "DNF, position lists, cheat-sheet totalization, and source finding.", "last_checked": "2026-10-17", "witness": "F_n", "parameters": { "input_bits": "Theta(n^3 log(n)^2)", "n": "construction parameter, not input length" }, "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "C-R1-2-sep-AIK", "kind": "separation", "A": "C", "B": "R1", "exponent": 2, "statement": "C(f_n)=Theta(n^2), R_1(f_n)=O(n)", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "source_checked", "source": "AIK26", "locator": "v3 Theorem 2.2", "proof_summary": "Unambiguous DNF with randomized elimination.", "last_checked": "2026-10-17", "witness": "underlying DNF", "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "C-R2-sep-AIK", "kind": "separation", "A": "C", "B": "R", "exponent": 2, "statement": "C(f_n)=Theta(n^2), R(f_n)=O(n)", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "derived", "source": "AIK26", "locator": "v3 Theorem 2.2", "proof_summary": "One-sided algorithm is also two-sided.", "last_checked": "2026-10-17", "witness": "underlying DNF", "depends_on": [ "C-R1-2-sep-AIK" ], "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "sym-D-Q2", "kind": "upper_bound", "A": "D", "B": "Q", "exponent": 2, "statement": "D(f) = O(Q(f)^2) for symmetric total Boolean f", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "source_checked", "source": "BBCMW01", "locator": "Symmetric-function characterization", "proof_summary": "Symmetrization plus polynomial lower bounds.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "sym-deg-rdeg", "kind": "upper_bound", "A": "deg", "B": "rdeg", "exponent": 1, "statement": "deg(f)+1 <= 3 rdeg(f) for nonconstant symmetric total f", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [ "f is nonconstant" ] }, "loss": "none", "status": "source_checked", "source": "IJKKSWW23", "locator": "2310.08004v2 Proposition 6", "proof_summary": "Symmetrize the rational representation and count roots.", "last_checked": "2026-10-17", "constant": { "value": 3, "normalization": "v2 main-text proposition; nonconstant f" }, "best_known_certified": false, "proof_audit": "not_independently_audited", "notes": "Abstract search metadata says deg/2; v2 Proposition 6 states (deg+1)/3. Use the versioned main text, pending latest-version reconciliation." }, { "id": "mono-deg-rdeg", "kind": "upper_bound", "A": "deg", "B": "rdeg", "exponent": 2, "statement": "deg(f) <= rdeg(f)^2 for monotone total f", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "monotone" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "source": "IJKKSWW23", "locator": "v2 Section 3.2, Corollary 14", "proof_summary": "For unate functions, rational degree equals sensitivity: monotone local changes force a nondeterministic-degree lower bound, and certificates supply the converse. Combine with Huang degree<=sensitivity squared.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "proof_route_checked_not_independent_full_proof" }, { "id": "partial-R-Q", "kind": "no_polynomial_bound", "A": "R", "B": "Q", "exponent": null, "statement": "For each integer k>=1: Q(f_n)<=k, R(f_n)=Omega_tilde(n^(1-1/(2k)))", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "polylog", "status": "source_checked", "source": "SSW20", "locator": "Abstract / main application", "proof_summary": "Decision-tree Fourier-growth bounds applied to forrelation.", "last_checked": "2026-10-17", "parameters": { "n": "input bits", "k": "fixed positive integer" }, "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "partial-Q-rdeg", "kind": "no_polynomial_bound", "A": "Q", "B": "rdeg", "exponent": null, "statement": "Imbalance_n: rdeg=O(1), adeg=Omega(n), hence Q=Omega(n)", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "source": "IJKKSWW23", "locator": "v2 Section 6.2 (Imbalance)", "proof_summary": "The promise makes an exact rational representation low-degree, while symmetrization and the univariate approximation bound force linear approximate degree, and hence linear quantum complexity.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "proof_route_checked_not_independent_full_proof", "witness": "Imbalance_n" }, { "id": "search-Q-adeg", "kind": "no_polynomial_bound", "A": "Q", "B": "rel_adeg", "exponent": null, "statement": "There are total query-TFNP relations on poly(n) input bits with rel_adeg and rel_conical both O(polylog n), while Q=tildeOmega(sqrt(n)).", "scope": { "object_kind": "search_relation", "totality": "total", "classes": [ "query_TFNP" ], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "see_source", "status": "primary_statement_and_proof_route_checked", "source": "BK24", "locator": "Corollary 40, from Theorem 39, Theorem 31 and Corollary 32", "proof_summary": "Construct repeated AND-OR search with a small family of XOR shifts: conical polynomials describe short witnesses, but a quantum direct-product lower bound and union bound over shifts give Q=Omega_tilde(sqrt(n)).", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "proof_route_checked_not_independent_full_proof", "parameters": { "input_bits": "poly(n)", "Q": "Omega_tilde(sqrt(n))", "rel_adeg": "polylog(n)" }, "notes": "Here n is the construction parameter, not necessarily the input length. These are bounded polynomial systems and two-sided bounded conical systems." }, { "id": "search-Q-conical", "kind": "no_polynomial_bound", "A": "Q", "B": "rel_conical", "exponent": null, "statement": "There are total query-TFNP relations on poly(n) input bits with rel_adeg and rel_conical both O(polylog n), while Q=tildeOmega(sqrt(n)).", "scope": { "object_kind": "search_relation", "totality": "total", "classes": [ "query_TFNP" ], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "see_source", "status": "primary_statement_and_proof_route_checked", "source": "BK24", "locator": "Corollary 40, from Theorem 39, Theorem 31 and Corollary 32", "proof_summary": "The low-degree witness polynomials remain conical juntas under the total-search transformation; the shifted repeated-search quantum lower bound is unchanged.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "proof_route_checked_not_independent_full_proof", "parameters": { "input_bits": "poly(n)", "Q": "Omega_tilde(sqrt(n))", "rel_conical": "polylog(n)" }, "notes": "Here n is the construction parameter, not necessarily the input length. These are bounded polynomial systems and two-sided bounded conical systems." }, { "id": "search-D-psQ", "kind": "upper_bound", "A": "D", "B": "psQ", "exponent": 5, "statement": "D(S)=O(psQ(S)^5 log N) for total search S", "scope": { "object_kind": "search_relation", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "explicit_log", "status": "primary_statement_and_proof_route_checked", "source": "AGL26", "locator": "Preprint Theorem 1.4; ICALP Theorem 4", "proof_summary": "Use pseudo-determinism to bound how many canonical outputs can arise; binary search for the canonical output reduces to total Boolean predicates, each simulated deterministically using the quartic D/Q relation.", "last_checked": "2026-10-17", "log_factor": "log N", "parameters": { "N": "input bits" }, "best_known_certified": false, "proof_audit": "proof_route_checked_not_independent_full_proof" }, { "id": "bs-R", "kind": "upper_bound", "A": "bs", "B": "R", "exponent": 1, "statement": "bs(f) <= 3 R(f)", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "definition_proof", "source": "definitions", "locator": "Coupling argument", "proof_summary": "For each disjoint sensitive block, correctness on the base input and its flip forces query probability at least 1/3; sum over blocks.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "constant": { "value": 3, "normalization": "two-sided error at most 1/3; fixed query cap" } }, { "id": "bs-adeg2", "kind": "upper_bound", "A": "bs", "B": "adeg", "exponent": 2, "statement": "bs(f) <= 6 adeg(f)^2", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "source": "NS94", "locator": "Lemma 3.8, combined with Lemma 3.5", "proof_summary": "Map one new variable to each disjoint sensitive block, fixing the rest. Symmetrize the resulting polynomial, then use a univariate derivative inequality to force quadratic degree.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "proof_route_checked_not_independent_full_proof", "constant": { "value": 6, "normalization": "0/1 target, absolute error 1/3" } }, { "id": "bs-deg2", "kind": "upper_bound", "A": "bs", "B": "deg", "exponent": 2, "statement": "bs(f) <= 2 deg(f)^2", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "secondary_checked", "source": "Mid04", "locator": "Theorem 2, attributed to Nisan–Szegedy", "proof_summary": "Degree lower bound from disjoint sensitive blocks.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "original_source": "NS94", "constant": { "value": 2, "normalization": "real exact degree" } }, { "id": "bs-s4", "kind": "upper_bound", "A": "bs", "B": "s", "exponent": 4, "statement": "bs(f) <= 2 s(f)^4", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "derived", "source": "definitions", "locator": "Composition of inequalities", "proof_summary": "Substitute deg <= s^2 into bs <= 2 deg^2.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "depends_on": [ "bs-deg2", "deg-s2" ] }, { "id": "D-R3", "kind": "upper_bound", "A": "D", "B": "R", "exponent": 3, "statement": "D(f) = O(R(f)^3)", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "derived", "source": "definitions", "locator": "Composition of inequalities", "proof_summary": "Use D <= C bs, C=O(R^2), and bs=O(R).", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "depends_on": [ "D-Cbs", "C-R2", "bs-R" ], "proof_sources": [ "Nis89" ], "original_locator": "Theorem 4, p.334", "original_normalization_note": "Original constant 8 uses expected R_2 at error 1/4. Exponent transfers to capped error-1/3 R by constant-error amplification; the constant 8 does not transfer unchanged." }, { "id": "D-adeg4", "kind": "upper_bound", "A": "D", "B": "adeg", "exponent": 4, "statement": "D(f) = O(adeg(f)^4)", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "derived", "source": "definitions", "locator": "Composition of inequalities", "proof_summary": "Use D <= deg bs and the two quadratic degree bounds.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "depends_on": [ "D-deg-bs", "deg-adeg2", "bs-adeg2" ] }, { "id": "Q-adeg4-upper", "kind": "upper_bound", "A": "Q", "B": "adeg", "exponent": 4, "statement": "Q(f) = O(adeg(f)^4)", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "derived", "source": "definitions", "locator": "Classical simulation", "proof_summary": "Q <= D <= O(adeg^4).", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "depends_on": [ "D-adeg4", "Q-QE", "QE-D" ] }, { "id": "R-Q4", "kind": "upper_bound", "A": "R", "B": "Q", "exponent": 4, "statement": "R(f) = O(Q(f)^4)", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "derived", "source": "definitions", "locator": "Classical simulation", "proof_summary": "R <= D <= O(Q^4).", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "depends_on": [ "R-D", "D-Q4" ] }, { "id": "search-D-psR3", "kind": "upper_bound", "A": "D", "B": "psR", "exponent": 3, "statement": "For every total search relation S, D(S)<=54 psR(S)^3 under the CDM23 error-1/3 convention.", "scope": { "object_kind": "search_relation", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "source": "CDM23", "locator": "CDM23 Theorem 3.2; full report Theorem 3.2(2) and proof", "proof_summary": "Choose the canonical multi-output function computed by the pseudo-deterministic algorithm, then apply the cubic deterministic/randomized relation for multi-output functions.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "proof_route_checked_not_independent_full_proof", "proof_sources": [ "CDM23full", "CDM25" ], "notes": "The complete proof of the multi-output bound is in the ECCC report; the proceedings omits some proofs. This cubic 2023 bound predates AGL26’s discussion of an older quartic classical bound.", "constant_factor": "54 (derived from the displayed inequalities; not asserted optimal)", "journal_locator": "CDM25 Theorem 3.2" }, { "id": "D-Rank-no-bound", "kind": "no_polynomial_bound", "A": "D", "B": "DTRank", "exponent": null, "statement": "OR_n has Rank=1 and D=n, so no bound on D in terms of Rank alone", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "source": "DM23", "locator": "Section 1; Section 5 depth/rank discussion", "proof_summary": "A chain computes OR with rank one, while the zero input forces n queries. Thus even a general function of rank alone cannot upper-bound deterministic depth.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "proof_route_checked_not_independent_full_proof", "witness": "OR_n" }, { "id": "partial-rdeg-Q", "kind": "no_polynomial_bound", "A": "rdeg", "B": "Q", "exponent": null, "statement": "MajOrNone_n has Q=O(1) and rdeg=Omega(n)", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "source_checked", "source": "IJKKSWW23", "locator": "v2 Section 6.1", "proof_summary": "Hamming-slice promise and rational-polynomial root counting.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "witness": "MajOrNone_n" }, { "id": "D-C2", "kind": "upper_bound", "A": "D", "B": "C", "exponent": 2, "statement": "D(f) <= C(f)^2", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "definition_proof", "source": "definitions", "locator": "Repository derivation / explicit family", "proof_summary": "Use D <= C_0 C_1 and C_b <= C.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "depends_on": [ "C-C0C1" ], "proof_sources": [ "Nis89" ], "original_locator": "Lemma 2.5, p.332" }, { "id": "Q-C2-upper", "kind": "upper_bound", "A": "Q", "B": "C", "exponent": 2, "statement": "Q(f) <= C(f)^2", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "definition_proof", "source": "definitions", "locator": "Repository derivation / explicit family", "proof_summary": "Q <= Q_E <= D <= C^2.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "depends_on": [ "Q-QE", "QE-D", "D-C2" ] }, { "id": "D-bs3", "kind": "upper_bound", "A": "D", "B": "bs", "exponent": 3, "statement": "D(f) <= bs(f)^3", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "definition_proof", "source": "definitions", "locator": "Repository derivation / explicit family", "proof_summary": "Use D <= C bs and C <= bs^2.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "depends_on": [ "D-Cbs", "C-bs2" ] }, { "id": "OR-D-Q", "kind": "separation", "A": "D", "B": "Q", "exponent": 2, "statement": "OR_n witnesses D(OR_n)=Theta(Q(OR_n)^2)", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric", "monotone" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "source_checked", "source": "BBCMW01", "locator": "AND/OR query characterization; approximate-degree lower bound via polynomial method", "proof_summary": "OR needs n classical queries or exact degree; Grover gives O(sqrt(n)), and polynomial lower bounds give Omega(sqrt(n)).", "witness": "OR_n", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "OR-R-Q", "kind": "separation", "A": "R", "B": "Q", "exponent": 2, "statement": "OR_n witnesses R(OR_n)=Theta(Q(OR_n)^2)", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric", "monotone" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "source_checked", "source": "BBCMW01", "locator": "AND/OR query characterization; approximate-degree lower bound via polynomial method", "proof_summary": "OR needs n classical queries or exact degree; Grover gives O(sqrt(n)), and polynomial lower bounds give Omega(sqrt(n)).", "witness": "OR_n", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "OR-deg-Q", "kind": "separation", "A": "deg", "B": "Q", "exponent": 2, "statement": "OR_n witnesses deg(OR_n)=Theta(Q(OR_n)^2)", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric", "monotone" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "source_checked", "source": "BBCMW01", "locator": "AND/OR query characterization; approximate-degree lower bound via polynomial method", "proof_summary": "OR needs n classical queries or exact degree; Grover gives O(sqrt(n)), and polynomial lower bounds give Omega(sqrt(n)).", "witness": "OR_n", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "OR-deg-adeg", "kind": "separation", "A": "deg", "B": "adeg", "exponent": 2, "statement": "OR_n witnesses deg(OR_n)=Theta(adeg(OR_n)^2)", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric", "monotone" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "source_checked", "source": "BBCMW01", "locator": "AND/OR query characterization; approximate-degree lower bound via polynomial method", "proof_summary": "OR needs n classical queries or exact degree; Grover gives O(sqrt(n)), and polynomial lower bounds give Omega(sqrt(n)).", "witness": "OR_n", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "bs-deg2-Tal", "A": "bs", "B": "deg", "exponent": 2, "statement": "bs(f) <= deg(f)^2", "source": "Tal13", "locator": "Theorem 4; Section 6", "proof_summary": "Apply a fractional-block-sensitive LP argument and polynomial composition to remove the factor two in the Nisan–Szegedy inequality.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "constant": { "value": 1, "normalization": "real exact degree" }, "historical": true }, { "id": "bs-deg2-Pro", "A": "bs", "B": "deg", "exponent": 2, "statement": "bs(f) <= deg(f)^2/(sqrt(10)-2)", "source": "Pro21", "locator": "Theorem 1", "proof_summary": "A sharper analysis of the symmetrized univariate polynomial improves the derivative estimate.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "constant": { "value": "1/(sqrt(10)-2)", "normalization": "no additive term" } }, { "id": "bs-adeg2-Pro", "A": "bs", "B": "adeg", "exponent": 2, "statement": "bs(f) <= sqrt(101/6) adeg(f)^2", "source": "Pro21", "locator": "Theorem 2", "proof_summary": "Analyze the symmetrized approximating polynomial more precisely than the original Markov bound.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "constant": { "value": "sqrt(101/6)", "normalization": "unrestricted absolute-error 1/3 approximation" } }, { "id": "bs-s4-Pro", "A": "bs", "B": "s", "exponent": 4, "statement": "bs(f) <= s(f)^4/(sqrt(10)-2)", "source": "Pro21", "locator": "Corollary following Theorem 1", "proof_summary": "Substitute Huang deg<=s^2 into the improved block/degree inequality.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "constant": { "value": "1/(sqrt(10)-2)", "normalization": "no additive term" } }, { "id": "fbs-deg2", "A": "fbs", "B": "deg", "exponent": 2, "statement": "fbs(f) <= deg(f)^2", "source": "KT13", "locator": "Corollary 3.2(1)", "proof_summary": "The composition-limit transfer lemma upgrades a block-sensitivity/degree bound to fractional block sensitivity with the same constant.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "fbs-adeg2", "A": "fbs", "B": "adeg", "exponent": 2, "statement": "fbs(f) = O(adeg(f)^2)", "source": "KT13", "locator": "Theorem 3.4", "proof_summary": "Composition amplification transfers the quadratic integral block bound to the fractional relaxation.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "QC-adeg", "A": "QC", "B": "adeg", "exponent": 1, "statement": "QC(f) = O(adeg(f))", "source": "KT13", "locator": "Corollary 1.3", "proof_summary": "Combine fbs=Theta(RC), fbs=O(adeg^2), and QC=Theta(sqrt(RC)).", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "R0-R-s-RC", "A": "R0", "B": null, "exponent": null, "statement": "R_0(f)=O(R(f) s(f) log(2+RC(f)))", "source": "KT13", "locator": "Proof of Theorem 5.3", "proof_summary": "Count minimal blocks by fractional block sensitivity, then use Midrijanis block-number derandomization.", "kind": "multivariate_upper_bound", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "explicit_log", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "factors": [ { "measure": "R", "power": 1 }, { "measure": "s", "power": 1 } ], "log_factor": "log(2+RC)" }, { "id": "C-bs-s", "A": "C", "B": null, "exponent": null, "statement": "C(f) <= bs(f) s(f)", "source": "BBCMW01", "locator": "Lemma 5.2 proof, with singleton sensitivity retained", "proof_summary": "The union of a maximal disjoint family of minimal sensitive blocks is a certificate; each minimal block has size at most sensitivity at its flip.", "kind": "multivariate_upper_bound", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "factors": [ { "measure": "bs", "power": 1 }, { "measure": "s", "power": 1 } ], "original_source": "Nis91" }, { "id": "R-Q4-sep-2026", "A": "R", "B": "Q", "exponent": 4, "statement": "For each fixed k>=2: Q(F_{k,m})<=C_k sqrt(m) log(m)^{b_k}, R(F_{k,m})>=c_k m^{2-1/k}/log(m)^2. Thus every exponent below 4 is excluded.", "source": "OAI26", "locator": "Theorem 1.1 and Sections 2–5", "proof_summary": "Addressed cheat sheets with outward pointers give a unique winner. A quantum tournament handles unreliable off-promise competitors; a planted hidden-zero argument and adaptive direct product force randomized address cost.", "kind": "separation", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "arbitrarily_small_power_and_polylog", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "witness": "addressed outward-pointer family F_{k,m}", "parameters": { "k": "fixed integer >=2", "m": "sufficiently large power of four", "constants": "may depend on k" }, "publication_status": "repository_preprint", "endpoint_attained": false }, { "id": "bs-lambda-sep-2026", "A": "bs", "B": "lambda", "exponent": "2.127", "statement": "A total family satisfies bs(f_t)=Omega(lambda(f_t)^2.127)", "source": "Mei26", "locator": "Theorem 6.1 and its composition corollary", "proof_summary": "Take a union of subcubes indexed by a doubly regular tournament, use the Lovasz local lemma to control spectral sensitivity, and amplify the finite example by composition.", "kind": "separation", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "witness": "tournament-indexed subcube union", "publication_status": "arxiv_preprint", "notes": "The certified finite example has 2017584 variables, bs>=14011, lambda<=89.0162. The numerical exponent near 2.20 elsewhere in the paper is not recorded as a proved lower bound." }, { "id": "QD-Q", "A": "QD", "B": "Q", "exponent": 1, "statement": "QD(f) <= Q(f)", "source": "BK19", "locator": "Proposition 6", "proof_summary": "Retain the output state of a bounded-error algorithm; opposite outputs have separated measurement distributions.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "QD-QSZK", "A": "QD", "B": "QSZK", "exponent": 1, "statement": "QD(f) <= QSZK(f)", "source": "BK19", "locator": "Theorem 2", "proof_summary": "Run both state-preparation procedures and use the near/far trace-distance promise to separate opposite inputs.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "QSZK-Q", "A": "QSZK", "B": "Q", "exponent": 1, "statement": "QSZK(f) <= Q(f)", "source": "BK19", "locator": "Proposition 15", "proof_summary": "Compare the output-bit state of the quantum algorithm with the constant zero state.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "QC-QD", "A": "QC", "B": "QD", "exponent": 1, "statement": "QC(f) = O(QD(f))", "source": "BK19", "locator": "Proposition 8", "proof_summary": "Fix an input and distinguish its output state from all opposite-output states.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "AdvPlus-QD", "A": "AdvPlus", "B": "QD", "exponent": 1, "statement": "AdvPlus(f) = O(QD(f))", "source": "BK19", "locator": "Section 3.2, equation (7)", "proof_summary": "The adversary progress proof only needs separated final states, not a common output measurement.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "Q0-QD5-history", "A": "Q0", "B": "QD", "exponent": 5, "statement": "Q_0(f)=O(QD(f)^5 log(2+QD(f)))", "source": "BK19", "locator": "Theorem 1; Section 4", "proof_summary": "Hybrid query probabilities yield fractional certificates; repeatedly sample them to find a verified certificate.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "explicit_log", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "log_factor": "log(2+QD)", "historical": false, "notes": "The later quartic D/Q bound supersedes the corresponding Q_0/Q consequence, not automatically the Q_0/QD statement." }, { "id": "fbs-QS2", "A": "fbs", "B": "QS", "exponent": 2, "statement": "fbs(f)=O(QS_weak(f)^2)", "source": "CMP24", "locator": "Theorem 5", "proof_summary": "Apply a hybrid argument against a fractional packing of differing blocks.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "QSstrong-Q", "A": "QSstrong", "B": "Q", "exponent": 1, "statement": "QS_str(f)=O(min(Q(f),sqrt(n)))", "source": "CMP24", "locator": "Main results / strong-input distinguishing algorithm", "proof_summary": "Run a quantum algorithm with access to both underlying inputs, or search directly for a differing coordinate.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "QSstrongIndex-Q", "A": "QSstrongIndex", "B": "Q", "exponent": "3/2", "statement": "QS_str^ind(f)=O(min(Q(f)^(3/2),sqrt(n)))", "source": "CMP24", "locator": "Main results / index-output algorithm", "proof_summary": "Convert distinguishing progress into finding a differing index via search.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "CGns-CGent", "A": "CGns", "B": "CGent", "exponent": 1, "statement": "CGns(f)=O(CGent(f))", "source": "CG22", "locator": "Proposition 2.7", "proof_summary": "Entangled strategies are nonsignaling.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "CGent-CGpub", "A": "CGent", "B": "CGpub", "exponent": 1, "statement": "CGent(f)=O(CGpub(f))", "source": "CG22", "locator": "Proposition 2.7", "proof_summary": "Shared entanglement can realize shared random bits.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "CGpub-CGpriv", "A": "CGpub", "B": "CGpriv", "exponent": 1, "statement": "CGpub(f)=O(CGpriv(f))", "source": "CG22", "locator": "Proposition 2.7", "proof_summary": "A public source can supply independent random seeds.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "CAdv-CGns", "A": "CAdv", "B": "CGns", "exponent": 1, "statement": "CAdv(f)=O(CGns(f))", "source": "CG22", "locator": "Theorem 6.2", "proof_summary": "Nonsignaling output marginals supply a feasible classical-adversary solution.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "CGpub-CAdv", "A": "CGpub", "B": "CAdv", "exponent": 1, "statement": "CGpub(f)=O(CAdv(f))", "source": "CG22", "locator": "Theorem 7.1", "proof_summary": "Correlated sampling of input-index distributions attains the minimax adversary overlap.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "CGpub-R", "A": "CGpub", "B": "R", "exponent": 1, "statement": "CGpub(f)=O(R(f))", "source": "CG22", "locator": "Theorem 5.7", "proof_summary": "Use shared coins to simulate a randomized tree, then sample a common query position.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "CGpub-EC", "A": "CGpub", "B": "EC", "exponent": 1, "statement": "CGpub(f)=O(EC(f))", "source": "CG22", "locator": "Theorem 5.5", "proof_summary": "Sample weighted certificates with public hashing to coordinate an intersecting index.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "CGpub-C", "A": "CGpub", "B": "C", "exponent": 1, "statement": "CGpub(f)=O(C(f))", "source": "CG22", "locator": "Theorem 5.4", "proof_summary": "Public hashing coordinates the two short opposite certificates.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "R0-CGpriv", "A": "R0", "B": "CGpriv", "exponent": 1, "statement": "R_0(f) <= CGpriv(f)", "source": "CG22", "locator": "Theorem 5.10(2)", "proof_summary": "Private strategies induce randomized certificate queries; verify their intersection progress to obtain a zero-error algorithm.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "CGpriv-EC2", "A": "CGpriv", "B": "EC", "exponent": 2, "statement": "CGpriv(f)=O(EC(f)^2)", "source": "CG22", "locator": "Theorem 5.10(2)", "proof_summary": "Normalize expectational certificate weights to independent output distributions; their winning overlap is inverse-quadratic.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "CGpriv-C0C1", "A": "CGpriv", "B": null, "exponent": null, "statement": "CGpriv(f) <= C_0(f) C_1(f)", "source": "CG22", "locator": "Theorem 5.10(4)", "proof_summary": "Each player samples uniformly from a shortest certificate; opposite certificates share a differing index.", "kind": "multivariate_upper_bound", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "factors": [ { "measure": "C0", "power": 1 }, { "measure": "C1", "power": 1 } ], "notes": "Total-function scope entered conservatively. Partial-function certificate conventions in CG22 Section 4.4 need separate records before extending the scope." }, { "id": "AdvPlus-CGpriv", "A": "AdvPlus", "B": "CGpriv", "exponent": "1/2", "statement": "AdvPlus(f)^2 <= CGpriv(f)", "source": "CG22", "locator": "Theorem 5.10(1)", "proof_summary": "Use the private strategy distributions in the minimax quantum adversary bound and compare squared sum of square roots with sum of products.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "CGpubSingle-s", "A": "CGpubSingle", "B": "s", "exponent": 1, "statement": "CGpub_{[1]}(f)=Theta(s(f))", "source": "CG22", "locator": "Theorem 8.4", "proof_summary": "A public strategy coordinates a sensitive neighbor index; the degree of the sensitivity graph bounds the achievable success probability.", "kind": "theta_bound", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "CGprivSingle-lambda", "A": "CGprivSingle", "B": "lambda", "exponent": 2, "statement": "CGpriv_{[1]}(f)=lambda(f)^2", "source": "CG22", "locator": "Theorem 8.8", "proof_summary": "The private single-bit game is a matrix scaling/spectral optimization on the sensitivity graph.", "kind": "identity", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "Rexp-literaldeg", "A": "Rexp", "B": "literaldeg", "exponent": 1, "statement": "RE(f)=ldeg_+(f)", "source": "KLW14", "locator": "Theorem 9", "proof_summary": "Tree leaves give nonnegative literal monomials; conversely sample a conjunction and scale its nonnegative contribution.", "kind": "identity", "scope": { "object_kind": "real_valued_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "nonnegative_reals", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "Qexp-sosdeg", "A": "Qexp", "B": "sosdeg", "exponent": 1, "statement": "QE_in_expectation(f)=deg_sos(f)", "source": "KLW14", "locator": "Theorem 12", "proof_summary": "Quantum output probabilities are sums of squared low-degree amplitudes; a sum-of-squares representation can be implemented as a quantum expectation algorithm.", "kind": "identity", "scope": { "object_kind": "real_valued_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "nonnegative_reals", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "Rexp-Qexp-no-bound", "A": "Rexp", "B": "Qexp", "exponent": null, "statement": "f(x)=(sum_i x_i-1)^2 has RE(f)=n and QE_in_expectation(f)=1", "source": "KLW14", "locator": "Section 5", "proof_summary": "A single square has SOS degree one, while the zero set forces degree n in every nonnegative literal representation.", "kind": "no_polynomial_bound", "scope": { "object_kind": "real_valued_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "witness": "squared Hamming-weight-minus-one" }, { "id": "hazardD-R2", "A": "hazardD", "B": "hazardR", "exponent": 2, "statement": "D_u(f)=O(R_u(f)^2)", "source": "BKMNSS25", "locator": "Theorem 4.4", "proof_summary": "Prime implicants/implicates make u-sensitivity, u-block sensitivity and u-certificates linearly equivalent; certificate rounds give quadratic deterministic simulation.", "kind": "upper_bound", "scope": { "object_kind": "hazard_free_extension", "totality": "total", "classes": [], "input_alphabet": "ternary_0_u_1", "output_alphabet": "ternary_0_u_1", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "hazardD-Q4", "A": "hazardD", "B": "hazardQ", "exponent": 4, "statement": "D_u(f)=O(Q_u(f)^4)", "source": "BKMNSS25", "locator": "Theorem 4.4", "proof_summary": "Use the same certificate-round algorithm and the quantum square-root block-sensitivity lower bound.", "kind": "upper_bound", "scope": { "object_kind": "hazard_free_extension", "totality": "total", "classes": [], "input_alphabet": "ternary_0_u_1", "output_alphabet": "ternary_0_u_1", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "hazardC-hazardS", "A": "hazardC", "B": "hazardS", "exponent": 1, "statement": "cc_u(f) <= 2 s_u(f)-1 for nonconstant f", "source": "BKMNSS25", "locator": "Theorem 3.1", "proof_summary": "A prime implicant and prime implicate certify an unknown output; they intersect, so at most k_1+k_2-1 positions suffice.", "kind": "upper_bound", "scope": { "object_kind": "hazard_free_extension", "totality": "total", "classes": [], "input_alphabet": "ternary_0_u_1", "output_alphabet": "ternary_0_u_1", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [ "f is nonconstant" ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "constant": { "value": 2, "additive": -1, "normalization": "ternary hazard-free extension" } }, { "id": "brick-s-exp", "A": "bricknumber_l", "B": "s", "exponent": null, "statement": "N_br^ell(f) <= (6 s(f))^ell", "source": "CG26", "locator": "Theorem 9", "proof_summary": "A weighted double-count over coordinate flips bounds bricks by induction on block size.", "kind": "parameterized_upper_bound", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "parameters": { "ell": "1<=ell<=n" } }, { "id": "block-s-exp", "A": "blocknumber_l", "B": "s", "exponent": null, "statement": "N_bs^ell(f) <= (6 s(f))^ell", "source": "CG26", "locator": "Theorem 5", "proof_summary": "Every minimal sensitive block is a brick, so apply the brick-count theorem.", "kind": "parameterized_upper_bound", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "parameters": { "ell": "1<=ell<=n" } }, { "id": "block-brick", "A": "blocknumber", "B": "bricknumber", "exponent": 1, "statement": "N_bs(f) <= N_br(f)", "source": "CG26", "locator": "Definitions 3,7", "proof_summary": "Every coordinate in a minimal sensitive block flips the value back at the block-flipped input.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "fbs-Qdeph", "A": "fbs", "B": "Qdephasing", "exponent": 1, "statement": "Q_deph_{p,epsilon}(f)=Omega_epsilon(p fbs(f))", "source": "ARS26", "locator": "Theorem 1.3; Theorem 5.1", "proof_summary": "Track a fractional-block-sensitive progress measure and bound its expected growth under independent index dephasing.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "parameters": { "p": "0= 3 bs_0(f)/(2 s_0(f))", "source": "AP14", "locator": "Main theorem", "proof_summary": "Analyze pairwise intersections of minimal sensitive blocks and charge the conflicts to 0-sensitivity.", "kind": "multivariate_lower_bound", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "factors": [ { "measure": "bs0", "power": 1 }, { "measure": "s0", "power": -1 } ], "constant": { "value": "3/2", "normalization": "nonconstant total f" } }, { "id": "sym-C-s-constant", "A": "C", "B": "s", "exponent": 1, "statement": "C(f) <= 2 s(f); a family has C(f)=2s(f)-4", "source": "MNP24", "locator": "Theorems 4(1),8; Lemma 7", "proof_summary": "Certify a Hamming-weight interval by its boundary ones and zeros; the boundary sensitivities pay for both ends.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "constant": { "value": 2, "normalization": "total symmetric f" } }, { "id": "sym-bs-s-constant", "A": "bs", "B": "s", "exponent": 1, "statement": "bs(f) <= (3/2) s(f), with asymptotically matching families", "source": "MNP24", "locator": "Theorems 4(2),9", "proof_summary": "Compute local block sensitivity from the two endpoints of a constant Hamming interval, then optimize their ratio to sensitivity.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "constant": { "value": "3/2", "normalization": "total symmetric f" } }, { "id": "sym-Q-lambda", "A": "Q", "B": "lambda", "exponent": 1, "statement": "Q(f)=Theta(lambda(f))", "source": "MNP24", "locator": "Theorems 1–2; Section 1.1", "proof_summary": "Analyze adjacent Hamming layers for spectral sensitivity and construct explicit minimax weights; compare with the known quantum and approximation characterization.", "kind": "theta_bound", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "sym-lambda-adeg", "A": "lambda", "B": "adeg", "exponent": 1, "statement": "lambda(f)=Theta(adeg(f))", "source": "MNP24", "locator": "Theorems 1–2; Section 1.1", "proof_summary": "Analyze adjacent Hamming layers for spectral sensitivity and construct explicit minimax weights; compare with the known quantum and approximation characterization.", "kind": "theta_bound", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "sym-Adv-lambda", "A": "AdvPlus", "B": "lambda", "exponent": 1, "statement": "AdvPlus(f)=Theta(lambda(f))", "source": "MNP24", "locator": "Theorems 1–2; Section 1.1", "proof_summary": "Analyze adjacent Hamming layers for spectral sensitivity and construct explicit minimax weights; compare with the known quantum and approximation characterization.", "kind": "theta_bound", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "sym-CGpriv-Adv2", "A": "CGpriv", "B": "AdvPlus", "exponent": 2, "statement": "CGpriv(f)=Theta(AdvPlus(f)^2)", "source": "MNP24", "locator": "Theorem 1", "proof_summary": "The explicit minimax adversary weights also solve the square-root private certificate-game optimization.", "kind": "theta_bound", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "graph_property-D-bs2", "A": "D", "B": "bs", "exponent": 2, "statement": "D(f)=O(bs(f)^2)", "source": "CPS22", "locator": "Theorem 1.1(1)", "proof_summary": "For graph properties, bs is Omega(number of vertices), while D is at most the number of edges; for zebra functions, monochromatic stripe structure bounds certificate-query progress.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "graph_property" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "zebra-D-bs2", "A": "D", "B": "bs", "exponent": 2, "statement": "D(f)=O(bs(f)^2)", "source": "CPS22", "locator": "Theorem 1.3(1)", "proof_summary": "For graph properties, bs is Omega(number of vertices), while D is at most the number of edges; for zebra functions, monochromatic stripe structure bounds certificate-query progress.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "zebra" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "graph-D-deg2", "A": "D", "B": "deg", "exponent": 2, "statement": "D(f)=O(deg(f)^2) for graph properties", "source": "CPS22", "locator": "Theorem 1.1(2)", "proof_summary": "Graph invariance forces real degree Omega(number of vertices); query every possible edge.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "graph_property" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "Dcert-C", "A": "D_cert", "B": "C", "exponent": 1, "statement": "D^cert(f)=C(f)", "source": "KLLPV26", "locator": "Theorem 3(1)", "proof_summary": "Certification restricts to one accepting input against all opposite-value inputs; deterministic certificates characterize the task, and the certification sandwich lemma gives the other equalities.", "kind": "identity", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "R0cert-C", "A": "R0_cert", "B": "C", "exponent": 1, "statement": "R0^cert(f)=C(f)", "source": "KLLPV26", "locator": "Theorem 3(1)", "proof_summary": "Certification restricts to one accepting input against all opposite-value inputs; deterministic certificates characterize the task, and the certification sandwich lemma gives the other equalities.", "kind": "identity", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "UCcert-C", "A": "UC_cert", "B": "C", "exponent": 1, "statement": "UC^cert(f)=C(f)", "source": "KLLPV26", "locator": "Theorem 3(1)", "proof_summary": "Certification restricts to one accepting input against all opposite-value inputs; deterministic certificates characterize the task, and the certification sandwich lemma gives the other equalities.", "kind": "identity", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "Rcert-RC", "A": "R_cert", "B": "RC", "exponent": 1, "statement": "R^cert(f)=Theta(RC(f))", "source": "KLLPV26", "locator": "Theorem 3(2)", "proof_summary": "Use stability and idempotence of certification to sandwich the transformed measure between randomized certification and fractional certificates.", "kind": "theta_bound", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "RScert-RC", "A": "RS_cert", "B": "RC", "exponent": 1, "statement": "RS^cert(f)=Theta(RC(f))", "source": "KLLPV26", "locator": "Theorem 3(2)", "proof_summary": "Use stability and idempotence of certification to sandwich the transformed measure between randomized certification and fractional certificates.", "kind": "theta_bound", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "ECcert-RC", "A": "EC_cert", "B": "RC", "exponent": 1, "statement": "EC^cert(f)=Theta(RC(f))", "source": "KLLPV26", "locator": "Theorem 3(2)", "proof_summary": "Use stability and idempotence of certification to sandwich the transformed measure between randomized certification and fractional certificates.", "kind": "theta_bound", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "CAdvcert-RC", "A": "CAdv_cert", "B": "RC", "exponent": 1, "statement": "CAdv^cert(f)=Theta(RC(f))", "source": "KLLPV26", "locator": "Theorem 3(2)", "proof_summary": "Use stability and idempotence of certification to sandwich the transformed measure between randomized certification and fractional certificates.", "kind": "theta_bound", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "Qcert-QC", "A": "Q_cert", "B": "QC", "exponent": 1, "statement": "Q^cert(f)=Theta(QC(f))", "source": "KLLPV26", "locator": "Theorems 4 and 6", "proof_summary": "Apply the certification sandwich lemma between quantum certification and a lower bound that already captures quantum certificates.", "kind": "theta_bound", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "QDcert-QC", "A": "QD_cert", "B": "QC", "exponent": 1, "statement": "QD^cert(f)=Theta(QC(f))", "source": "KLLPV26", "locator": "Theorems 4 and 6", "proof_summary": "Apply the certification sandwich lemma between quantum certification and a lower bound that already captures quantum certificates.", "kind": "theta_bound", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "AdvPluscert-QC", "A": "AdvPlus_cert", "B": "QC", "exponent": 1, "statement": "AdvPlus^cert(f)=Theta(QC(f))", "source": "KLLPV26", "locator": "Theorems 4 and 6", "proof_summary": "Apply the certification sandwich lemma between quantum certification and a lower bound that already captures quantum certificates.", "kind": "theta_bound", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "QScert-QC", "A": "QS_cert", "B": "QC", "exponent": 1, "statement": "QS^cert(f)=Theta(QC(f))", "source": "KLLPV26", "locator": "Theorems 4 and 6", "proof_summary": "Apply the certification sandwich lemma between quantum certification and a lower bound that already captures quantum certificates.", "kind": "theta_bound", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "cubeBoundedAdegcert-QC", "A": "cubeBoundedAdeg_cert", "B": "QC", "exponent": 1, "statement": "cubeBoundedAdeg^cert(f)=Theta(QC(f))", "source": "KLLPV26", "locator": "Theorems 4 and 6", "proof_summary": "Apply the certification sandwich lemma between quantum certification and a lower bound that already captures quantum certificates.", "kind": "theta_bound", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "QEcert-Q0cert", "A": "QE_cert", "B": "Q0_cert", "exponent": 1, "statement": "Q_E^cert(f)=Theta(Q_0^cert(f))", "source": "KLLPV26", "locator": "Theorem 5", "proof_summary": "Exact amplitude amplification on an input-fixed certification problem removes the inconclusive outcome with constant-factor query overhead.", "kind": "theta_bound", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "lambdaCert-s", "A": "lambda_cert", "B": "s", "exponent": "1/2", "statement": "lambda^cert(f)=sqrt(s(f))", "source": "KLLPV26", "locator": "Theorem 11(1); Theorem 42", "proof_summary": "Every input-fixed certification sensitivity graph is a star with s_x(f) leaves, whose adjacency norm is sqrt(s_x(f)).", "kind": "identity", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "osdeg-Q0", "A": "osdeg", "B": "Q0", "exponent": 1, "statement": "osdeg(f)=O_epsilon(Q_0(f))", "source": "KLLPV26", "locator": "Theorem 8", "proof_summary": "The no-error acceptance probabilities yield globally bounded one-sided approximants after adjusting inconclusive mass.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "rdegCert-rdeg", "A": "rdeg_cert", "B": "rdeg", "exponent": 1, "statement": "rdeg^cert(f)=rdeg(f)", "source": "KLLPV26", "locator": "Theorem 11(2); Lemma 50", "proof_summary": "Restrict nondeterministic representations to an input-fixed certification task and use rdeg=max(ndeg(f),ndeg(not f)).", "kind": "identity", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "bs-boundedDegCert2", "A": "bs", "B": "cubeBoundedDeg_cert", "exponent": 2, "statement": "bs(f)=O((globally-bounded-deg^cert(f))^2)", "source": "KLLPV26", "locator": "Theorem 12(1)", "proof_summary": "The globally bounded degree/block-sensitivity inequality survives the certification transform; block sensitivity is certification-invariant.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "C-boundedDegCert3", "A": "C", "B": "cubeBoundedDeg_cert", "exponent": 3, "statement": "C(f)=O((globally-bounded-deg^cert(f))^3)", "source": "KLLPV26", "locator": "Theorem 12(2)", "proof_summary": "The maxonomial algorithm still works on certification promises inherited from total f, then combine the certified block-degree bound.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "RC-C", "A": "RC", "B": "C", "exponent": 1, "statement": "RC(f)=O(C(f))", "source": "definitions", "locator": "Direct model simulation", "proof_summary": "Read a shortest fixed-input certificate.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "derived", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "RC-R", "A": "RC", "B": "R", "exponent": 1, "statement": "RC(f)=O(R(f))", "source": "definitions", "locator": "Direct model simulation", "proof_summary": "Restrict the decision algorithm to its input-fixed certification promise.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "derived", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "C-R0", "A": "C", "B": "R0", "exponent": 1, "statement": "C(f)=O(R0(f))", "source": "definitions", "locator": "Direct model simulation", "proof_summary": "Every terminating zero-error computation path is a certificate for its input.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "derived", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "R0-D", "A": "R0", "B": "D", "exponent": 1, "statement": "R0(f)=O(D(f))", "source": "definitions", "locator": "Direct model simulation", "proof_summary": "A deterministic tree is a zero-error randomized tree.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "derived", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "R-R0", "A": "R", "B": "R0", "exponent": 1, "statement": "R(f)=O(R0(f))", "source": "definitions", "locator": "Direct model simulation", "proof_summary": "Truncate an expected-time zero-error tree at a constant multiple of its worst-input expectation.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "derived", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "Q0-QE", "A": "Q0", "B": "QE", "exponent": 1, "statement": "Q0(f)=O(QE(f))", "source": "definitions", "locator": "Direct model simulation", "proof_summary": "An exact fixed-cap algorithm satisfies zero-error requirements.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "derived", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "Q0-R0", "A": "Q0", "B": "R0", "exponent": 1, "statement": "Q0(f)=O(R0(f))", "source": "definitions", "locator": "Direct model simulation", "proof_summary": "Truncate a randomized zero-error algorithm and simulate it quantumly.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "derived", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "C-adeg4-upper", "A": "C", "B": "adeg", "exponent": 4, "statement": "C(f)=O(adeg(f)^4)", "source": "definitions", "locator": "Composition of recorded claims: C-bs2, bs-adeg2-Pro", "proof_summary": "Combine C<=bs^2 with the source-checked quadratic upper bound on bs in terms of approximate degree.", "depends_on": [ "C-bs2", "bs-adeg2-Pro" ], "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "derived", "proof_audit": "explicit_dependency_argument", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "C-adeg4-sep", "A": "C", "B": "adeg", "exponent": 4, "statement": "For the AIK26 family F_n, C(F_n)>=n(ceil(n/2)+1), adeg(F_n)=O(sqrt(n) log(n)^3); this excludes every exponent below 4.", "source": "definitions", "locator": "Composition of recorded claims: C-Q4-sep-AIK, adeg-Q", "proof_summary": "Use adeg<=2Q on the original AIK26 witness. If C<=O(adeg^p) held universally for p<4, the parameterized bounds would imply n^2<=O(n^(p/2) log(n)^(3p)), a contradiction. No conversion of log(n) into log(adeg) is needed.", "depends_on": [ "C-Q4-sep-AIK", "adeg-Q" ], "kind": "separation", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "polylog", "status": "derived", "proof_audit": "explicit_dependency_argument", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "R0-Q4-upper", "A": "R0", "B": "Q", "exponent": 4, "statement": "R0(f)=O(Q(f)^4)", "source": "definitions", "locator": "Composition of recorded claims: R0-D, D-Q4", "proof_summary": "A deterministic algorithm is an always-correct randomized algorithm; apply the quartic deterministic-versus-quantum upper bound.", "depends_on": [ "R0-D", "D-Q4" ], "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "derived", "proof_audit": "explicit_dependency_argument", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "R0-Q4-sep", "A": "R0", "B": "Q", "exponent": 4, "statement": "For the AIK26 family F_n, R0(F_n)=Omega(n^2), Q(F_n)=O(sqrt(n) log(n)^3); every exponent below 4 is impossible.", "source": "definitions", "locator": "Composition of recorded claims: C-R0, C-Q4-sep-AIK", "proof_summary": "Certificate complexity lower bounds zero-error randomized complexity up to a constant. Apply this to the AIK26 witness, retaining its parameterized quantum upper bound.", "depends_on": [ "C-R0", "C-Q4-sep-AIK" ], "kind": "separation", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "polylog", "status": "derived", "proof_audit": "explicit_dependency_argument", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "Q-s3-sep-BHT", "A": "Q", "B": "s", "exponent": 3, "statement": "A total family has Q(f)=Omega_tilde(s(f)^3).", "source": "BHT17", "locator": "Theorem 1 and Corollary 2", "proof_summary": "Make one side of the sensitivity constant using unambiguous certificates, then compose with an AND/OR gadget to balance the sensitivities while preserving the cheat-sheet quantum lower bound.", "kind": "separation", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "polylog", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "C-UCmin-BHT", "A": "C", "B": "UCmin", "exponent": "1.22", "statement": "A total family has C(f)>=UCmin(f)^(log(38/3)/log(8)-o(1)); in particular exponent 1.22 is excluded.", "source": "BHT17", "locator": "Theorem 4", "proof_summary": "Compose weighted projective-plane gadgets with fractional intermediate weights, then round and encode them into Boolean variables.", "kind": "separation", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "subpolynomial", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "exponent_expression": { "op": "log_ratio", "numerator": "38/3", "denominator": 8 }, "historical": true, "superseded_by": [ "C-UC1-2-BBGJK" ] }, { "id": "C-s-BHT", "A": "C", "B": "s", "exponent": "2.22", "statement": "A total family excludes every exponent below 1+log(38/3)/log(8) for C versus s.", "source": "BHT17", "locator": "Theorem 1 applied to Theorem 4; Corollary 3", "proof_summary": "Apply the sensitivity transformation to the fractional-weight unambiguous-certificate separation.", "kind": "separation", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "subpolynomial", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "exponent_expression": { "op": "add", "left": 1, "right": { "op": "log_ratio", "numerator": "38/3", "denominator": 8 } }, "historical": true, "superseded_by": [ "C-s3-BBGJK" ] }, { "id": "s-QE-Amb13", "A": "s", "B": "QE", "exponent": "1.15", "statement": "For iterated ternary non-equality NE_d, s=3^d and QE=O(2048^(d/8)); the excluded exponent is 8 log(3)/log(2048), not merely the rounded 1.15.", "source": "Amb13", "locator": "Theorem 1; Section 3.4, eight-level construction", "proof_summary": "Design an exact quantum subroutine whose amplitude is transformed under composition; an eight-level algorithm uses 2048 queries and composes recursively.", "kind": "separation", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "exponent_expression": { "op": "mul", "left": 8, "right": { "op": "log_ratio", "numerator": 3, "denominator": 2048 } } }, { "id": "adeg-C2-BT17", "A": "adeg", "B": "C", "exponent": 2, "statement": "A total family has adeg(F)>=C(F)^(2-o(1)).", "source": "BT17", "locator": "Theorem 38; Section 5.2.2, equations (63)–(65)", "proof_summary": "Use dual block composition and hardness amplification, then a balanced-pair encoding that supplies cheap certificates.", "kind": "separation", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "subpolynomial", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "C-fbs2-GSS", "A": "C", "B": "fbs", "exponent": 2, "statement": "There are total functions on n^2 bits with C=Omega(n^2) and fbs=O(n).", "source": "GSS13", "locator": "Theorem 32; Section 5.1.1", "proof_summary": "Choose sparse random codewords far apart but hitting large subcubes, then OR-compose n copies. Distance bounds fractional block packing; subcube hitting forces large certificates.", "kind": "separation", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "D-UC1-GPW", "A": "D", "B": "UC1", "exponent": 2, "statement": "A total family has D=Omega_tilde(UC1^2).", "source": "GPW18", "locator": "Theorem 5; Section 2.1", "proof_summary": "A marked all-one column contains a pointer chain certifying zeros in all other columns. An adaptive adversary forces inspection of the whole matrix.", "kind": "separation", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "polylog", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "R-deg2-GJPW", "A": "R", "B": "deg", "exponent": 2, "statement": "A total family has R(f)=Omega_tilde(deg(f)^2).", "source": "GJPW18", "locator": "Lemma 7 (coWAPPdt versus UPdt); Section 4", "proof_summary": "A two-level pointer function has cheap unambiguous positive certificates, hence low exact degree. A distributional conical-junta argument forces quadratic randomized cost.", "kind": "separation", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "polylog", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "audit_note": "Uses deg<=UC1: sum the disjoint positive certificate indicators. The complement orientation of coWAPP does not affect ordinary two-sided R." }, { "id": "D-UC-sep-AKK", "A": "D", "B": "UC", "exponent": 2, "statement": "D(f)>=UC(f)^(2-o(1)) for a total family.", "source": "AKK16", "locator": "Theorem 1.2; Section 2 and recursive construction", "proof_summary": "Alternate AND/OR composition with cheat-sheet totalization, controlling unambiguous certificates while retaining query lower bounds.", "kind": "separation", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "subpolynomial", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "R-UC-sep-AKK", "A": "R", "B": "UC", "exponent": 2, "statement": "R(f)>=UC(f)^(2-o(1)) for a total family.", "source": "AKK16", "locator": "Theorem 1.3; Section 2 and recursive construction", "proof_summary": "Alternate AND/OR composition with cheat-sheet totalization, controlling unambiguous certificates while retaining query lower bounds.", "kind": "separation", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "subpolynomial", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "Q-UC-sep-AKK", "A": "Q", "B": "UC", "exponent": "3/2", "statement": "Q(f)>=UC(f)^(3/2-o(1)) for a total family.", "source": "AKK16", "locator": "Theorem 1.4; Section 2 and recursive construction", "proof_summary": "Alternate AND/OR composition with cheat-sheet totalization, controlling unambiguous certificates while retaining query lower bounds.", "kind": "separation", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "subpolynomial", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "Q-UC1-sep-AKK", "A": "Q", "B": "UC1", "exponent": 2, "statement": "A total family has Q(f)=Omega_tilde(UC1(f)^2).", "source": "AKK16", "locator": "Theorem 1.5", "proof_summary": "Use a cheat sheet to make the accepting certificate unique while preserving quantum hardness.", "kind": "separation", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "polylog", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "deg-UC1", "A": "deg", "B": "UC1", "exponent": 1, "statement": "deg(f)<=UC1(f) for total f.", "source": "GPW18", "locator": "Discussion after Theorems 4–5", "proof_summary": "Sum the monomial indicators of an unambiguous 1-certificate partition; the sum equals f exactly.", "kind": "upper_bound", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "C-UC1-2-BBGJK", "A": "C", "B": "UC1", "exponent": 2, "statement": "A total family satisfies C0(f)>=Omega_tilde(UC1(f)^2).", "source": "BBGJK21", "locator": "Theorems 1–2; Section 3 and Section 5.1", "proof_summary": "Random row-association functions create a partial-function certificate gap; a cheat-sheet reduction turns it into a total function with cheap unambiguous positive certificates.", "kind": "separation", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "polylog", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "historical": true, "superseded_by": [ "C0-UC1-exact2-Pab" ] }, { "id": "C-deg2-BBGJK", "A": "C", "B": "deg", "exponent": 2, "statement": "A total family has C(f)>=Omega_tilde(deg(f)^2).", "source": "BBGJK21", "locator": "Corollary 4; also Bal21 Corollary 6", "proof_summary": "The total unambiguous-DNF witness has C0 large while deg<=UC1, by summing its disjoint accepting-certificate indicators.", "kind": "separation", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "polylog", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "historical": true, "superseded_by": [ "C-deg2-Pab" ] }, { "id": "C-s3-BBGJK", "A": "C", "B": "s", "exponent": 3, "statement": "A total family has C(f)>=Omega_tilde(s(f)^3).", "source": "BBGJK21", "locator": "Corollary 5; BHT17 Theorem 1; also Bal21 Corollary 5", "proof_summary": "Apply the BHT sensitivity transformation to the quadratic certificate/unambiguous-certificate gap.", "kind": "separation", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "polylog", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "historical": true, "superseded_by": [ "C-s3-Pab" ] }, { "id": "C-adeg3-BBGJK-history", "A": "C", "B": "adeg", "exponent": 3, "statement": "A total family has C(f)>=Omega_tilde(adeg(f)^3).", "source": "BBGJK21", "locator": "Corollary 6; Section 6.1", "proof_summary": "Use the Hex partial function, cheat-sheet encoding of paths, and low-degree AND approximations for checking a selected certificate cell.", "kind": "separation", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "polylog", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "historical": true, "superseded_by": [ "C-adeg4-sep" ] }, { "id": "C-lambda4-BBGJK", "A": "C", "B": "lambda", "exponent": 4, "statement": "A total family excludes every exponent below 4 for C versus lambda.", "source": "definitions", "locator": "BBGJK21 Theorem 1; BHT17 sensitivity transformation; ABKBRT21 Appendix A.1", "proof_summary": "The intermediate BHT transformation preserves C and has s1=O(1), s0=O(UC1), so lambda<=sqrt(s0*s1)=O(sqrt(UC1)). Apply the quadratic C/UC1 gap.", "kind": "separation", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "polylog", "status": "derived", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "depends_on": [ "C-UC1-2-BBGJK" ], "proof_sources": [ "BBGJK21", "BHT17", "ABKBRT21" ] }, { "id": "Q-deg2-ABK", "A": "Q", "B": "deg", "exponent": 2, "statement": "A total family has Q(f)=Omega_tilde(deg(f)^2).", "source": "ABK16", "locator": "Theorem 4; Section 4.2", "proof_summary": "Apply a cheat sheet to a quantum-hard low-certificate function. The correct cheat-sheet cell is unique, so summing its exact checking polynomials gives small degree.", "kind": "separation", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "polylog", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "D-R0-sep-ABB", "A": "D", "B": "R0", "exponent": 2, "statement": "A total family has D(f)=Omega_tilde(R0(f)^2).", "source": "ABBLS17", "locator": "Corollary 7", "proof_summary": "A pointer matrix has an adversarially hidden all-one column; random sampling and back pointers find its certificate quickly.", "kind": "separation", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "polylog", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "R0-R-sep-ABB", "A": "R0", "B": "R", "exponent": 2, "statement": "A total family has R0(f)=Omega_tilde(R(f)^2).", "source": "ABBLS17", "locator": "Corollary 14", "proof_summary": "A back-pointer variant makes one-sided randomized evaluation cheap while zero-error evaluation must locate a full certificate.", "kind": "separation", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "polylog", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "R0-QE-sep-ABB", "A": "R0", "B": "QE", "exponent": 2, "statement": "A total family has R0(f)=Omega_tilde(QE(f)^2).", "source": "ABBLS17", "locator": "Corollary 14", "proof_summary": "Use exact quantum search in the same pointer construction with a zero-error randomized lower bound.", "kind": "separation", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "polylog", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "R-QE-sep-ABB", "A": "R", "B": "QE", "exponent": "3/2", "statement": "A total family has R(f)=Omega_tilde(QE(f)^(3/2)).", "source": "ABBLS17", "locator": "Corollary 21", "proof_summary": "Several marked columns permit exact quantum search while planted distributions force many classical queries.", "kind": "separation", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "polylog", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "R-adeg-sep-ABB", "A": "R", "B": "adeg", "exponent": 4, "statement": "A total family has R(f)=Omega_tilde(adeg(f)^4).", "source": "ABBLS17", "locator": "Corollary 23", "proof_summary": "Represent the pointer verification by low-degree polynomials while retaining the randomized lower bound.", "kind": "separation", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "polylog", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "deg-C2-ANDOR", "A": "deg", "B": "C", "exponent": 2, "statement": "For AND_m composed with OR_m, deg=m^2 and C=m; hence deg=C^2.", "source": "definitions", "locator": "Elementary polynomial and certificate calculation", "proof_summary": "The exact polynomial has a nonzero full-degree monomial. Certificates are one zero row or one selected 1 in each row, of maximum size m.", "kind": "separation", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "monotone", "read_once" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "derived", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "s-deg-Kushilevitz", "A": "s", "B": "deg", "exponent": "1.63", "statement": "For the iterated six-variable Kushilevitz gadget, s(f_k)=6^k and deg(f_k)=3^k, giving exact exponent log(6)/log(3).", "source": "NW95", "locator": "Footnote 1, manuscript p.5 (journal p.560)", "proof_summary": "The footnote gives an explicit degree-3 polynomial on six variables. Its value is 0 at zero and 1 at every unit vector. Disjoint recursive composition preserves full sensitivity and multiplies degree.", "kind": "separation", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "exponent_expression": { "op": "log_ratio", "numerator": 6, "denominator": 3 }, "finite_gadget_verification": "tests/test_witnesses.py" }, { "id": "OR-s-Q", "A": "s", "B": "Q", "exponent": 2, "statement": "For OR_n, s=n and Q=Theta(sqrt(n)), so the exponent is 2.", "source": "BBCMW01", "locator": "OR bounds in Section 6.1", "proof_summary": "OR has n sensitive bits at zero. Quantum search and approximate-degree bounds give square-root complexity; the star adjacency matrix has norm sqrt(n).", "kind": "separation", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric", "monotone" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "OR-s-adeg", "A": "s", "B": "adeg", "exponent": 2, "statement": "For OR_n, s=n and adeg=Theta(sqrt(n)), so the exponent is 2.", "source": "NS94", "locator": "Example 3.11 and Lemma 3.8", "proof_summary": "OR has n sensitive bits at zero. Quantum search and approximate-degree bounds give square-root complexity; the star adjacency matrix has norm sqrt(n).", "kind": "separation", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric", "monotone" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "OR-s-lambda", "A": "s", "B": "lambda", "exponent": 2, "statement": "For OR_n, s=n and lambda=Theta(sqrt(n)), so the exponent is 2.", "source": "definitions", "locator": "Sensitivity graph of OR is a star", "proof_summary": "OR has n sensitive bits at zero. Quantum search and approximate-degree bounds give square-root complexity; the star adjacency matrix has norm sqrt(n).", "kind": "separation", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric", "monotone" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "C0-UC1-exact2-Pab", "A": "C0", "B": "UC1", "exponent": 2, "statement": "There are total functions with UC1=O(n) and C0=Omega(n^2).", "source": "Pab26", "locator": "Theorem 1; Section 2", "proof_summary": "Construct pairwise-incompatible signed set pairs using a fixed-size subset in each block. Their associated terms are unambiguous, while a 0-certificate must hit many coordinates.", "kind": "separation", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "C-deg2-Pab", "A": "C", "B": "deg", "exponent": 2, "statement": "A total family satisfies C(f)=Omega(deg(f)^2), without a polylogarithmic lower loss.", "source": "Pab26", "locator": "Corollary 1.2", "proof_summary": "The log-free C0/UC1 witness has deg<=UC1, giving the quadratic separation.", "kind": "separation", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "C-s3-Pab", "A": "C", "B": "s", "exponent": 3, "statement": "A total family satisfies C(f)=Omega(s(f)^3), without a polylogarithmic lower loss.", "source": "Pab26", "locator": "Corollary 1.3; Appendix C", "proof_summary": "Apply the BHT desensitization and balancing transformation to the log-free C0/UC1 witness.", "kind": "separation", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "C-adeg4-Bal26", "A": "C", "B": "adeg", "exponent": 4, "statement": "A total family Gn has C(Gn)=Omega(n^2), adeg(Gn)=Theta_tilde(sqrt(n)), on Theta_tilde(n^3) input bits.", "source": "Bal26", "locator": "Theorem 1.1; Sections 3–5", "proof_summary": "Restrict the ambiguous negative certificates to a structured family. Verify their validity using a low-degree count of compatible positive certificates, then apply the cheat-sheet transform.", "kind": "separation", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "polylog", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "C-adeg4-Pab", "A": "C", "B": "adeg", "exponent": 4, "statement": "A total family on O(n^3 log n) bits satisfies C(f)>=Omega(adeg(f)^4/log(n)^6).", "source": "Pab26", "locator": "Theorem 4; Appendix D", "proof_summary": "Attach pointers and certificate cells to the unambiguous-DNF construction. Check each selected certificate with O(n) low-degree consistency tests, then combine their approximants.", "kind": "separation", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "polylog", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "The Appendix D construction also has C(G)=Omega(n^2) and adeg(G)=O_tilde(sqrt(n)); these parameterized bounds, not erasing log(input length) in isolation, justify the excluded exponent 4." }, { "id": "deg-R0-NAND-SW", "A": "deg", "B": "R0", "exponent": "1.32", "exponent_expression": { "op": "log_ratio", "numerator": 2, "denominator": { "op": "div", "left": { "op": "add", "left": 1, "right": { "op": "sqrt", "arg": 33 } }, "right": 4 } }, "statement": "On the complete binary NAND tree of height h, deg=2^h, QE>=2^(h-1), and R0=O(((1+sqrt(33))/4)^h). The lower exponent is log(2)/log((1+sqrt(33))/4), approximately 1.32674.", "source": "SW86", "locator": "Example 1.1, p.30; Theorem 1.5, p.31; polynomial/query translation is elementary", "proof_summary": "Choose the first child uniformly and stop whenever its value determines the parent. The two output-conditioned costs satisfy a 2-by-2 linear recurrence with dominant root (1+sqrt(33))/4. Exact degree doubles at each NAND level; the polynomial method bounds exact quantum queries.", "kind": "separation", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "read_once" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "original_scan_and_recurrence_checked", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "QE-R0-NAND-SW", "A": "QE", "B": "R0", "exponent": "1.32", "exponent_expression": { "op": "log_ratio", "numerator": 2, "denominator": { "op": "div", "left": { "op": "add", "left": 1, "right": { "op": "sqrt", "arg": 33 } }, "right": 4 } }, "statement": "On the complete binary NAND tree of height h, deg=2^h, QE>=2^(h-1), and R0=O(((1+sqrt(33))/4)^h). The lower exponent is log(2)/log((1+sqrt(33))/4), approximately 1.32674.", "source": "SW86", "locator": "Example 1.1, p.30; Theorem 1.5, p.31; polynomial/query translation is elementary", "proof_summary": "Choose the first child uniformly and stop whenever its value determines the parent. The two output-conditioned costs satisfy a 2-by-2 linear recurrence with dominant root (1+sqrt(33))/4. Exact degree doubles at each NAND level; the polynomial method bounds exact quantum queries.", "kind": "separation", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "read_once" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "original_scan_and_recurrence_checked", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "Q-AdvGeneral", "A": "Q", "B": "AdvGeneral", "exponent": 1, "kind": "theta_bound", "statement": "Q(f)=Theta(AdvGeneral(f)) for every partial or total Boolean function.", "source": "LMRSS11", "locator": "Theorem 1.1; Definition 3.3; Theorems 3.4 and 4.1", "proof_summary": "The filtered gamma_2 norm of the state-conversion task is comparable to the general adversary bound. A query algorithm built from its factorization supplies the upper bound; the adversary progress argument supplies the lower bound.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "AdvPlus-AdvGeneral", "A": "AdvPlus", "B": "AdvGeneral", "exponent": 1, "kind": "upper_bound", "statement": "AdvPlus(f)<=AdvGeneral(f).", "source": "HLS07", "locator": "Definitions 1–2", "proof_summary": "Every nonnegative adversary matrix is feasible in the signed optimization.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "AdvGeneral-Qeps-constant", "A": "AdvGeneral", "B": "Q_eps", "exponent": 1, "kind": "upper_bound", "statement": "AdvGeneral(f)<=2 Q_epsilon(f)/(1-2 sqrt(epsilon(1-epsilon))) for Boolean output and 0<=epsilon<1/2.", "source": "HLS07", "locator": "Theorem 2, Boolean-output case; Section 3", "proof_summary": "Track a weighted progress quantity across oracle queries, and use the final measurement’s output correctness to bound it, including signed adversary weights.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [ "epsilon in [0,1/2), oracle as in HLS07" ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "constant_factor": "2/(1-2 sqrt(epsilon(1-epsilon)))" }, { "id": "partial-R0-D", "A": "R0", "B": "D", "exponent": 1, "kind": "upper_bound", "statement": "R0(f)<=D(f).", "source": "definitions", "locator": "Promise-valid elementary derivation; docs/partial-functions.md", "proof_summary": "Run an optimal deterministic tree.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "Applies to weak promise certificates and unrestricted promise degree; it does not import a total-function certificate-product bound." }, { "id": "partial-R-R0", "A": "R", "B": "R0", "exponent": 1, "kind": "upper_bound", "statement": "R(f)=O(R0(f)) for worst-case expected Las Vegas cost.", "source": "definitions", "locator": "Promise-valid elementary derivation; docs/partial-functions.md", "proof_summary": "Truncate after three times the maximum expected cost; Markov bounds the failure probability.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "Applies to weak promise certificates and unrestricted promise degree; it does not import a total-function certificate-product bound." }, { "id": "partial-Q-R", "A": "Q", "B": "R", "exponent": 1, "kind": "upper_bound", "statement": "Q(f)<=R(f) in the standard bit-query model.", "source": "definitions", "locator": "Promise-valid elementary derivation; docs/partial-functions.md", "proof_summary": "A quantum algorithm can simulate classical randomness and queries.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "Applies to weak promise certificates and unrestricted promise degree; it does not import a total-function certificate-product bound." }, { "id": "partial-Q-QE", "A": "Q", "B": "QE", "exponent": 1, "kind": "upper_bound", "statement": "Q(f)<=QE(f).", "source": "definitions", "locator": "Promise-valid elementary derivation; docs/partial-functions.md", "proof_summary": "An exact algorithm is a bounded-error algorithm.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "Applies to weak promise certificates and unrestricted promise degree; it does not import a total-function certificate-product bound." }, { "id": "partial-QE-D", "A": "QE", "B": "D", "exponent": 1, "kind": "upper_bound", "statement": "QE(f)<=D(f).", "source": "definitions", "locator": "Promise-valid elementary derivation; docs/partial-functions.md", "proof_summary": "Simulate the deterministic tree reversibly.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "Applies to weak promise certificates and unrestricted promise degree; it does not import a total-function certificate-product bound." }, { "id": "partial-C-R0", "A": "C", "B": "R0", "exponent": 1, "kind": "upper_bound", "statement": "C(f)<=R0(f) for weak promise certificates.", "source": "definitions", "locator": "Promise-valid elementary derivation; docs/partial-functions.md", "proof_summary": "Every terminating zero-error path identifies the output on the promise, hence contains a weak certificate; its length is at least C(f,x) at each x.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "Applies to weak promise certificates and unrestricted promise degree; it does not import a total-function certificate-product bound." }, { "id": "partial-RC-C", "A": "RC", "B": "C", "exponent": 1, "kind": "upper_bound", "statement": "RC(f)<=C(f), under the stated verification convention.", "source": "definitions", "locator": "Promise-valid elementary derivation; docs/partial-functions.md", "proof_summary": "Read a smallest certificate for the fixed target input and reject a mismatch.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "Applies to weak promise certificates and unrestricted promise degree; it does not import a total-function certificate-product bound." }, { "id": "partial-bs-RC", "A": "bs", "B": "RC", "exponent": 1, "kind": "upper_bound", "statement": "bs(f)=O(RC(f)).", "source": "definitions", "locator": "Promise-valid elementary derivation; docs/partial-functions.md", "proof_summary": "A successful verifier must hit each sensitive disjoint block with constant probability; sum the expected query indicators.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "Applies to weak promise certificates and unrestricted promise degree; it does not import a total-function certificate-product bound." }, { "id": "partial-s-bs", "A": "s", "B": "bs", "exponent": 1, "kind": "upper_bound", "statement": "s(f)<=bs(f).", "source": "definitions", "locator": "Promise-valid elementary derivation; docs/partial-functions.md", "proof_summary": "Each sensitive bit is a disjoint sensitive singleton block.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "Applies to weak promise certificates and unrestricted promise degree; it does not import a total-function certificate-product bound." }, { "id": "partial-lambda-s", "A": "lambda", "B": "s", "exponent": 1, "kind": "upper_bound", "statement": "lambda(f)<=s(f).", "source": "definitions", "locator": "Promise-valid elementary derivation; docs/partial-functions.md", "proof_summary": "The promised sensitivity graph has maximum degree s; its adjacency norm is at most its maximum degree.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "Applies to weak promise certificates and unrestricted promise degree; it does not import a total-function certificate-product bound." }, { "id": "partial-s-lambda2", "A": "s", "B": "lambda", "exponent": 2, "kind": "upper_bound", "statement": "s(f)<=lambda(f)^2.", "source": "definitions", "locator": "Promise-valid elementary derivation; docs/partial-functions.md", "proof_summary": "A vertex with s sensitive neighbors gives a star subgraph with adjacency norm sqrt(s).", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "Applies to weak promise certificates and unrestricted promise degree; it does not import a total-function certificate-product bound." }, { "id": "partial-adeg-deg", "A": "adeg", "B": "deg", "exponent": 1, "kind": "upper_bound", "statement": "adeg(f)<=deg(f) for unrestricted promise polynomials.", "source": "definitions", "locator": "Promise-valid elementary derivation; docs/partial-functions.md", "proof_summary": "An exact interpolant is also an approximant.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "Applies to weak promise certificates and unrestricted promise degree; it does not import a total-function certificate-product bound." }, { "id": "partial-adeg-R", "A": "adeg", "B": "R", "exponent": 1, "kind": "upper_bound", "statement": "adeg(f)<=R(f) for fixed worst-case query cap.", "source": "definitions", "locator": "Promise-valid elementary derivation; docs/partial-functions.md", "proof_summary": "The acceptance probability of a randomized depth-T decision tree is a polynomial of degree at most T; average its leaf conjunctions.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "Applies to weak promise certificates and unrestricted promise degree; it does not import a total-function certificate-product bound." }, { "id": "partial-deg-QE", "A": "deg", "B": "QE", "exponent": 1, "kind": "upper_bound", "statement": "deg(f)<=2 QE(f).", "source": "definitions", "locator": "Promise-valid elementary derivation; docs/partial-functions.md", "proof_summary": "The acceptance probability of a T-query exact quantum algorithm is degree at most 2T and equals f on its promise.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "Applies to weak promise certificates and unrestricted promise degree; it does not import a total-function certificate-product bound.", "proof_sources": [ "BBCMW01" ] }, { "id": "partial-adeg-Q", "A": "adeg", "B": "Q", "exponent": 1, "kind": "upper_bound", "statement": "adeg(f)<=2 Q(f).", "source": "definitions", "locator": "Promise-valid elementary derivation; docs/partial-functions.md", "proof_summary": "The same acceptance polynomial approximates f on the promise.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "Applies to weak promise certificates and unrestricted promise degree; it does not import a total-function certificate-product bound.", "proof_sources": [ "BBCMW01" ] }, { "id": "partial-RC-R", "A": "RC", "B": "R", "exponent": 1, "kind": "upper_bound", "statement": "RC(f)=O(R(f)).", "source": "definitions", "locator": "Promise-valid elementary derivation; docs/partial-functions.md", "proof_summary": "For a fixed target x, repeat a bounded-error decision algorithm a constant number of times, replay its paths using x, and reject when an actual queried bit disagrees. Distinguishability of opposite outputs forces constant probability of finding a mismatch; accept x with certainty.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "Applies to weak promise certificates and unrestricted promise degree; it does not import a total-function certificate-product bound." }, { "id": "partial-RC-Q2", "A": "RC", "B": "Q", "exponent": 2, "kind": "upper_bound", "statement": "RC(f)=O(Q(f)^2).", "source": "definitions", "locator": "Promise-valid elementary derivation; docs/partial-functions.md", "proof_summary": "At a fixed target x, the quantum hybrid argument gives a fractional certificate of weight O(T^2). Sampling that fractional certificate a constant number of times yields a target verifier.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "Applies to weak promise certificates and unrestricted promise degree; it does not import a total-function certificate-product bound.", "proof_sources": [ "Aar02" ] }, { "id": "partial-lambda-Q", "A": "lambda", "B": "Q", "exponent": 1, "kind": "upper_bound", "statement": "lambda(f)=O(Q(f)).", "source": "definitions", "locator": "Promise-valid elementary derivation; docs/partial-functions.md", "proof_summary": "Use the promised sensitivity adjacency matrix as a positive adversary matrix; each coordinate-filtered matrix is a matching with norm at most one.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "Applies to weak promise certificates and unrestricted promise degree; it does not import a total-function certificate-product bound." }, { "id": "partial-fbs-bs-n", "A": "fbs", "B": "bs", "exponent": null, "kind": "upper_bound", "statement": "For an n-bit partial function, fbs(f)<=sqrt(n bs(f)); the bound is asymptotically tight, with constant approaching 1.", "source": "AKPVZ18", "locator": "Theorems 15 and 17", "proof_summary": "For the upper bound, split sensitive blocks by size and construct a fractional certificate. For tightness, use lines of a finite projective plane (pairwise intersecting, uniformly incident) and OR composition.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "input_length_factor": "sqrt(n)", "notes": "An input-length factor is retained; this is not a pure power bound in bs." }, { "id": "partial-fbs-bs-unbounded", "A": "fbs", "B": "bs", "exponent": null, "kind": "no_polynomial_bound", "statement": "There are partial Boolean functions with bs=1 and fbs unbounded; consequently no finite power of max(1,bs) bounds fbs.", "source": "AKPVZ18", "locator": "Theorem 17, take k=1", "proof_summary": "Use the promise consisting of the zero string and incidence vectors of lines of a finite projective plane. Two lines intersect, so bs=1. Give every line fractional weight 1/(t+1), obtaining fbs=Theta(t).", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "partial-CAdv-fbs-unbounded", "A": "CAdv", "B": "fbs", "exponent": null, "kind": "no_polynomial_bound", "statement": "There are partial Boolean functions with fbs=O(1) but CAdv=Omega(n).", "source": "AKPVZ18", "locator": "Theorems 2 and 9", "proof_summary": "Promise a unique 1 and ask which half contains it. A 1-query weak certificate gives fbs<=1, whereas the complete bipartite relation between the two halves yields a linear classical adversary.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "partial-CAdv-AdvPlus2", "A": "CAdv", "B": "AdvPlus", "exponent": 2, "kind": "upper_bound", "statement": "CAdv(f)<=2 AdvPlus(f)^2 for partial Boolean functions.", "source": "ABK21", "locator": "Theorem 30", "proof_summary": "Apply Cauchy–Schwarz to the positive-adversary weight constraints, split each product into minimum and maximum, and bound the latter sum by twice the objective.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "constant_factor": "2" }, { "id": "partial-AdvPlus-CAdv", "A": "AdvPlus", "B": "CAdv", "exponent": 1, "kind": "upper_bound", "statement": "AdvPlus(f)<=CAdv(f).", "source": "ABK21", "locator": "Section 6; definitions of the two optimization problems", "proof_summary": "The geometric mean dominates the minimum, so every classical-adversary feasible weight scheme is positive-adversary feasible.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "constant_factor": "1" }, { "id": "partial-AdvPlus-boundedadeg2", "A": "AdvPlus", "B": "cubeBoundedAdeg", "exponent": 2, "kind": "upper_bound", "statement": "AdvPlus(f)=O(cubeBoundedAdeg(f)^2) at fixed error below 1/2, even for partial Boolean functions.", "source": "ABK21", "locator": "Final Theorem 7; Section 6, Theorem 29 and Lemmas 26–27", "proof_summary": "Use a globally bounded approximant to choose a total completion; its critical fractional block sensitivity is O(degree^2). Compare that completion measure with CAdv and then positive Adv.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "Global boundedness is essential. The final version repairs the Lemma 27 weight-construction typo in the inspected v1 preprint." }, { "id": "partial-Q-boundeddeg-AB23", "A": "Q", "B": "cubeBoundedDeg", "exponent": null, "kind": "no_polynomial_bound", "statement": "For partial Boolean functions, cubeBoundedDeg=O(log N) while Q=tildeOmega(N^(1/3)); no finite polynomial in cubeBoundedDeg upper-bounds Q.", "source": "AB23", "locator": "Corollary 2, Theorems 1, 6 and 8; definition in Section 2.1", "proof_summary": "Construct threshold satisfiability with a degree-nine globally bounded representing polynomial over a large alphabet. Prove a quantum lower bound for the contained tripartite shift problem using a negative-weight adversary; encode alphabet symbols in O(log N) bits.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "polylogarithmic", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "partial-R-QE-GS26", "A": "R", "B": "QE", "exponent": null, "kind": "no_polynomial_bound", "statement": "Extremal Forrelation has QE=1 in the paper’s joint phase-oracle convention and R=tildeOmega(N^(1/4)), for truth-table input length N=2^(n+1). A standard bit-oracle convention changes the quantum cost by at most a constant.", "source": "GS26", "locator": "Theorem 4; Definition 3 and Section 3.2", "proof_summary": "One interference experiment determines the sign when the forrelation is promised to be exactly plus or minus one. For the classical lower bound, construct bent-function instances and use a linear-algebraic analysis of limited transcripts.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "polylogarithmic", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "relation-Adv-CAdv", "A": "rel_Adv", "B": "rel_CAdv", "exponent": 1, "kind": "upper_bound", "statement": "Adv(S)<=CAdv(S).", "source": "BK24", "locator": "Section 5.1, before Theorem 42", "proof_summary": "Use sqrt(ab)>=min(a,b) in each disjoint-valid-output constraint.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "relation-fbs-CAdv", "A": "rel_fbs", "B": "rel_CAdv", "exponent": 1, "kind": "upper_bound", "statement": "fbs(S)<=CAdv(S).", "source": "BK24", "locator": "Section 5.1, before Theorem 42", "proof_summary": "For a fixed input, the classical-adversary weights form a feasible fractional certificate against inputs with disjoint valid-output sets.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "relation-bs-fbs", "A": "rel_bs", "B": "rel_fbs", "exponent": 1, "kind": "upper_bound", "statement": "bs(S)<=fbs(S).", "source": "BK24", "locator": "Section 5.1", "proof_summary": "Assign unit weight to a packing of pairwise disjoint sensitive blocks.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "relation-fcbs-CAdv", "A": "rel_fcbs", "B": "rel_CAdv", "exponent": 1, "kind": "upper_bound", "statement": "fcbs(S)<=2 CAdv(S).", "source": "BK24", "locator": "Theorem 42", "proof_summary": "Choose a valid-output selector using a weighted nearest critical input, then double the critical input weights to construct its fractional certificate.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "relation-CAdv-Adv2", "A": "rel_CAdv", "B": "rel_Adv", "exponent": 2, "kind": "upper_bound", "statement": "CAdv(S)<=2 Adv(S)^2.", "source": "BK24", "locator": "Section 5.1, paragraph after Theorem 42", "proof_summary": "The partial-function Cauchy–Schwarz proof of ABK21 Theorem 30 uses only constrained pairs, so it extends to disjoint valid-output sets.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "relation-fcbs-adeg2", "A": "rel_fcbs", "B": "rel_adeg", "exponent": 2, "kind": "upper_bound", "statement": "fcbs(S)=O(rel_adeg(S)^2) at fixed error below 1/2.", "source": "BK24", "locator": "Theorem 43", "proof_summary": "Choose the largest valid-output polynomial as selector. At a critical input its polynomial exceeds one half; reduce to the partial one-versus-rest problem and apply the bounded-polynomial fractional-sensitivity bound.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "relation-fbs-adeg2", "A": "rel_fbs", "B": "rel_adeg", "exponent": 2, "kind": "upper_bound", "statement": "fbs(S)=O(rel_adeg(S)^2) at fixed error below 1/2.", "source": "BK24", "locator": "Theorem 43", "proof_summary": "At a fixed input, sum the polynomial mass over its valid outputs. On every disjoint-output input the mass is small, yielding a bounded polynomial for a partial decision problem.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "relation-Adv-Q", "A": "rel_Adv", "B": "Q", "exponent": 1, "kind": "upper_bound", "statement": "Adv(S)=O(Q(S)).", "source": "BK24", "locator": "Section 5.1", "proof_summary": "Apply the positive adversary progress argument only to inputs whose valid-output sets are disjoint.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "relation-CAdv-R", "A": "rel_CAdv", "B": "R", "exponent": 1, "kind": "upper_bound", "statement": "CAdv(S)=O(R(S)).", "source": "BK24", "locator": "Section 5.1", "proof_summary": "A randomized solver distinguishes inputs with disjoint valid-output sets; its query weights satisfy the adversary constraints after constant rescaling.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "relation-adeg-CAdv-unbounded", "A": "rel_adeg", "B": "rel_CAdv", "exponent": null, "kind": "no_polynomial_bound", "statement": "For the total pigeonhole search problem PH_n, CAdv<=2 log(n/2) but rel_adeg=tildeOmega(n^(1/3)). Thus no finite power of CAdv, or of positive Adv, upper-bounds relation approximate degree.", "source": "BK24", "locator": "Definition 44, Lemma 45 and Theorem 46", "proof_summary": "Any two inputs with disjoint collision certificates differ in at least n/2 bits, so uniform weights give a logarithmic classical adversary. Symmetrize the output-polynomial system and reduce to the collision lower bound.", "scope": { "object_kind": "search_relation", "totality": "total", "classes": [ "query_TFNP" ], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "polylogarithmic", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "Collision certificates include both colliding indices and their shared values. Do not apply the decision-function inequality Adv=O(approximate-degree^2) in the reverse direction, or confuse relation polynomial systems with a single polynomial." }, { "id": "partial-search-conversion-cubeBoundedAdeg", "A": "Q", "B": "cubeBoundedAdeg", "exponent": null, "kind": "no_polynomial_bound", "statement": "BK24 constructs partial decision functions with bounded approximate degree and two-sided conical degree O(polylog n), but Q=tildeOmega(n^(1/6)); input length is at most 2^polylog(n).", "source": "BK24", "locator": "Corollary 41 and Theorem 19", "proof_summary": "Index an auxiliary bit y_c by every solution certificate c. Promise that all bits y_c for certificates valid on x have the same value, and ask for that value. Polynomial systems transfer via sum_c p_c(x)y_c. In the reverse direction, measure a decision algorithm’s query register on auxiliary input all zero and verify sampled certificates. The quantum hybrid argument gives Q(search)<=tildeO(Q(decision)^3), hence the cube-root lower-bound transfer.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "polylogarithmic", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "partial-search-conversion-conical", "A": "Q", "B": "conical", "exponent": null, "kind": "no_polynomial_bound", "statement": "BK24 constructs partial decision functions with bounded approximate degree and two-sided conical degree O(polylog n), but Q=tildeOmega(n^(1/6)); input length is at most 2^polylog(n).", "source": "BK24", "locator": "Corollary 41 and Theorem 19", "proof_summary": "Index an auxiliary bit y_c by every solution certificate c. Promise that all bits y_c for certificates valid on x have the same value, and ask for that value. Polynomial systems transfer via sum_c p_c(x)y_c. In the reverse direction, measure a decision algorithm’s query register on auxiliary input all zero and verify sampled certificates. The quantum hybrid argument gives Q(search)<=tildeO(Q(decision)^3), hence the cube-root lower-bound transfer.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "polylogarithmic", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "search-psQ-R-V", "A": "psQ", "B": "R", "exponent": null, "kind": "upper_bound", "statement": "For every N-bit search relation S, psQ(S)=O((R(S)+V(S))*sqrt(N)).", "source": "AGL26", "locator": "ICALP Theorem 5", "proof_summary": "Use the completeness reduction to Find1 with randomized generation and deterministic verification, then apply the O(sqrt(N)) pseudo-deterministic quantum Find1 algorithm.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "input_length_factor": "sqrt(N)", "notes": "This is multivariate and input-length dependent; it is excluded from scalar exponent closure. Verification cost V can be large even for a total relation.", "bound_expression": { "op": "mul", "args": [ { "op": "add", "args": [ "R", "V" ] }, { "op": "power", "base": "input_length", "exponent": "1/2" } ] }, "scalar_inference_allowed": false }, { "id": "search-psQ-R-unbounded", "A": "psQ", "B": "R", "exponent": null, "kind": "no_polynomial_bound", "statement": "AOES is a total N-bit search relation with R=O(1) and psQ=Omega(N).", "source": "AGL26", "locator": "ICALP Theorem 1 and Definition 22", "proof_summary": "The input hides a string through XOR blocks; any other string is a valid answer, so blind random output succeeds. A canonical solver can be amplified and converted into solving an XOR instance, giving the linear lower bound.", "scope": { "object_kind": "search_relation", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "Arbitrary total relation, not a claim of a short-verification query-TFNP separation." }, { "id": "search-psR-psQ-QL", "A": "psR", "B": "psQ", "exponent": null, "kind": "no_polynomial_bound", "statement": "QL-Estimation has R=O(1), psQ=O(log N), and psR=Omega(sqrt(N)) in its stated mixed-oracle model. The classical algorithm that is canonized uses O(log N) queries.", "source": "AGL26", "locator": "ICALP Definition 36 and Theorem 37; introduction Theorem 3", "proof_summary": "Use Simon’s hidden period as a key selecting a canonical approximate Hamming weight. The quantum algorithm learns the key, whereas a classical canonical solver either solves Simon or stabilizes Hamming-weight estimation.", "scope": { "object_kind": "search_relation", "totality": "partial", "classes": [], "input_alphabet": "mixed_bit_and_symbol_oracles", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [ "One query to f returns an n-bit symbol; X queries return one bit; N=2^n is a length parameter." ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This is a promised relation with mixed alphabet queries. Bit-expanding the f oracle changes costs and must be stated explicitly." }, { "id": "search-Find1-psQ", "A": "psQ", "B": "R", "exponent": null, "kind": "no_polynomial_bound", "statement": "On the promise |x|>=n/2, Find1 has R=O(1) and psQ=Omega(sqrt(n)); hence it separates randomized solving from quantum canonical solving.", "source": "GIPS21", "locator": "Theorems 3, 4 and 7", "proof_summary": "Show any low-query quantum canonical solver for Find1 would violate a hybrid distinguishability bound; the paper also establishes completeness for search with efficient verification.", "scope": { "object_kind": "search_relation", "totality": "partial", "classes": [ "Find1" ], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "partial-fbs-RC", "A": "fbs", "B": "RC", "exponent": 1, "kind": "theta_bound", "statement": "For weak promise certificates, fbs(f)=FC(f)=Theta(RC(f)), including partial Boolean functions.", "source": "definitions", "locator": "LP duality and fixed-target verifier sampling; docs/partial-functions.md", "proof_summary": "Fractional packing/cover LP duality gives fbs=FC. Replay a verifier on the fixed target x: its probabilities of querying coordinates, doubled, cover every opposite input because rejection requires a mismatch. Conversely, sample from a fractional certificate with a constant expected number of queries per unit total weight; every opposite input is rejected with constant probability.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This equivalence concerns fractional weak certificates, not classical CAdv. CAdv can be much larger on promises.", "proof_sources": [ "Aar02" ] }, { "id": "promise-unbounded-D-deg-promise-unit-OR", "A": "D", "B": "deg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unit-OR, D is unbounded while deg=O(1). Thus D is not O(max(1,deg)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Unrestricted polynomial sum_i x_i is exact and has degree one. Zero requires inspecting all coordinates classically; each unit position is a disjoint sensitive singleton at zero. The sensitivity graph is a star with norm sqrt(m). Grover search gives Q=O(sqrt(m)); the star adversary gives the matching lower bound. For exact search, apply BHMT02 Theorem 4 with hypothesized marked fraction 1/m and verify the returned index. When there is one mark it is found with certainty; when there are none verification rejects. This also proves QE=O(sqrt(m)).", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unit-OR", "proof_sources": [ "BHMT02", "BBCMW01" ] }, { "id": "promise-unbounded-D-adeg-promise-unit-OR", "A": "D", "B": "adeg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unit-OR, D is unbounded while adeg=O(1). Thus D is not O(max(1,adeg)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Unrestricted polynomial sum_i x_i is exact and has degree one. Zero requires inspecting all coordinates classically; each unit position is a disjoint sensitive singleton at zero. The sensitivity graph is a star with norm sqrt(m). Grover search gives Q=O(sqrt(m)); the star adversary gives the matching lower bound. For exact search, apply BHMT02 Theorem 4 with hypothesized marked fraction 1/m and verify the returned index. When there is one mark it is found with certainty; when there are none verification rejects. This also proves QE=O(sqrt(m)).", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unit-OR", "proof_sources": [ "BHMT02", "BBCMW01" ] }, { "id": "promise-unbounded-R0-deg-promise-unit-OR", "A": "R0", "B": "deg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unit-OR, R0 is unbounded while deg=O(1). Thus R0 is not O(max(1,deg)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Unrestricted polynomial sum_i x_i is exact and has degree one. Zero requires inspecting all coordinates classically; each unit position is a disjoint sensitive singleton at zero. The sensitivity graph is a star with norm sqrt(m). Grover search gives Q=O(sqrt(m)); the star adversary gives the matching lower bound. For exact search, apply BHMT02 Theorem 4 with hypothesized marked fraction 1/m and verify the returned index. When there is one mark it is found with certainty; when there are none verification rejects. This also proves QE=O(sqrt(m)).", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unit-OR", "proof_sources": [ "BHMT02", "BBCMW01" ] }, { "id": "promise-unbounded-R0-adeg-promise-unit-OR", "A": "R0", "B": "adeg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unit-OR, R0 is unbounded while adeg=O(1). Thus R0 is not O(max(1,adeg)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Unrestricted polynomial sum_i x_i is exact and has degree one. Zero requires inspecting all coordinates classically; each unit position is a disjoint sensitive singleton at zero. The sensitivity graph is a star with norm sqrt(m). Grover search gives Q=O(sqrt(m)); the star adversary gives the matching lower bound. For exact search, apply BHMT02 Theorem 4 with hypothesized marked fraction 1/m and verify the returned index. When there is one mark it is found with certainty; when there are none verification rejects. This also proves QE=O(sqrt(m)).", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unit-OR", "proof_sources": [ "BHMT02", "BBCMW01" ] }, { "id": "promise-unbounded-R-deg-promise-unit-OR", "A": "R", "B": "deg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unit-OR, R is unbounded while deg=O(1). Thus R is not O(max(1,deg)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Unrestricted polynomial sum_i x_i is exact and has degree one. Zero requires inspecting all coordinates classically; each unit position is a disjoint sensitive singleton at zero. The sensitivity graph is a star with norm sqrt(m). Grover search gives Q=O(sqrt(m)); the star adversary gives the matching lower bound. For exact search, apply BHMT02 Theorem 4 with hypothesized marked fraction 1/m and verify the returned index. When there is one mark it is found with certainty; when there are none verification rejects. This also proves QE=O(sqrt(m)).", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unit-OR", "proof_sources": [ "BHMT02", "BBCMW01" ] }, { "id": "promise-unbounded-R-adeg-promise-unit-OR", "A": "R", "B": "adeg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unit-OR, R is unbounded while adeg=O(1). Thus R is not O(max(1,adeg)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Unrestricted polynomial sum_i x_i is exact and has degree one. Zero requires inspecting all coordinates classically; each unit position is a disjoint sensitive singleton at zero. The sensitivity graph is a star with norm sqrt(m). Grover search gives Q=O(sqrt(m)); the star adversary gives the matching lower bound. For exact search, apply BHMT02 Theorem 4 with hypothesized marked fraction 1/m and verify the returned index. When there is one mark it is found with certainty; when there are none verification rejects. This also proves QE=O(sqrt(m)).", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unit-OR", "proof_sources": [ "BHMT02", "BBCMW01" ] }, { "id": "promise-unbounded-C-deg-promise-unit-OR", "A": "C", "B": "deg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unit-OR, C is unbounded while deg=O(1). Thus C is not O(max(1,deg)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Unrestricted polynomial sum_i x_i is exact and has degree one. Zero requires inspecting all coordinates classically; each unit position is a disjoint sensitive singleton at zero. The sensitivity graph is a star with norm sqrt(m). Grover search gives Q=O(sqrt(m)); the star adversary gives the matching lower bound. For exact search, apply BHMT02 Theorem 4 with hypothesized marked fraction 1/m and verify the returned index. When there is one mark it is found with certainty; when there are none verification rejects. This also proves QE=O(sqrt(m)).", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unit-OR", "proof_sources": [ "BHMT02", "BBCMW01" ] }, { "id": "promise-unbounded-C-adeg-promise-unit-OR", "A": "C", "B": "adeg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unit-OR, C is unbounded while adeg=O(1). Thus C is not O(max(1,adeg)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Unrestricted polynomial sum_i x_i is exact and has degree one. Zero requires inspecting all coordinates classically; each unit position is a disjoint sensitive singleton at zero. The sensitivity graph is a star with norm sqrt(m). Grover search gives Q=O(sqrt(m)); the star adversary gives the matching lower bound. For exact search, apply BHMT02 Theorem 4 with hypothesized marked fraction 1/m and verify the returned index. When there is one mark it is found with certainty; when there are none verification rejects. This also proves QE=O(sqrt(m)).", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unit-OR", "proof_sources": [ "BHMT02", "BBCMW01" ] }, { "id": "promise-unbounded-RC-deg-promise-unit-OR", "A": "RC", "B": "deg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unit-OR, RC is unbounded while deg=O(1). Thus RC is not O(max(1,deg)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Unrestricted polynomial sum_i x_i is exact and has degree one. Zero requires inspecting all coordinates classically; each unit position is a disjoint sensitive singleton at zero. The sensitivity graph is a star with norm sqrt(m). Grover search gives Q=O(sqrt(m)); the star adversary gives the matching lower bound. For exact search, apply BHMT02 Theorem 4 with hypothesized marked fraction 1/m and verify the returned index. When there is one mark it is found with certainty; when there are none verification rejects. This also proves QE=O(sqrt(m)).", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unit-OR", "proof_sources": [ "BHMT02", "BBCMW01" ] }, { "id": "promise-unbounded-RC-adeg-promise-unit-OR", "A": "RC", "B": "adeg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unit-OR, RC is unbounded while adeg=O(1). Thus RC is not O(max(1,adeg)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Unrestricted polynomial sum_i x_i is exact and has degree one. Zero requires inspecting all coordinates classically; each unit position is a disjoint sensitive singleton at zero. The sensitivity graph is a star with norm sqrt(m). Grover search gives Q=O(sqrt(m)); the star adversary gives the matching lower bound. For exact search, apply BHMT02 Theorem 4 with hypothesized marked fraction 1/m and verify the returned index. When there is one mark it is found with certainty; when there are none verification rejects. This also proves QE=O(sqrt(m)).", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unit-OR", "proof_sources": [ "BHMT02", "BBCMW01" ] }, { "id": "promise-unbounded-bs-deg-promise-unit-OR", "A": "bs", "B": "deg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unit-OR, bs is unbounded while deg=O(1). Thus bs is not O(max(1,deg)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Unrestricted polynomial sum_i x_i is exact and has degree one. Zero requires inspecting all coordinates classically; each unit position is a disjoint sensitive singleton at zero. The sensitivity graph is a star with norm sqrt(m). Grover search gives Q=O(sqrt(m)); the star adversary gives the matching lower bound. For exact search, apply BHMT02 Theorem 4 with hypothesized marked fraction 1/m and verify the returned index. When there is one mark it is found with certainty; when there are none verification rejects. This also proves QE=O(sqrt(m)).", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unit-OR", "proof_sources": [ "BHMT02", "BBCMW01" ] }, { "id": "promise-unbounded-bs-adeg-promise-unit-OR", "A": "bs", "B": "adeg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unit-OR, bs is unbounded while adeg=O(1). Thus bs is not O(max(1,adeg)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Unrestricted polynomial sum_i x_i is exact and has degree one. Zero requires inspecting all coordinates classically; each unit position is a disjoint sensitive singleton at zero. The sensitivity graph is a star with norm sqrt(m). Grover search gives Q=O(sqrt(m)); the star adversary gives the matching lower bound. For exact search, apply BHMT02 Theorem 4 with hypothesized marked fraction 1/m and verify the returned index. When there is one mark it is found with certainty; when there are none verification rejects. This also proves QE=O(sqrt(m)).", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unit-OR", "proof_sources": [ "BHMT02", "BBCMW01" ] }, { "id": "promise-unbounded-s-deg-promise-unit-OR", "A": "s", "B": "deg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unit-OR, s is unbounded while deg=O(1). Thus s is not O(max(1,deg)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Unrestricted polynomial sum_i x_i is exact and has degree one. Zero requires inspecting all coordinates classically; each unit position is a disjoint sensitive singleton at zero. The sensitivity graph is a star with norm sqrt(m). Grover search gives Q=O(sqrt(m)); the star adversary gives the matching lower bound. For exact search, apply BHMT02 Theorem 4 with hypothesized marked fraction 1/m and verify the returned index. When there is one mark it is found with certainty; when there are none verification rejects. This also proves QE=O(sqrt(m)).", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unit-OR", "proof_sources": [ "BHMT02", "BBCMW01" ] }, { "id": "promise-unbounded-s-adeg-promise-unit-OR", "A": "s", "B": "adeg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unit-OR, s is unbounded while adeg=O(1). Thus s is not O(max(1,adeg)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Unrestricted polynomial sum_i x_i is exact and has degree one. Zero requires inspecting all coordinates classically; each unit position is a disjoint sensitive singleton at zero. The sensitivity graph is a star with norm sqrt(m). Grover search gives Q=O(sqrt(m)); the star adversary gives the matching lower bound. For exact search, apply BHMT02 Theorem 4 with hypothesized marked fraction 1/m and verify the returned index. When there is one mark it is found with certainty; when there are none verification rejects. This also proves QE=O(sqrt(m)).", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unit-OR", "proof_sources": [ "BHMT02", "BBCMW01" ] }, { "id": "promise-unbounded-lambda-deg-promise-unit-OR", "A": "lambda", "B": "deg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unit-OR, lambda is unbounded while deg=O(1). Thus lambda is not O(max(1,deg)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Unrestricted polynomial sum_i x_i is exact and has degree one. Zero requires inspecting all coordinates classically; each unit position is a disjoint sensitive singleton at zero. The sensitivity graph is a star with norm sqrt(m). Grover search gives Q=O(sqrt(m)); the star adversary gives the matching lower bound. For exact search, apply BHMT02 Theorem 4 with hypothesized marked fraction 1/m and verify the returned index. When there is one mark it is found with certainty; when there are none verification rejects. This also proves QE=O(sqrt(m)).", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unit-OR", "proof_sources": [ "BHMT02", "BBCMW01" ] }, { "id": "promise-unbounded-lambda-adeg-promise-unit-OR", "A": "lambda", "B": "adeg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unit-OR, lambda is unbounded while adeg=O(1). Thus lambda is not O(max(1,adeg)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Unrestricted polynomial sum_i x_i is exact and has degree one. Zero requires inspecting all coordinates classically; each unit position is a disjoint sensitive singleton at zero. The sensitivity graph is a star with norm sqrt(m). Grover search gives Q=O(sqrt(m)); the star adversary gives the matching lower bound. For exact search, apply BHMT02 Theorem 4 with hypothesized marked fraction 1/m and verify the returned index. When there is one mark it is found with certainty; when there are none verification rejects. This also proves QE=O(sqrt(m)).", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unit-OR", "proof_sources": [ "BHMT02", "BBCMW01" ] }, { "id": "promise-unbounded-Q-deg-promise-unit-OR", "A": "Q", "B": "deg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unit-OR, Q is unbounded while deg=O(1). Thus Q is not O(max(1,deg)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Unrestricted polynomial sum_i x_i is exact and has degree one. Zero requires inspecting all coordinates classically; each unit position is a disjoint sensitive singleton at zero. The sensitivity graph is a star with norm sqrt(m). Grover search gives Q=O(sqrt(m)); the star adversary gives the matching lower bound. For exact search, apply BHMT02 Theorem 4 with hypothesized marked fraction 1/m and verify the returned index. When there is one mark it is found with certainty; when there are none verification rejects. This also proves QE=O(sqrt(m)).", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unit-OR", "proof_sources": [ "BHMT02", "BBCMW01" ] }, { "id": "promise-unbounded-Q-adeg-promise-unit-OR", "A": "Q", "B": "adeg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unit-OR, Q is unbounded while adeg=O(1). Thus Q is not O(max(1,adeg)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Unrestricted polynomial sum_i x_i is exact and has degree one. Zero requires inspecting all coordinates classically; each unit position is a disjoint sensitive singleton at zero. The sensitivity graph is a star with norm sqrt(m). Grover search gives Q=O(sqrt(m)); the star adversary gives the matching lower bound. For exact search, apply BHMT02 Theorem 4 with hypothesized marked fraction 1/m and verify the returned index. When there is one mark it is found with certainty; when there are none verification rejects. This also proves QE=O(sqrt(m)).", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unit-OR", "proof_sources": [ "BHMT02", "BBCMW01" ] }, { "id": "promise-unbounded-QE-deg-promise-unit-OR", "A": "QE", "B": "deg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unit-OR, QE is unbounded while deg=O(1). Thus QE is not O(max(1,deg)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Unrestricted polynomial sum_i x_i is exact and has degree one. Zero requires inspecting all coordinates classically; each unit position is a disjoint sensitive singleton at zero. The sensitivity graph is a star with norm sqrt(m). Grover search gives Q=O(sqrt(m)); the star adversary gives the matching lower bound. For exact search, apply BHMT02 Theorem 4 with hypothesized marked fraction 1/m and verify the returned index. When there is one mark it is found with certainty; when there are none verification rejects. This also proves QE=O(sqrt(m)).", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unit-OR", "proof_sources": [ "BHMT02", "BBCMW01" ] }, { "id": "promise-unbounded-QE-adeg-promise-unit-OR", "A": "QE", "B": "adeg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unit-OR, QE is unbounded while adeg=O(1). Thus QE is not O(max(1,adeg)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Unrestricted polynomial sum_i x_i is exact and has degree one. Zero requires inspecting all coordinates classically; each unit position is a disjoint sensitive singleton at zero. The sensitivity graph is a star with norm sqrt(m). Grover search gives Q=O(sqrt(m)); the star adversary gives the matching lower bound. For exact search, apply BHMT02 Theorem 4 with hypothesized marked fraction 1/m and verify the returned index. When there is one mark it is found with certainty; when there are none verification rejects. This also proves QE=O(sqrt(m)).", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unit-OR", "proof_sources": [ "BHMT02", "BBCMW01" ] }, { "id": "promise-unbounded-D-s-promise-doubled-parity", "A": "D", "B": "s", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-doubled-parity, D is unbounded while s=O(1). Thus D is not O(max(1,s)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "There are no promised pairs at Hamming distance one, hence s=lambda=0. Substitution x_(2i-1)=x_(2i)=z_i converts any interpolating/approximating polynomial to parity on m bits, giving degree m. Flip whole pairs for bs=m. One representative bit per pair supplies the classical upper bound; standard parity lower bounds and verification give linear remaining measures.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-doubled-parity" }, { "id": "promise-unbounded-D-lambda-promise-doubled-parity", "A": "D", "B": "lambda", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-doubled-parity, D is unbounded while lambda=O(1). Thus D is not O(max(1,lambda)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "There are no promised pairs at Hamming distance one, hence s=lambda=0. Substitution x_(2i-1)=x_(2i)=z_i converts any interpolating/approximating polynomial to parity on m bits, giving degree m. Flip whole pairs for bs=m. One representative bit per pair supplies the classical upper bound; standard parity lower bounds and verification give linear remaining measures.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-doubled-parity" }, { "id": "promise-unbounded-R0-s-promise-doubled-parity", "A": "R0", "B": "s", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-doubled-parity, R0 is unbounded while s=O(1). Thus R0 is not O(max(1,s)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "There are no promised pairs at Hamming distance one, hence s=lambda=0. Substitution x_(2i-1)=x_(2i)=z_i converts any interpolating/approximating polynomial to parity on m bits, giving degree m. Flip whole pairs for bs=m. One representative bit per pair supplies the classical upper bound; standard parity lower bounds and verification give linear remaining measures.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-doubled-parity" }, { "id": "promise-unbounded-R0-lambda-promise-doubled-parity", "A": "R0", "B": "lambda", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-doubled-parity, R0 is unbounded while lambda=O(1). Thus R0 is not O(max(1,lambda)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "There are no promised pairs at Hamming distance one, hence s=lambda=0. Substitution x_(2i-1)=x_(2i)=z_i converts any interpolating/approximating polynomial to parity on m bits, giving degree m. Flip whole pairs for bs=m. One representative bit per pair supplies the classical upper bound; standard parity lower bounds and verification give linear remaining measures.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-doubled-parity" }, { "id": "promise-unbounded-R-s-promise-doubled-parity", "A": "R", "B": "s", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-doubled-parity, R is unbounded while s=O(1). Thus R is not O(max(1,s)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "There are no promised pairs at Hamming distance one, hence s=lambda=0. Substitution x_(2i-1)=x_(2i)=z_i converts any interpolating/approximating polynomial to parity on m bits, giving degree m. Flip whole pairs for bs=m. One representative bit per pair supplies the classical upper bound; standard parity lower bounds and verification give linear remaining measures.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-doubled-parity" }, { "id": "promise-unbounded-R-lambda-promise-doubled-parity", "A": "R", "B": "lambda", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-doubled-parity, R is unbounded while lambda=O(1). Thus R is not O(max(1,lambda)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "There are no promised pairs at Hamming distance one, hence s=lambda=0. Substitution x_(2i-1)=x_(2i)=z_i converts any interpolating/approximating polynomial to parity on m bits, giving degree m. Flip whole pairs for bs=m. One representative bit per pair supplies the classical upper bound; standard parity lower bounds and verification give linear remaining measures.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-doubled-parity" }, { "id": "promise-unbounded-C-s-promise-doubled-parity", "A": "C", "B": "s", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-doubled-parity, C is unbounded while s=O(1). Thus C is not O(max(1,s)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "There are no promised pairs at Hamming distance one, hence s=lambda=0. Substitution x_(2i-1)=x_(2i)=z_i converts any interpolating/approximating polynomial to parity on m bits, giving degree m. Flip whole pairs for bs=m. One representative bit per pair supplies the classical upper bound; standard parity lower bounds and verification give linear remaining measures.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-doubled-parity" }, { "id": "promise-unbounded-C-lambda-promise-doubled-parity", "A": "C", "B": "lambda", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-doubled-parity, C is unbounded while lambda=O(1). Thus C is not O(max(1,lambda)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "There are no promised pairs at Hamming distance one, hence s=lambda=0. Substitution x_(2i-1)=x_(2i)=z_i converts any interpolating/approximating polynomial to parity on m bits, giving degree m. Flip whole pairs for bs=m. One representative bit per pair supplies the classical upper bound; standard parity lower bounds and verification give linear remaining measures.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-doubled-parity" }, { "id": "promise-unbounded-RC-s-promise-doubled-parity", "A": "RC", "B": "s", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-doubled-parity, RC is unbounded while s=O(1). Thus RC is not O(max(1,s)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "There are no promised pairs at Hamming distance one, hence s=lambda=0. Substitution x_(2i-1)=x_(2i)=z_i converts any interpolating/approximating polynomial to parity on m bits, giving degree m. Flip whole pairs for bs=m. One representative bit per pair supplies the classical upper bound; standard parity lower bounds and verification give linear remaining measures.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-doubled-parity" }, { "id": "promise-unbounded-RC-lambda-promise-doubled-parity", "A": "RC", "B": "lambda", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-doubled-parity, RC is unbounded while lambda=O(1). Thus RC is not O(max(1,lambda)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "There are no promised pairs at Hamming distance one, hence s=lambda=0. Substitution x_(2i-1)=x_(2i)=z_i converts any interpolating/approximating polynomial to parity on m bits, giving degree m. Flip whole pairs for bs=m. One representative bit per pair supplies the classical upper bound; standard parity lower bounds and verification give linear remaining measures.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-doubled-parity" }, { "id": "promise-unbounded-bs-s-promise-doubled-parity", "A": "bs", "B": "s", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-doubled-parity, bs is unbounded while s=O(1). Thus bs is not O(max(1,s)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "There are no promised pairs at Hamming distance one, hence s=lambda=0. Substitution x_(2i-1)=x_(2i)=z_i converts any interpolating/approximating polynomial to parity on m bits, giving degree m. Flip whole pairs for bs=m. One representative bit per pair supplies the classical upper bound; standard parity lower bounds and verification give linear remaining measures.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-doubled-parity" }, { "id": "promise-unbounded-bs-lambda-promise-doubled-parity", "A": "bs", "B": "lambda", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-doubled-parity, bs is unbounded while lambda=O(1). Thus bs is not O(max(1,lambda)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "There are no promised pairs at Hamming distance one, hence s=lambda=0. Substitution x_(2i-1)=x_(2i)=z_i converts any interpolating/approximating polynomial to parity on m bits, giving degree m. Flip whole pairs for bs=m. One representative bit per pair supplies the classical upper bound; standard parity lower bounds and verification give linear remaining measures.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-doubled-parity" }, { "id": "promise-unbounded-QE-s-promise-doubled-parity", "A": "QE", "B": "s", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-doubled-parity, QE is unbounded while s=O(1). Thus QE is not O(max(1,s)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "There are no promised pairs at Hamming distance one, hence s=lambda=0. Substitution x_(2i-1)=x_(2i)=z_i converts any interpolating/approximating polynomial to parity on m bits, giving degree m. Flip whole pairs for bs=m. One representative bit per pair supplies the classical upper bound; standard parity lower bounds and verification give linear remaining measures.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-doubled-parity" }, { "id": "promise-unbounded-QE-lambda-promise-doubled-parity", "A": "QE", "B": "lambda", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-doubled-parity, QE is unbounded while lambda=O(1). Thus QE is not O(max(1,lambda)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "There are no promised pairs at Hamming distance one, hence s=lambda=0. Substitution x_(2i-1)=x_(2i)=z_i converts any interpolating/approximating polynomial to parity on m bits, giving degree m. Flip whole pairs for bs=m. One representative bit per pair supplies the classical upper bound; standard parity lower bounds and verification give linear remaining measures.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-doubled-parity" }, { "id": "promise-unbounded-deg-s-promise-doubled-parity", "A": "deg", "B": "s", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-doubled-parity, deg is unbounded while s=O(1). Thus deg is not O(max(1,s)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "There are no promised pairs at Hamming distance one, hence s=lambda=0. Substitution x_(2i-1)=x_(2i)=z_i converts any interpolating/approximating polynomial to parity on m bits, giving degree m. Flip whole pairs for bs=m. One representative bit per pair supplies the classical upper bound; standard parity lower bounds and verification give linear remaining measures.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-doubled-parity" }, { "id": "promise-unbounded-deg-lambda-promise-doubled-parity", "A": "deg", "B": "lambda", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-doubled-parity, deg is unbounded while lambda=O(1). Thus deg is not O(max(1,lambda)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "There are no promised pairs at Hamming distance one, hence s=lambda=0. Substitution x_(2i-1)=x_(2i)=z_i converts any interpolating/approximating polynomial to parity on m bits, giving degree m. Flip whole pairs for bs=m. One representative bit per pair supplies the classical upper bound; standard parity lower bounds and verification give linear remaining measures.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-doubled-parity" }, { "id": "promise-unbounded-Q-s-promise-doubled-parity", "A": "Q", "B": "s", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-doubled-parity, Q is unbounded while s=O(1). Thus Q is not O(max(1,s)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "There are no promised pairs at Hamming distance one, hence s=lambda=0. Substitution x_(2i-1)=x_(2i)=z_i converts any interpolating/approximating polynomial to parity on m bits, giving degree m. Flip whole pairs for bs=m. One representative bit per pair supplies the classical upper bound; standard parity lower bounds and verification give linear remaining measures.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-doubled-parity" }, { "id": "promise-unbounded-Q-lambda-promise-doubled-parity", "A": "Q", "B": "lambda", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-doubled-parity, Q is unbounded while lambda=O(1). Thus Q is not O(max(1,lambda)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "There are no promised pairs at Hamming distance one, hence s=lambda=0. Substitution x_(2i-1)=x_(2i)=z_i converts any interpolating/approximating polynomial to parity on m bits, giving degree m. Flip whole pairs for bs=m. One representative bit per pair supplies the classical upper bound; standard parity lower bounds and verification give linear remaining measures.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-doubled-parity" }, { "id": "promise-unbounded-adeg-s-promise-doubled-parity", "A": "adeg", "B": "s", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-doubled-parity, adeg is unbounded while s=O(1). Thus adeg is not O(max(1,s)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "There are no promised pairs at Hamming distance one, hence s=lambda=0. Substitution x_(2i-1)=x_(2i)=z_i converts any interpolating/approximating polynomial to parity on m bits, giving degree m. Flip whole pairs for bs=m. One representative bit per pair supplies the classical upper bound; standard parity lower bounds and verification give linear remaining measures.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-doubled-parity" }, { "id": "promise-unbounded-adeg-lambda-promise-doubled-parity", "A": "adeg", "B": "lambda", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-doubled-parity, adeg is unbounded while lambda=O(1). Thus adeg is not O(max(1,lambda)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "There are no promised pairs at Hamming distance one, hence s=lambda=0. Substitution x_(2i-1)=x_(2i)=z_i converts any interpolating/approximating polynomial to parity on m bits, giving degree m. Flip whole pairs for bs=m. One representative bit per pair supplies the classical upper bound; standard parity lower bounds and verification give linear remaining measures.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-doubled-parity" }, { "id": "promise-unbounded-D-R0-promise-dense-two-axes", "A": "D", "B": "R0", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-dense-two-axes, D is unbounded while R0=O(1). Thus D is not O(max(1,R0)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Each promised input has a one-bit weak certificate: any of its 1 positions identifies the half. Sampling uniformly until a 1 is found is zero-error and uses at most four queries in expectation, so R0,R,Q,RC,bs are O(1). There are no opposite-output adjacent inputs. To lower-bound exact degree, symmetrize p(a,0) and p(0,b) separately. If degree is less than floor(m/2)+1, the first univariate polynomial is identically its required constant and the second is identically the other constant, because both agree at at least floor(m/2)+1 consecutive weights. Their values at zero must agree, a contradiction. Thus deg=Omega(m), QE>=deg/2, and D>=deg. Deterministic evaluation gives O(m) upper bounds.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-dense-two-axes" }, { "id": "promise-unbounded-D-R-promise-dense-two-axes", "A": "D", "B": "R", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-dense-two-axes, D is unbounded while R=O(1). Thus D is not O(max(1,R)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Each promised input has a one-bit weak certificate: any of its 1 positions identifies the half. Sampling uniformly until a 1 is found is zero-error and uses at most four queries in expectation, so R0,R,Q,RC,bs are O(1). There are no opposite-output adjacent inputs. To lower-bound exact degree, symmetrize p(a,0) and p(0,b) separately. If degree is less than floor(m/2)+1, the first univariate polynomial is identically its required constant and the second is identically the other constant, because both agree at at least floor(m/2)+1 consecutive weights. Their values at zero must agree, a contradiction. Thus deg=Omega(m), QE>=deg/2, and D>=deg. Deterministic evaluation gives O(m) upper bounds.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-dense-two-axes" }, { "id": "promise-unbounded-D-C-promise-dense-two-axes", "A": "D", "B": "C", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-dense-two-axes, D is unbounded while C=O(1). Thus D is not O(max(1,C)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Each promised input has a one-bit weak certificate: any of its 1 positions identifies the half. Sampling uniformly until a 1 is found is zero-error and uses at most four queries in expectation, so R0,R,Q,RC,bs are O(1). There are no opposite-output adjacent inputs. To lower-bound exact degree, symmetrize p(a,0) and p(0,b) separately. If degree is less than floor(m/2)+1, the first univariate polynomial is identically its required constant and the second is identically the other constant, because both agree at at least floor(m/2)+1 consecutive weights. Their values at zero must agree, a contradiction. Thus deg=Omega(m), QE>=deg/2, and D>=deg. Deterministic evaluation gives O(m) upper bounds.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-dense-two-axes" }, { "id": "promise-unbounded-D-RC-promise-dense-two-axes", "A": "D", "B": "RC", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-dense-two-axes, D is unbounded while RC=O(1). Thus D is not O(max(1,RC)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Each promised input has a one-bit weak certificate: any of its 1 positions identifies the half. Sampling uniformly until a 1 is found is zero-error and uses at most four queries in expectation, so R0,R,Q,RC,bs are O(1). There are no opposite-output adjacent inputs. To lower-bound exact degree, symmetrize p(a,0) and p(0,b) separately. If degree is less than floor(m/2)+1, the first univariate polynomial is identically its required constant and the second is identically the other constant, because both agree at at least floor(m/2)+1 consecutive weights. Their values at zero must agree, a contradiction. Thus deg=Omega(m), QE>=deg/2, and D>=deg. Deterministic evaluation gives O(m) upper bounds.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-dense-two-axes" }, { "id": "promise-unbounded-D-bs-promise-dense-two-axes", "A": "D", "B": "bs", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-dense-two-axes, D is unbounded while bs=O(1). Thus D is not O(max(1,bs)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Each promised input has a one-bit weak certificate: any of its 1 positions identifies the half. Sampling uniformly until a 1 is found is zero-error and uses at most four queries in expectation, so R0,R,Q,RC,bs are O(1). There are no opposite-output adjacent inputs. To lower-bound exact degree, symmetrize p(a,0) and p(0,b) separately. If degree is less than floor(m/2)+1, the first univariate polynomial is identically its required constant and the second is identically the other constant, because both agree at at least floor(m/2)+1 consecutive weights. Their values at zero must agree, a contradiction. Thus deg=Omega(m), QE>=deg/2, and D>=deg. Deterministic evaluation gives O(m) upper bounds.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-dense-two-axes" }, { "id": "promise-unbounded-D-s-promise-dense-two-axes", "A": "D", "B": "s", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-dense-two-axes, D is unbounded while s=O(1). Thus D is not O(max(1,s)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Each promised input has a one-bit weak certificate: any of its 1 positions identifies the half. Sampling uniformly until a 1 is found is zero-error and uses at most four queries in expectation, so R0,R,Q,RC,bs are O(1). There are no opposite-output adjacent inputs. To lower-bound exact degree, symmetrize p(a,0) and p(0,b) separately. If degree is less than floor(m/2)+1, the first univariate polynomial is identically its required constant and the second is identically the other constant, because both agree at at least floor(m/2)+1 consecutive weights. Their values at zero must agree, a contradiction. Thus deg=Omega(m), QE>=deg/2, and D>=deg. Deterministic evaluation gives O(m) upper bounds.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-dense-two-axes" }, { "id": "promise-unbounded-D-lambda-promise-dense-two-axes", "A": "D", "B": "lambda", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-dense-two-axes, D is unbounded while lambda=O(1). Thus D is not O(max(1,lambda)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Each promised input has a one-bit weak certificate: any of its 1 positions identifies the half. Sampling uniformly until a 1 is found is zero-error and uses at most four queries in expectation, so R0,R,Q,RC,bs are O(1). There are no opposite-output adjacent inputs. To lower-bound exact degree, symmetrize p(a,0) and p(0,b) separately. If degree is less than floor(m/2)+1, the first univariate polynomial is identically its required constant and the second is identically the other constant, because both agree at at least floor(m/2)+1 consecutive weights. Their values at zero must agree, a contradiction. Thus deg=Omega(m), QE>=deg/2, and D>=deg. Deterministic evaluation gives O(m) upper bounds.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-dense-two-axes" }, { "id": "promise-unbounded-D-Q-promise-dense-two-axes", "A": "D", "B": "Q", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-dense-two-axes, D is unbounded while Q=O(1). Thus D is not O(max(1,Q)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Each promised input has a one-bit weak certificate: any of its 1 positions identifies the half. Sampling uniformly until a 1 is found is zero-error and uses at most four queries in expectation, so R0,R,Q,RC,bs are O(1). There are no opposite-output adjacent inputs. To lower-bound exact degree, symmetrize p(a,0) and p(0,b) separately. If degree is less than floor(m/2)+1, the first univariate polynomial is identically its required constant and the second is identically the other constant, because both agree at at least floor(m/2)+1 consecutive weights. Their values at zero must agree, a contradiction. Thus deg=Omega(m), QE>=deg/2, and D>=deg. Deterministic evaluation gives O(m) upper bounds.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-dense-two-axes" }, { "id": "promise-unbounded-D-adeg-promise-dense-two-axes", "A": "D", "B": "adeg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-dense-two-axes, D is unbounded while adeg=O(1). Thus D is not O(max(1,adeg)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Each promised input has a one-bit weak certificate: any of its 1 positions identifies the half. Sampling uniformly until a 1 is found is zero-error and uses at most four queries in expectation, so R0,R,Q,RC,bs are O(1). There are no opposite-output adjacent inputs. To lower-bound exact degree, symmetrize p(a,0) and p(0,b) separately. If degree is less than floor(m/2)+1, the first univariate polynomial is identically its required constant and the second is identically the other constant, because both agree at at least floor(m/2)+1 consecutive weights. Their values at zero must agree, a contradiction. Thus deg=Omega(m), QE>=deg/2, and D>=deg. Deterministic evaluation gives O(m) upper bounds.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-dense-two-axes" }, { "id": "promise-unbounded-QE-R0-promise-dense-two-axes", "A": "QE", "B": "R0", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-dense-two-axes, QE is unbounded while R0=O(1). Thus QE is not O(max(1,R0)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Each promised input has a one-bit weak certificate: any of its 1 positions identifies the half. Sampling uniformly until a 1 is found is zero-error and uses at most four queries in expectation, so R0,R,Q,RC,bs are O(1). There are no opposite-output adjacent inputs. To lower-bound exact degree, symmetrize p(a,0) and p(0,b) separately. If degree is less than floor(m/2)+1, the first univariate polynomial is identically its required constant and the second is identically the other constant, because both agree at at least floor(m/2)+1 consecutive weights. Their values at zero must agree, a contradiction. Thus deg=Omega(m), QE>=deg/2, and D>=deg. Deterministic evaluation gives O(m) upper bounds.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-dense-two-axes" }, { "id": "promise-unbounded-QE-R-promise-dense-two-axes", "A": "QE", "B": "R", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-dense-two-axes, QE is unbounded while R=O(1). Thus QE is not O(max(1,R)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Each promised input has a one-bit weak certificate: any of its 1 positions identifies the half. Sampling uniformly until a 1 is found is zero-error and uses at most four queries in expectation, so R0,R,Q,RC,bs are O(1). There are no opposite-output adjacent inputs. To lower-bound exact degree, symmetrize p(a,0) and p(0,b) separately. If degree is less than floor(m/2)+1, the first univariate polynomial is identically its required constant and the second is identically the other constant, because both agree at at least floor(m/2)+1 consecutive weights. Their values at zero must agree, a contradiction. Thus deg=Omega(m), QE>=deg/2, and D>=deg. Deterministic evaluation gives O(m) upper bounds.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-dense-two-axes" }, { "id": "promise-unbounded-QE-C-promise-dense-two-axes", "A": "QE", "B": "C", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-dense-two-axes, QE is unbounded while C=O(1). Thus QE is not O(max(1,C)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Each promised input has a one-bit weak certificate: any of its 1 positions identifies the half. Sampling uniformly until a 1 is found is zero-error and uses at most four queries in expectation, so R0,R,Q,RC,bs are O(1). There are no opposite-output adjacent inputs. To lower-bound exact degree, symmetrize p(a,0) and p(0,b) separately. If degree is less than floor(m/2)+1, the first univariate polynomial is identically its required constant and the second is identically the other constant, because both agree at at least floor(m/2)+1 consecutive weights. Their values at zero must agree, a contradiction. Thus deg=Omega(m), QE>=deg/2, and D>=deg. Deterministic evaluation gives O(m) upper bounds.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-dense-two-axes" }, { "id": "promise-unbounded-QE-RC-promise-dense-two-axes", "A": "QE", "B": "RC", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-dense-two-axes, QE is unbounded while RC=O(1). Thus QE is not O(max(1,RC)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Each promised input has a one-bit weak certificate: any of its 1 positions identifies the half. Sampling uniformly until a 1 is found is zero-error and uses at most four queries in expectation, so R0,R,Q,RC,bs are O(1). There are no opposite-output adjacent inputs. To lower-bound exact degree, symmetrize p(a,0) and p(0,b) separately. If degree is less than floor(m/2)+1, the first univariate polynomial is identically its required constant and the second is identically the other constant, because both agree at at least floor(m/2)+1 consecutive weights. Their values at zero must agree, a contradiction. Thus deg=Omega(m), QE>=deg/2, and D>=deg. Deterministic evaluation gives O(m) upper bounds.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-dense-two-axes" }, { "id": "promise-unbounded-QE-bs-promise-dense-two-axes", "A": "QE", "B": "bs", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-dense-two-axes, QE is unbounded while bs=O(1). Thus QE is not O(max(1,bs)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Each promised input has a one-bit weak certificate: any of its 1 positions identifies the half. Sampling uniformly until a 1 is found is zero-error and uses at most four queries in expectation, so R0,R,Q,RC,bs are O(1). There are no opposite-output adjacent inputs. To lower-bound exact degree, symmetrize p(a,0) and p(0,b) separately. If degree is less than floor(m/2)+1, the first univariate polynomial is identically its required constant and the second is identically the other constant, because both agree at at least floor(m/2)+1 consecutive weights. Their values at zero must agree, a contradiction. Thus deg=Omega(m), QE>=deg/2, and D>=deg. Deterministic evaluation gives O(m) upper bounds.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-dense-two-axes" }, { "id": "promise-unbounded-QE-s-promise-dense-two-axes", "A": "QE", "B": "s", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-dense-two-axes, QE is unbounded while s=O(1). Thus QE is not O(max(1,s)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Each promised input has a one-bit weak certificate: any of its 1 positions identifies the half. Sampling uniformly until a 1 is found is zero-error and uses at most four queries in expectation, so R0,R,Q,RC,bs are O(1). There are no opposite-output adjacent inputs. To lower-bound exact degree, symmetrize p(a,0) and p(0,b) separately. If degree is less than floor(m/2)+1, the first univariate polynomial is identically its required constant and the second is identically the other constant, because both agree at at least floor(m/2)+1 consecutive weights. Their values at zero must agree, a contradiction. Thus deg=Omega(m), QE>=deg/2, and D>=deg. Deterministic evaluation gives O(m) upper bounds.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-dense-two-axes" }, { "id": "promise-unbounded-QE-lambda-promise-dense-two-axes", "A": "QE", "B": "lambda", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-dense-two-axes, QE is unbounded while lambda=O(1). Thus QE is not O(max(1,lambda)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Each promised input has a one-bit weak certificate: any of its 1 positions identifies the half. Sampling uniformly until a 1 is found is zero-error and uses at most four queries in expectation, so R0,R,Q,RC,bs are O(1). There are no opposite-output adjacent inputs. To lower-bound exact degree, symmetrize p(a,0) and p(0,b) separately. If degree is less than floor(m/2)+1, the first univariate polynomial is identically its required constant and the second is identically the other constant, because both agree at at least floor(m/2)+1 consecutive weights. Their values at zero must agree, a contradiction. Thus deg=Omega(m), QE>=deg/2, and D>=deg. Deterministic evaluation gives O(m) upper bounds.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-dense-two-axes" }, { "id": "promise-unbounded-QE-Q-promise-dense-two-axes", "A": "QE", "B": "Q", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-dense-two-axes, QE is unbounded while Q=O(1). Thus QE is not O(max(1,Q)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Each promised input has a one-bit weak certificate: any of its 1 positions identifies the half. Sampling uniformly until a 1 is found is zero-error and uses at most four queries in expectation, so R0,R,Q,RC,bs are O(1). There are no opposite-output adjacent inputs. To lower-bound exact degree, symmetrize p(a,0) and p(0,b) separately. If degree is less than floor(m/2)+1, the first univariate polynomial is identically its required constant and the second is identically the other constant, because both agree at at least floor(m/2)+1 consecutive weights. Their values at zero must agree, a contradiction. Thus deg=Omega(m), QE>=deg/2, and D>=deg. Deterministic evaluation gives O(m) upper bounds.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-dense-two-axes" }, { "id": "promise-unbounded-QE-adeg-promise-dense-two-axes", "A": "QE", "B": "adeg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-dense-two-axes, QE is unbounded while adeg=O(1). Thus QE is not O(max(1,adeg)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Each promised input has a one-bit weak certificate: any of its 1 positions identifies the half. Sampling uniformly until a 1 is found is zero-error and uses at most four queries in expectation, so R0,R,Q,RC,bs are O(1). There are no opposite-output adjacent inputs. To lower-bound exact degree, symmetrize p(a,0) and p(0,b) separately. If degree is less than floor(m/2)+1, the first univariate polynomial is identically its required constant and the second is identically the other constant, because both agree at at least floor(m/2)+1 consecutive weights. Their values at zero must agree, a contradiction. Thus deg=Omega(m), QE>=deg/2, and D>=deg. Deterministic evaluation gives O(m) upper bounds.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-dense-two-axes" }, { "id": "promise-unbounded-deg-R0-promise-dense-two-axes", "A": "deg", "B": "R0", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-dense-two-axes, deg is unbounded while R0=O(1). Thus deg is not O(max(1,R0)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Each promised input has a one-bit weak certificate: any of its 1 positions identifies the half. Sampling uniformly until a 1 is found is zero-error and uses at most four queries in expectation, so R0,R,Q,RC,bs are O(1). There are no opposite-output adjacent inputs. To lower-bound exact degree, symmetrize p(a,0) and p(0,b) separately. If degree is less than floor(m/2)+1, the first univariate polynomial is identically its required constant and the second is identically the other constant, because both agree at at least floor(m/2)+1 consecutive weights. Their values at zero must agree, a contradiction. Thus deg=Omega(m), QE>=deg/2, and D>=deg. Deterministic evaluation gives O(m) upper bounds.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-dense-two-axes" }, { "id": "promise-unbounded-deg-R-promise-dense-two-axes", "A": "deg", "B": "R", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-dense-two-axes, deg is unbounded while R=O(1). Thus deg is not O(max(1,R)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Each promised input has a one-bit weak certificate: any of its 1 positions identifies the half. Sampling uniformly until a 1 is found is zero-error and uses at most four queries in expectation, so R0,R,Q,RC,bs are O(1). There are no opposite-output adjacent inputs. To lower-bound exact degree, symmetrize p(a,0) and p(0,b) separately. If degree is less than floor(m/2)+1, the first univariate polynomial is identically its required constant and the second is identically the other constant, because both agree at at least floor(m/2)+1 consecutive weights. Their values at zero must agree, a contradiction. Thus deg=Omega(m), QE>=deg/2, and D>=deg. Deterministic evaluation gives O(m) upper bounds.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-dense-two-axes" }, { "id": "promise-unbounded-deg-C-promise-dense-two-axes", "A": "deg", "B": "C", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-dense-two-axes, deg is unbounded while C=O(1). Thus deg is not O(max(1,C)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Each promised input has a one-bit weak certificate: any of its 1 positions identifies the half. Sampling uniformly until a 1 is found is zero-error and uses at most four queries in expectation, so R0,R,Q,RC,bs are O(1). There are no opposite-output adjacent inputs. To lower-bound exact degree, symmetrize p(a,0) and p(0,b) separately. If degree is less than floor(m/2)+1, the first univariate polynomial is identically its required constant and the second is identically the other constant, because both agree at at least floor(m/2)+1 consecutive weights. Their values at zero must agree, a contradiction. Thus deg=Omega(m), QE>=deg/2, and D>=deg. Deterministic evaluation gives O(m) upper bounds.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-dense-two-axes" }, { "id": "promise-unbounded-deg-RC-promise-dense-two-axes", "A": "deg", "B": "RC", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-dense-two-axes, deg is unbounded while RC=O(1). Thus deg is not O(max(1,RC)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Each promised input has a one-bit weak certificate: any of its 1 positions identifies the half. Sampling uniformly until a 1 is found is zero-error and uses at most four queries in expectation, so R0,R,Q,RC,bs are O(1). There are no opposite-output adjacent inputs. To lower-bound exact degree, symmetrize p(a,0) and p(0,b) separately. If degree is less than floor(m/2)+1, the first univariate polynomial is identically its required constant and the second is identically the other constant, because both agree at at least floor(m/2)+1 consecutive weights. Their values at zero must agree, a contradiction. Thus deg=Omega(m), QE>=deg/2, and D>=deg. Deterministic evaluation gives O(m) upper bounds.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-dense-two-axes" }, { "id": "promise-unbounded-deg-bs-promise-dense-two-axes", "A": "deg", "B": "bs", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-dense-two-axes, deg is unbounded while bs=O(1). Thus deg is not O(max(1,bs)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Each promised input has a one-bit weak certificate: any of its 1 positions identifies the half. Sampling uniformly until a 1 is found is zero-error and uses at most four queries in expectation, so R0,R,Q,RC,bs are O(1). There are no opposite-output adjacent inputs. To lower-bound exact degree, symmetrize p(a,0) and p(0,b) separately. If degree is less than floor(m/2)+1, the first univariate polynomial is identically its required constant and the second is identically the other constant, because both agree at at least floor(m/2)+1 consecutive weights. Their values at zero must agree, a contradiction. Thus deg=Omega(m), QE>=deg/2, and D>=deg. Deterministic evaluation gives O(m) upper bounds.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-dense-two-axes" }, { "id": "promise-unbounded-deg-s-promise-dense-two-axes", "A": "deg", "B": "s", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-dense-two-axes, deg is unbounded while s=O(1). Thus deg is not O(max(1,s)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Each promised input has a one-bit weak certificate: any of its 1 positions identifies the half. Sampling uniformly until a 1 is found is zero-error and uses at most four queries in expectation, so R0,R,Q,RC,bs are O(1). There are no opposite-output adjacent inputs. To lower-bound exact degree, symmetrize p(a,0) and p(0,b) separately. If degree is less than floor(m/2)+1, the first univariate polynomial is identically its required constant and the second is identically the other constant, because both agree at at least floor(m/2)+1 consecutive weights. Their values at zero must agree, a contradiction. Thus deg=Omega(m), QE>=deg/2, and D>=deg. Deterministic evaluation gives O(m) upper bounds.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-dense-two-axes" }, { "id": "promise-unbounded-deg-lambda-promise-dense-two-axes", "A": "deg", "B": "lambda", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-dense-two-axes, deg is unbounded while lambda=O(1). Thus deg is not O(max(1,lambda)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Each promised input has a one-bit weak certificate: any of its 1 positions identifies the half. Sampling uniformly until a 1 is found is zero-error and uses at most four queries in expectation, so R0,R,Q,RC,bs are O(1). There are no opposite-output adjacent inputs. To lower-bound exact degree, symmetrize p(a,0) and p(0,b) separately. If degree is less than floor(m/2)+1, the first univariate polynomial is identically its required constant and the second is identically the other constant, because both agree at at least floor(m/2)+1 consecutive weights. Their values at zero must agree, a contradiction. Thus deg=Omega(m), QE>=deg/2, and D>=deg. Deterministic evaluation gives O(m) upper bounds.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-dense-two-axes" }, { "id": "promise-unbounded-deg-Q-promise-dense-two-axes", "A": "deg", "B": "Q", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-dense-two-axes, deg is unbounded while Q=O(1). Thus deg is not O(max(1,Q)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Each promised input has a one-bit weak certificate: any of its 1 positions identifies the half. Sampling uniformly until a 1 is found is zero-error and uses at most four queries in expectation, so R0,R,Q,RC,bs are O(1). There are no opposite-output adjacent inputs. To lower-bound exact degree, symmetrize p(a,0) and p(0,b) separately. If degree is less than floor(m/2)+1, the first univariate polynomial is identically its required constant and the second is identically the other constant, because both agree at at least floor(m/2)+1 consecutive weights. Their values at zero must agree, a contradiction. Thus deg=Omega(m), QE>=deg/2, and D>=deg. Deterministic evaluation gives O(m) upper bounds.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-dense-two-axes" }, { "id": "promise-unbounded-deg-adeg-promise-dense-two-axes", "A": "deg", "B": "adeg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-dense-two-axes, deg is unbounded while adeg=O(1). Thus deg is not O(max(1,adeg)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Each promised input has a one-bit weak certificate: any of its 1 positions identifies the half. Sampling uniformly until a 1 is found is zero-error and uses at most four queries in expectation, so R0,R,Q,RC,bs are O(1). There are no opposite-output adjacent inputs. To lower-bound exact degree, symmetrize p(a,0) and p(0,b) separately. If degree is less than floor(m/2)+1, the first univariate polynomial is identically its required constant and the second is identically the other constant, because both agree at at least floor(m/2)+1 consecutive weights. Their values at zero must agree, a contradiction. Thus deg=Omega(m), QE>=deg/2, and D>=deg. Deterministic evaluation gives O(m) upper bounds.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-dense-two-axes" }, { "id": "promise-unbounded-D-C-promise-nonempty-two-axes", "A": "D", "B": "C", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-nonempty-two-axes, D is unbounded while C=O(1). Thus D is not O(max(1,C)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Again C=RC=fbs=bs=1 and s=lambda=0. If p approximates the side label on the promise, amplify it on its two promised ranges to error 1/12 using a constant-degree polynomial. Then q(z)=p(0,z)-p(z,0), with sign chosen for the label convention, is zero at z=0 and within 1/6 of 1 for every nonzero z. It approximates total OR, so degree(p)=Omega(sqrt(m)). A bounded-error algorithm searches for a marked bit; the lower bound follows from this degree argument. Linear exact degree follows because the restrictions p(a,0) and p(0,b) would otherwise force incompatible values at their common origin.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-nonempty-two-axes" }, { "id": "promise-unbounded-D-RC-promise-nonempty-two-axes", "A": "D", "B": "RC", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-nonempty-two-axes, D is unbounded while RC=O(1). Thus D is not O(max(1,RC)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Again C=RC=fbs=bs=1 and s=lambda=0. If p approximates the side label on the promise, amplify it on its two promised ranges to error 1/12 using a constant-degree polynomial. Then q(z)=p(0,z)-p(z,0), with sign chosen for the label convention, is zero at z=0 and within 1/6 of 1 for every nonzero z. It approximates total OR, so degree(p)=Omega(sqrt(m)). A bounded-error algorithm searches for a marked bit; the lower bound follows from this degree argument. Linear exact degree follows because the restrictions p(a,0) and p(0,b) would otherwise force incompatible values at their common origin.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-nonempty-two-axes" }, { "id": "promise-unbounded-D-bs-promise-nonempty-two-axes", "A": "D", "B": "bs", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-nonempty-two-axes, D is unbounded while bs=O(1). Thus D is not O(max(1,bs)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Again C=RC=fbs=bs=1 and s=lambda=0. If p approximates the side label on the promise, amplify it on its two promised ranges to error 1/12 using a constant-degree polynomial. Then q(z)=p(0,z)-p(z,0), with sign chosen for the label convention, is zero at z=0 and within 1/6 of 1 for every nonzero z. It approximates total OR, so degree(p)=Omega(sqrt(m)). A bounded-error algorithm searches for a marked bit; the lower bound follows from this degree argument. Linear exact degree follows because the restrictions p(a,0) and p(0,b) would otherwise force incompatible values at their common origin.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-nonempty-two-axes" }, { "id": "promise-unbounded-D-s-promise-nonempty-two-axes", "A": "D", "B": "s", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-nonempty-two-axes, D is unbounded while s=O(1). Thus D is not O(max(1,s)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Again C=RC=fbs=bs=1 and s=lambda=0. If p approximates the side label on the promise, amplify it on its two promised ranges to error 1/12 using a constant-degree polynomial. Then q(z)=p(0,z)-p(z,0), with sign chosen for the label convention, is zero at z=0 and within 1/6 of 1 for every nonzero z. It approximates total OR, so degree(p)=Omega(sqrt(m)). A bounded-error algorithm searches for a marked bit; the lower bound follows from this degree argument. Linear exact degree follows because the restrictions p(a,0) and p(0,b) would otherwise force incompatible values at their common origin.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-nonempty-two-axes" }, { "id": "promise-unbounded-D-lambda-promise-nonempty-two-axes", "A": "D", "B": "lambda", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-nonempty-two-axes, D is unbounded while lambda=O(1). Thus D is not O(max(1,lambda)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Again C=RC=fbs=bs=1 and s=lambda=0. If p approximates the side label on the promise, amplify it on its two promised ranges to error 1/12 using a constant-degree polynomial. Then q(z)=p(0,z)-p(z,0), with sign chosen for the label convention, is zero at z=0 and within 1/6 of 1 for every nonzero z. It approximates total OR, so degree(p)=Omega(sqrt(m)). A bounded-error algorithm searches for a marked bit; the lower bound follows from this degree argument. Linear exact degree follows because the restrictions p(a,0) and p(0,b) would otherwise force incompatible values at their common origin.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-nonempty-two-axes" }, { "id": "promise-unbounded-QE-C-promise-nonempty-two-axes", "A": "QE", "B": "C", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-nonempty-two-axes, QE is unbounded while C=O(1). Thus QE is not O(max(1,C)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Again C=RC=fbs=bs=1 and s=lambda=0. If p approximates the side label on the promise, amplify it on its two promised ranges to error 1/12 using a constant-degree polynomial. Then q(z)=p(0,z)-p(z,0), with sign chosen for the label convention, is zero at z=0 and within 1/6 of 1 for every nonzero z. It approximates total OR, so degree(p)=Omega(sqrt(m)). A bounded-error algorithm searches for a marked bit; the lower bound follows from this degree argument. Linear exact degree follows because the restrictions p(a,0) and p(0,b) would otherwise force incompatible values at their common origin.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-nonempty-two-axes" }, { "id": "promise-unbounded-QE-RC-promise-nonempty-two-axes", "A": "QE", "B": "RC", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-nonempty-two-axes, QE is unbounded while RC=O(1). Thus QE is not O(max(1,RC)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Again C=RC=fbs=bs=1 and s=lambda=0. If p approximates the side label on the promise, amplify it on its two promised ranges to error 1/12 using a constant-degree polynomial. Then q(z)=p(0,z)-p(z,0), with sign chosen for the label convention, is zero at z=0 and within 1/6 of 1 for every nonzero z. It approximates total OR, so degree(p)=Omega(sqrt(m)). A bounded-error algorithm searches for a marked bit; the lower bound follows from this degree argument. Linear exact degree follows because the restrictions p(a,0) and p(0,b) would otherwise force incompatible values at their common origin.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-nonempty-two-axes" }, { "id": "promise-unbounded-QE-bs-promise-nonempty-two-axes", "A": "QE", "B": "bs", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-nonempty-two-axes, QE is unbounded while bs=O(1). Thus QE is not O(max(1,bs)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Again C=RC=fbs=bs=1 and s=lambda=0. If p approximates the side label on the promise, amplify it on its two promised ranges to error 1/12 using a constant-degree polynomial. Then q(z)=p(0,z)-p(z,0), with sign chosen for the label convention, is zero at z=0 and within 1/6 of 1 for every nonzero z. It approximates total OR, so degree(p)=Omega(sqrt(m)). A bounded-error algorithm searches for a marked bit; the lower bound follows from this degree argument. Linear exact degree follows because the restrictions p(a,0) and p(0,b) would otherwise force incompatible values at their common origin.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-nonempty-two-axes" }, { "id": "promise-unbounded-QE-s-promise-nonempty-two-axes", "A": "QE", "B": "s", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-nonempty-two-axes, QE is unbounded while s=O(1). Thus QE is not O(max(1,s)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Again C=RC=fbs=bs=1 and s=lambda=0. If p approximates the side label on the promise, amplify it on its two promised ranges to error 1/12 using a constant-degree polynomial. Then q(z)=p(0,z)-p(z,0), with sign chosen for the label convention, is zero at z=0 and within 1/6 of 1 for every nonzero z. It approximates total OR, so degree(p)=Omega(sqrt(m)). A bounded-error algorithm searches for a marked bit; the lower bound follows from this degree argument. Linear exact degree follows because the restrictions p(a,0) and p(0,b) would otherwise force incompatible values at their common origin.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-nonempty-two-axes" }, { "id": "promise-unbounded-QE-lambda-promise-nonempty-two-axes", "A": "QE", "B": "lambda", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-nonempty-two-axes, QE is unbounded while lambda=O(1). Thus QE is not O(max(1,lambda)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Again C=RC=fbs=bs=1 and s=lambda=0. If p approximates the side label on the promise, amplify it on its two promised ranges to error 1/12 using a constant-degree polynomial. Then q(z)=p(0,z)-p(z,0), with sign chosen for the label convention, is zero at z=0 and within 1/6 of 1 for every nonzero z. It approximates total OR, so degree(p)=Omega(sqrt(m)). A bounded-error algorithm searches for a marked bit; the lower bound follows from this degree argument. Linear exact degree follows because the restrictions p(a,0) and p(0,b) would otherwise force incompatible values at their common origin.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-nonempty-two-axes" }, { "id": "promise-unbounded-deg-C-promise-nonempty-two-axes", "A": "deg", "B": "C", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-nonempty-two-axes, deg is unbounded while C=O(1). Thus deg is not O(max(1,C)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Again C=RC=fbs=bs=1 and s=lambda=0. If p approximates the side label on the promise, amplify it on its two promised ranges to error 1/12 using a constant-degree polynomial. Then q(z)=p(0,z)-p(z,0), with sign chosen for the label convention, is zero at z=0 and within 1/6 of 1 for every nonzero z. It approximates total OR, so degree(p)=Omega(sqrt(m)). A bounded-error algorithm searches for a marked bit; the lower bound follows from this degree argument. Linear exact degree follows because the restrictions p(a,0) and p(0,b) would otherwise force incompatible values at their common origin.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-nonempty-two-axes" }, { "id": "promise-unbounded-deg-RC-promise-nonempty-two-axes", "A": "deg", "B": "RC", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-nonempty-two-axes, deg is unbounded while RC=O(1). Thus deg is not O(max(1,RC)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Again C=RC=fbs=bs=1 and s=lambda=0. If p approximates the side label on the promise, amplify it on its two promised ranges to error 1/12 using a constant-degree polynomial. Then q(z)=p(0,z)-p(z,0), with sign chosen for the label convention, is zero at z=0 and within 1/6 of 1 for every nonzero z. It approximates total OR, so degree(p)=Omega(sqrt(m)). A bounded-error algorithm searches for a marked bit; the lower bound follows from this degree argument. Linear exact degree follows because the restrictions p(a,0) and p(0,b) would otherwise force incompatible values at their common origin.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-nonempty-two-axes" }, { "id": "promise-unbounded-deg-bs-promise-nonempty-two-axes", "A": "deg", "B": "bs", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-nonempty-two-axes, deg is unbounded while bs=O(1). Thus deg is not O(max(1,bs)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Again C=RC=fbs=bs=1 and s=lambda=0. If p approximates the side label on the promise, amplify it on its two promised ranges to error 1/12 using a constant-degree polynomial. Then q(z)=p(0,z)-p(z,0), with sign chosen for the label convention, is zero at z=0 and within 1/6 of 1 for every nonzero z. It approximates total OR, so degree(p)=Omega(sqrt(m)). A bounded-error algorithm searches for a marked bit; the lower bound follows from this degree argument. Linear exact degree follows because the restrictions p(a,0) and p(0,b) would otherwise force incompatible values at their common origin.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-nonempty-two-axes" }, { "id": "promise-unbounded-deg-s-promise-nonempty-two-axes", "A": "deg", "B": "s", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-nonempty-two-axes, deg is unbounded while s=O(1). Thus deg is not O(max(1,s)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Again C=RC=fbs=bs=1 and s=lambda=0. If p approximates the side label on the promise, amplify it on its two promised ranges to error 1/12 using a constant-degree polynomial. Then q(z)=p(0,z)-p(z,0), with sign chosen for the label convention, is zero at z=0 and within 1/6 of 1 for every nonzero z. It approximates total OR, so degree(p)=Omega(sqrt(m)). A bounded-error algorithm searches for a marked bit; the lower bound follows from this degree argument. Linear exact degree follows because the restrictions p(a,0) and p(0,b) would otherwise force incompatible values at their common origin.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-nonempty-two-axes" }, { "id": "promise-unbounded-deg-lambda-promise-nonempty-two-axes", "A": "deg", "B": "lambda", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-nonempty-two-axes, deg is unbounded while lambda=O(1). Thus deg is not O(max(1,lambda)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Again C=RC=fbs=bs=1 and s=lambda=0. If p approximates the side label on the promise, amplify it on its two promised ranges to error 1/12 using a constant-degree polynomial. Then q(z)=p(0,z)-p(z,0), with sign chosen for the label convention, is zero at z=0 and within 1/6 of 1 for every nonzero z. It approximates total OR, so degree(p)=Omega(sqrt(m)). A bounded-error algorithm searches for a marked bit; the lower bound follows from this degree argument. Linear exact degree follows because the restrictions p(a,0) and p(0,b) would otherwise force incompatible values at their common origin.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-nonempty-two-axes" }, { "id": "promise-unbounded-Q-C-promise-nonempty-two-axes", "A": "Q", "B": "C", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-nonempty-two-axes, Q is unbounded while C=O(1). Thus Q is not O(max(1,C)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Again C=RC=fbs=bs=1 and s=lambda=0. If p approximates the side label on the promise, amplify it on its two promised ranges to error 1/12 using a constant-degree polynomial. Then q(z)=p(0,z)-p(z,0), with sign chosen for the label convention, is zero at z=0 and within 1/6 of 1 for every nonzero z. It approximates total OR, so degree(p)=Omega(sqrt(m)). A bounded-error algorithm searches for a marked bit; the lower bound follows from this degree argument. Linear exact degree follows because the restrictions p(a,0) and p(0,b) would otherwise force incompatible values at their common origin.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-nonempty-two-axes" }, { "id": "promise-unbounded-Q-RC-promise-nonempty-two-axes", "A": "Q", "B": "RC", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-nonempty-two-axes, Q is unbounded while RC=O(1). Thus Q is not O(max(1,RC)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Again C=RC=fbs=bs=1 and s=lambda=0. If p approximates the side label on the promise, amplify it on its two promised ranges to error 1/12 using a constant-degree polynomial. Then q(z)=p(0,z)-p(z,0), with sign chosen for the label convention, is zero at z=0 and within 1/6 of 1 for every nonzero z. It approximates total OR, so degree(p)=Omega(sqrt(m)). A bounded-error algorithm searches for a marked bit; the lower bound follows from this degree argument. Linear exact degree follows because the restrictions p(a,0) and p(0,b) would otherwise force incompatible values at their common origin.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-nonempty-two-axes" }, { "id": "promise-unbounded-Q-bs-promise-nonempty-two-axes", "A": "Q", "B": "bs", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-nonempty-two-axes, Q is unbounded while bs=O(1). Thus Q is not O(max(1,bs)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Again C=RC=fbs=bs=1 and s=lambda=0. If p approximates the side label on the promise, amplify it on its two promised ranges to error 1/12 using a constant-degree polynomial. Then q(z)=p(0,z)-p(z,0), with sign chosen for the label convention, is zero at z=0 and within 1/6 of 1 for every nonzero z. It approximates total OR, so degree(p)=Omega(sqrt(m)). A bounded-error algorithm searches for a marked bit; the lower bound follows from this degree argument. Linear exact degree follows because the restrictions p(a,0) and p(0,b) would otherwise force incompatible values at their common origin.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-nonempty-two-axes" }, { "id": "promise-unbounded-Q-s-promise-nonempty-two-axes", "A": "Q", "B": "s", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-nonempty-two-axes, Q is unbounded while s=O(1). Thus Q is not O(max(1,s)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Again C=RC=fbs=bs=1 and s=lambda=0. If p approximates the side label on the promise, amplify it on its two promised ranges to error 1/12 using a constant-degree polynomial. Then q(z)=p(0,z)-p(z,0), with sign chosen for the label convention, is zero at z=0 and within 1/6 of 1 for every nonzero z. It approximates total OR, so degree(p)=Omega(sqrt(m)). A bounded-error algorithm searches for a marked bit; the lower bound follows from this degree argument. Linear exact degree follows because the restrictions p(a,0) and p(0,b) would otherwise force incompatible values at their common origin.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-nonempty-two-axes" }, { "id": "promise-unbounded-Q-lambda-promise-nonempty-two-axes", "A": "Q", "B": "lambda", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-nonempty-two-axes, Q is unbounded while lambda=O(1). Thus Q is not O(max(1,lambda)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Again C=RC=fbs=bs=1 and s=lambda=0. If p approximates the side label on the promise, amplify it on its two promised ranges to error 1/12 using a constant-degree polynomial. Then q(z)=p(0,z)-p(z,0), with sign chosen for the label convention, is zero at z=0 and within 1/6 of 1 for every nonzero z. It approximates total OR, so degree(p)=Omega(sqrt(m)). A bounded-error algorithm searches for a marked bit; the lower bound follows from this degree argument. Linear exact degree follows because the restrictions p(a,0) and p(0,b) would otherwise force incompatible values at their common origin.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-nonempty-two-axes" }, { "id": "promise-unbounded-adeg-C-promise-nonempty-two-axes", "A": "adeg", "B": "C", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-nonempty-two-axes, adeg is unbounded while C=O(1). Thus adeg is not O(max(1,C)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Again C=RC=fbs=bs=1 and s=lambda=0. If p approximates the side label on the promise, amplify it on its two promised ranges to error 1/12 using a constant-degree polynomial. Then q(z)=p(0,z)-p(z,0), with sign chosen for the label convention, is zero at z=0 and within 1/6 of 1 for every nonzero z. It approximates total OR, so degree(p)=Omega(sqrt(m)). A bounded-error algorithm searches for a marked bit; the lower bound follows from this degree argument. Linear exact degree follows because the restrictions p(a,0) and p(0,b) would otherwise force incompatible values at their common origin.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-nonempty-two-axes" }, { "id": "promise-unbounded-adeg-RC-promise-nonempty-two-axes", "A": "adeg", "B": "RC", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-nonempty-two-axes, adeg is unbounded while RC=O(1). Thus adeg is not O(max(1,RC)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Again C=RC=fbs=bs=1 and s=lambda=0. If p approximates the side label on the promise, amplify it on its two promised ranges to error 1/12 using a constant-degree polynomial. Then q(z)=p(0,z)-p(z,0), with sign chosen for the label convention, is zero at z=0 and within 1/6 of 1 for every nonzero z. It approximates total OR, so degree(p)=Omega(sqrt(m)). A bounded-error algorithm searches for a marked bit; the lower bound follows from this degree argument. Linear exact degree follows because the restrictions p(a,0) and p(0,b) would otherwise force incompatible values at their common origin.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-nonempty-two-axes" }, { "id": "promise-unbounded-adeg-bs-promise-nonempty-two-axes", "A": "adeg", "B": "bs", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-nonempty-two-axes, adeg is unbounded while bs=O(1). Thus adeg is not O(max(1,bs)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Again C=RC=fbs=bs=1 and s=lambda=0. If p approximates the side label on the promise, amplify it on its two promised ranges to error 1/12 using a constant-degree polynomial. Then q(z)=p(0,z)-p(z,0), with sign chosen for the label convention, is zero at z=0 and within 1/6 of 1 for every nonzero z. It approximates total OR, so degree(p)=Omega(sqrt(m)). A bounded-error algorithm searches for a marked bit; the lower bound follows from this degree argument. Linear exact degree follows because the restrictions p(a,0) and p(0,b) would otherwise force incompatible values at their common origin.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-nonempty-two-axes" }, { "id": "promise-unbounded-adeg-s-promise-nonempty-two-axes", "A": "adeg", "B": "s", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-nonempty-two-axes, adeg is unbounded while s=O(1). Thus adeg is not O(max(1,s)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Again C=RC=fbs=bs=1 and s=lambda=0. If p approximates the side label on the promise, amplify it on its two promised ranges to error 1/12 using a constant-degree polynomial. Then q(z)=p(0,z)-p(z,0), with sign chosen for the label convention, is zero at z=0 and within 1/6 of 1 for every nonzero z. It approximates total OR, so degree(p)=Omega(sqrt(m)). A bounded-error algorithm searches for a marked bit; the lower bound follows from this degree argument. Linear exact degree follows because the restrictions p(a,0) and p(0,b) would otherwise force incompatible values at their common origin.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-nonempty-two-axes" }, { "id": "promise-unbounded-adeg-lambda-promise-nonempty-two-axes", "A": "adeg", "B": "lambda", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-nonempty-two-axes, adeg is unbounded while lambda=O(1). Thus adeg is not O(max(1,lambda)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "Again C=RC=fbs=bs=1 and s=lambda=0. If p approximates the side label on the promise, amplify it on its two promised ranges to error 1/12 using a constant-degree polynomial. Then q(z)=p(0,z)-p(z,0), with sign chosen for the label convention, is zero at z=0 and within 1/6 of 1 for every nonzero z. It approximates total OR, so degree(p)=Omega(sqrt(m)). A bounded-error algorithm searches for a marked bit; the lower bound follows from this degree argument. Linear exact degree follows because the restrictions p(a,0) and p(0,b) would otherwise force incompatible values at their common origin.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-nonempty-two-axes" }, { "id": "promise-unbounded-D-bs-promise-projective-plane", "A": "D", "B": "bs", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-projective-plane, D is unbounded while bs=O(1). Thus D is not O(max(1,bs)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 17 with k=1; LP/verifier and fixed-line consequences derived here", "proof_summary": "Any two lines intersect, so bs=1. Put weight 1/(t+1) on every line to obtain fbs=Theta(t), hence RC=Theta(t). A fixed line meets all lines, so querying its t+1 points decides the problem, and yields C,D,R,R0=O(t); C,RC lower bounds give matching growth. Lines have weight t+1, so unrestricted degree is one. Promised opposite pairs are never adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-projective-plane" }, { "id": "promise-unbounded-D-s-promise-projective-plane", "A": "D", "B": "s", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-projective-plane, D is unbounded while s=O(1). Thus D is not O(max(1,s)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 17 with k=1; LP/verifier and fixed-line consequences derived here", "proof_summary": "Any two lines intersect, so bs=1. Put weight 1/(t+1) on every line to obtain fbs=Theta(t), hence RC=Theta(t). A fixed line meets all lines, so querying its t+1 points decides the problem, and yields C,D,R,R0=O(t); C,RC lower bounds give matching growth. Lines have weight t+1, so unrestricted degree is one. Promised opposite pairs are never adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-projective-plane" }, { "id": "promise-unbounded-D-lambda-promise-projective-plane", "A": "D", "B": "lambda", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-projective-plane, D is unbounded while lambda=O(1). Thus D is not O(max(1,lambda)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 17 with k=1; LP/verifier and fixed-line consequences derived here", "proof_summary": "Any two lines intersect, so bs=1. Put weight 1/(t+1) on every line to obtain fbs=Theta(t), hence RC=Theta(t). A fixed line meets all lines, so querying its t+1 points decides the problem, and yields C,D,R,R0=O(t); C,RC lower bounds give matching growth. Lines have weight t+1, so unrestricted degree is one. Promised opposite pairs are never adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-projective-plane" }, { "id": "promise-unbounded-D-deg-promise-projective-plane", "A": "D", "B": "deg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-projective-plane, D is unbounded while deg=O(1). Thus D is not O(max(1,deg)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 17 with k=1; LP/verifier and fixed-line consequences derived here", "proof_summary": "Any two lines intersect, so bs=1. Put weight 1/(t+1) on every line to obtain fbs=Theta(t), hence RC=Theta(t). A fixed line meets all lines, so querying its t+1 points decides the problem, and yields C,D,R,R0=O(t); C,RC lower bounds give matching growth. Lines have weight t+1, so unrestricted degree is one. Promised opposite pairs are never adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-projective-plane" }, { "id": "promise-unbounded-D-adeg-promise-projective-plane", "A": "D", "B": "adeg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-projective-plane, D is unbounded while adeg=O(1). Thus D is not O(max(1,adeg)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 17 with k=1; LP/verifier and fixed-line consequences derived here", "proof_summary": "Any two lines intersect, so bs=1. Put weight 1/(t+1) on every line to obtain fbs=Theta(t), hence RC=Theta(t). A fixed line meets all lines, so querying its t+1 points decides the problem, and yields C,D,R,R0=O(t); C,RC lower bounds give matching growth. Lines have weight t+1, so unrestricted degree is one. Promised opposite pairs are never adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-projective-plane" }, { "id": "promise-unbounded-R0-bs-promise-projective-plane", "A": "R0", "B": "bs", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-projective-plane, R0 is unbounded while bs=O(1). Thus R0 is not O(max(1,bs)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 17 with k=1; LP/verifier and fixed-line consequences derived here", "proof_summary": "Any two lines intersect, so bs=1. Put weight 1/(t+1) on every line to obtain fbs=Theta(t), hence RC=Theta(t). A fixed line meets all lines, so querying its t+1 points decides the problem, and yields C,D,R,R0=O(t); C,RC lower bounds give matching growth. Lines have weight t+1, so unrestricted degree is one. Promised opposite pairs are never adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-projective-plane" }, { "id": "promise-unbounded-R0-s-promise-projective-plane", "A": "R0", "B": "s", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-projective-plane, R0 is unbounded while s=O(1). Thus R0 is not O(max(1,s)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 17 with k=1; LP/verifier and fixed-line consequences derived here", "proof_summary": "Any two lines intersect, so bs=1. Put weight 1/(t+1) on every line to obtain fbs=Theta(t), hence RC=Theta(t). A fixed line meets all lines, so querying its t+1 points decides the problem, and yields C,D,R,R0=O(t); C,RC lower bounds give matching growth. Lines have weight t+1, so unrestricted degree is one. Promised opposite pairs are never adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-projective-plane" }, { "id": "promise-unbounded-R0-lambda-promise-projective-plane", "A": "R0", "B": "lambda", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-projective-plane, R0 is unbounded while lambda=O(1). Thus R0 is not O(max(1,lambda)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 17 with k=1; LP/verifier and fixed-line consequences derived here", "proof_summary": "Any two lines intersect, so bs=1. Put weight 1/(t+1) on every line to obtain fbs=Theta(t), hence RC=Theta(t). A fixed line meets all lines, so querying its t+1 points decides the problem, and yields C,D,R,R0=O(t); C,RC lower bounds give matching growth. Lines have weight t+1, so unrestricted degree is one. Promised opposite pairs are never adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-projective-plane" }, { "id": "promise-unbounded-R0-deg-promise-projective-plane", "A": "R0", "B": "deg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-projective-plane, R0 is unbounded while deg=O(1). Thus R0 is not O(max(1,deg)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 17 with k=1; LP/verifier and fixed-line consequences derived here", "proof_summary": "Any two lines intersect, so bs=1. Put weight 1/(t+1) on every line to obtain fbs=Theta(t), hence RC=Theta(t). A fixed line meets all lines, so querying its t+1 points decides the problem, and yields C,D,R,R0=O(t); C,RC lower bounds give matching growth. Lines have weight t+1, so unrestricted degree is one. Promised opposite pairs are never adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-projective-plane" }, { "id": "promise-unbounded-R0-adeg-promise-projective-plane", "A": "R0", "B": "adeg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-projective-plane, R0 is unbounded while adeg=O(1). Thus R0 is not O(max(1,adeg)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 17 with k=1; LP/verifier and fixed-line consequences derived here", "proof_summary": "Any two lines intersect, so bs=1. Put weight 1/(t+1) on every line to obtain fbs=Theta(t), hence RC=Theta(t). A fixed line meets all lines, so querying its t+1 points decides the problem, and yields C,D,R,R0=O(t); C,RC lower bounds give matching growth. Lines have weight t+1, so unrestricted degree is one. Promised opposite pairs are never adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-projective-plane" }, { "id": "promise-unbounded-R-bs-promise-projective-plane", "A": "R", "B": "bs", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-projective-plane, R is unbounded while bs=O(1). Thus R is not O(max(1,bs)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 17 with k=1; LP/verifier and fixed-line consequences derived here", "proof_summary": "Any two lines intersect, so bs=1. Put weight 1/(t+1) on every line to obtain fbs=Theta(t), hence RC=Theta(t). A fixed line meets all lines, so querying its t+1 points decides the problem, and yields C,D,R,R0=O(t); C,RC lower bounds give matching growth. Lines have weight t+1, so unrestricted degree is one. Promised opposite pairs are never adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-projective-plane" }, { "id": "promise-unbounded-R-s-promise-projective-plane", "A": "R", "B": "s", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-projective-plane, R is unbounded while s=O(1). Thus R is not O(max(1,s)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 17 with k=1; LP/verifier and fixed-line consequences derived here", "proof_summary": "Any two lines intersect, so bs=1. Put weight 1/(t+1) on every line to obtain fbs=Theta(t), hence RC=Theta(t). A fixed line meets all lines, so querying its t+1 points decides the problem, and yields C,D,R,R0=O(t); C,RC lower bounds give matching growth. Lines have weight t+1, so unrestricted degree is one. Promised opposite pairs are never adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-projective-plane" }, { "id": "promise-unbounded-R-lambda-promise-projective-plane", "A": "R", "B": "lambda", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-projective-plane, R is unbounded while lambda=O(1). Thus R is not O(max(1,lambda)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 17 with k=1; LP/verifier and fixed-line consequences derived here", "proof_summary": "Any two lines intersect, so bs=1. Put weight 1/(t+1) on every line to obtain fbs=Theta(t), hence RC=Theta(t). A fixed line meets all lines, so querying its t+1 points decides the problem, and yields C,D,R,R0=O(t); C,RC lower bounds give matching growth. Lines have weight t+1, so unrestricted degree is one. Promised opposite pairs are never adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-projective-plane" }, { "id": "promise-unbounded-R-deg-promise-projective-plane", "A": "R", "B": "deg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-projective-plane, R is unbounded while deg=O(1). Thus R is not O(max(1,deg)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 17 with k=1; LP/verifier and fixed-line consequences derived here", "proof_summary": "Any two lines intersect, so bs=1. Put weight 1/(t+1) on every line to obtain fbs=Theta(t), hence RC=Theta(t). A fixed line meets all lines, so querying its t+1 points decides the problem, and yields C,D,R,R0=O(t); C,RC lower bounds give matching growth. Lines have weight t+1, so unrestricted degree is one. Promised opposite pairs are never adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-projective-plane" }, { "id": "promise-unbounded-R-adeg-promise-projective-plane", "A": "R", "B": "adeg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-projective-plane, R is unbounded while adeg=O(1). Thus R is not O(max(1,adeg)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 17 with k=1; LP/verifier and fixed-line consequences derived here", "proof_summary": "Any two lines intersect, so bs=1. Put weight 1/(t+1) on every line to obtain fbs=Theta(t), hence RC=Theta(t). A fixed line meets all lines, so querying its t+1 points decides the problem, and yields C,D,R,R0=O(t); C,RC lower bounds give matching growth. Lines have weight t+1, so unrestricted degree is one. Promised opposite pairs are never adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-projective-plane" }, { "id": "promise-unbounded-C-bs-promise-projective-plane", "A": "C", "B": "bs", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-projective-plane, C is unbounded while bs=O(1). Thus C is not O(max(1,bs)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 17 with k=1; LP/verifier and fixed-line consequences derived here", "proof_summary": "Any two lines intersect, so bs=1. Put weight 1/(t+1) on every line to obtain fbs=Theta(t), hence RC=Theta(t). A fixed line meets all lines, so querying its t+1 points decides the problem, and yields C,D,R,R0=O(t); C,RC lower bounds give matching growth. Lines have weight t+1, so unrestricted degree is one. Promised opposite pairs are never adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-projective-plane" }, { "id": "promise-unbounded-C-s-promise-projective-plane", "A": "C", "B": "s", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-projective-plane, C is unbounded while s=O(1). Thus C is not O(max(1,s)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 17 with k=1; LP/verifier and fixed-line consequences derived here", "proof_summary": "Any two lines intersect, so bs=1. Put weight 1/(t+1) on every line to obtain fbs=Theta(t), hence RC=Theta(t). A fixed line meets all lines, so querying its t+1 points decides the problem, and yields C,D,R,R0=O(t); C,RC lower bounds give matching growth. Lines have weight t+1, so unrestricted degree is one. Promised opposite pairs are never adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-projective-plane" }, { "id": "promise-unbounded-C-lambda-promise-projective-plane", "A": "C", "B": "lambda", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-projective-plane, C is unbounded while lambda=O(1). Thus C is not O(max(1,lambda)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 17 with k=1; LP/verifier and fixed-line consequences derived here", "proof_summary": "Any two lines intersect, so bs=1. Put weight 1/(t+1) on every line to obtain fbs=Theta(t), hence RC=Theta(t). A fixed line meets all lines, so querying its t+1 points decides the problem, and yields C,D,R,R0=O(t); C,RC lower bounds give matching growth. Lines have weight t+1, so unrestricted degree is one. Promised opposite pairs are never adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-projective-plane" }, { "id": "promise-unbounded-C-deg-promise-projective-plane", "A": "C", "B": "deg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-projective-plane, C is unbounded while deg=O(1). Thus C is not O(max(1,deg)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 17 with k=1; LP/verifier and fixed-line consequences derived here", "proof_summary": "Any two lines intersect, so bs=1. Put weight 1/(t+1) on every line to obtain fbs=Theta(t), hence RC=Theta(t). A fixed line meets all lines, so querying its t+1 points decides the problem, and yields C,D,R,R0=O(t); C,RC lower bounds give matching growth. Lines have weight t+1, so unrestricted degree is one. Promised opposite pairs are never adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-projective-plane" }, { "id": "promise-unbounded-C-adeg-promise-projective-plane", "A": "C", "B": "adeg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-projective-plane, C is unbounded while adeg=O(1). Thus C is not O(max(1,adeg)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 17 with k=1; LP/verifier and fixed-line consequences derived here", "proof_summary": "Any two lines intersect, so bs=1. Put weight 1/(t+1) on every line to obtain fbs=Theta(t), hence RC=Theta(t). A fixed line meets all lines, so querying its t+1 points decides the problem, and yields C,D,R,R0=O(t); C,RC lower bounds give matching growth. Lines have weight t+1, so unrestricted degree is one. Promised opposite pairs are never adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-projective-plane" }, { "id": "promise-unbounded-RC-bs-promise-projective-plane", "A": "RC", "B": "bs", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-projective-plane, RC is unbounded while bs=O(1). Thus RC is not O(max(1,bs)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 17 with k=1; LP/verifier and fixed-line consequences derived here", "proof_summary": "Any two lines intersect, so bs=1. Put weight 1/(t+1) on every line to obtain fbs=Theta(t), hence RC=Theta(t). A fixed line meets all lines, so querying its t+1 points decides the problem, and yields C,D,R,R0=O(t); C,RC lower bounds give matching growth. Lines have weight t+1, so unrestricted degree is one. Promised opposite pairs are never adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-projective-plane" }, { "id": "promise-unbounded-RC-s-promise-projective-plane", "A": "RC", "B": "s", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-projective-plane, RC is unbounded while s=O(1). Thus RC is not O(max(1,s)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 17 with k=1; LP/verifier and fixed-line consequences derived here", "proof_summary": "Any two lines intersect, so bs=1. Put weight 1/(t+1) on every line to obtain fbs=Theta(t), hence RC=Theta(t). A fixed line meets all lines, so querying its t+1 points decides the problem, and yields C,D,R,R0=O(t); C,RC lower bounds give matching growth. Lines have weight t+1, so unrestricted degree is one. Promised opposite pairs are never adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-projective-plane" }, { "id": "promise-unbounded-RC-lambda-promise-projective-plane", "A": "RC", "B": "lambda", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-projective-plane, RC is unbounded while lambda=O(1). Thus RC is not O(max(1,lambda)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 17 with k=1; LP/verifier and fixed-line consequences derived here", "proof_summary": "Any two lines intersect, so bs=1. Put weight 1/(t+1) on every line to obtain fbs=Theta(t), hence RC=Theta(t). A fixed line meets all lines, so querying its t+1 points decides the problem, and yields C,D,R,R0=O(t); C,RC lower bounds give matching growth. Lines have weight t+1, so unrestricted degree is one. Promised opposite pairs are never adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-projective-plane" }, { "id": "promise-unbounded-RC-deg-promise-projective-plane", "A": "RC", "B": "deg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-projective-plane, RC is unbounded while deg=O(1). Thus RC is not O(max(1,deg)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 17 with k=1; LP/verifier and fixed-line consequences derived here", "proof_summary": "Any two lines intersect, so bs=1. Put weight 1/(t+1) on every line to obtain fbs=Theta(t), hence RC=Theta(t). A fixed line meets all lines, so querying its t+1 points decides the problem, and yields C,D,R,R0=O(t); C,RC lower bounds give matching growth. Lines have weight t+1, so unrestricted degree is one. Promised opposite pairs are never adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-projective-plane" }, { "id": "promise-unbounded-RC-adeg-promise-projective-plane", "A": "RC", "B": "adeg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-projective-plane, RC is unbounded while adeg=O(1). Thus RC is not O(max(1,adeg)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 17 with k=1; LP/verifier and fixed-line consequences derived here", "proof_summary": "Any two lines intersect, so bs=1. Put weight 1/(t+1) on every line to obtain fbs=Theta(t), hence RC=Theta(t). A fixed line meets all lines, so querying its t+1 points decides the problem, and yields C,D,R,R0=O(t); C,RC lower bounds give matching growth. Lines have weight t+1, so unrestricted degree is one. Promised opposite pairs are never adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-projective-plane" }, { "id": "promise-unbounded-R-QE-promise-extremal-forrelation", "A": "R", "B": "QE", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-extremal-forrelation, R is unbounded while QE=O(1). Thus R is not O(max(1,QE)^p) for any fixed finite p over partial Boolean functions.", "source": "GS26", "locator": "Theorem 4", "proof_summary": "Use GS26 Theorem 4. Exact quantum cost is constant in standard bit-oracle conventions, while R=tildeOmega(m^(1/4)). The logarithmic loss does not affect the unboundedness consequence used here.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-extremal-forrelation" }, { "id": "promise-unbounded-R-Q-promise-extremal-forrelation", "A": "R", "B": "Q", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-extremal-forrelation, R is unbounded while Q=O(1). Thus R is not O(max(1,Q)^p) for any fixed finite p over partial Boolean functions.", "source": "GS26", "locator": "Theorem 4", "proof_summary": "Use GS26 Theorem 4. Exact quantum cost is constant in standard bit-oracle conventions, while R=tildeOmega(m^(1/4)). The logarithmic loss does not affect the unboundedness consequence used here.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-extremal-forrelation" }, { "id": "promise-unbounded-R-deg-promise-extremal-forrelation", "A": "R", "B": "deg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-extremal-forrelation, R is unbounded while deg=O(1). Thus R is not O(max(1,deg)^p) for any fixed finite p over partial Boolean functions.", "source": "GS26", "locator": "Theorem 4", "proof_summary": "Use GS26 Theorem 4. Exact quantum cost is constant in standard bit-oracle conventions, while R=tildeOmega(m^(1/4)). The logarithmic loss does not affect the unboundedness consequence used here.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-extremal-forrelation" }, { "id": "promise-unbounded-R-adeg-promise-extremal-forrelation", "A": "R", "B": "adeg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-extremal-forrelation, R is unbounded while adeg=O(1). Thus R is not O(max(1,adeg)^p) for any fixed finite p over partial Boolean functions.", "source": "GS26", "locator": "Theorem 4", "proof_summary": "Use GS26 Theorem 4. Exact quantum cost is constant in standard bit-oracle conventions, while R=tildeOmega(m^(1/4)). The logarithmic loss does not affect the unboundedness consequence used here.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-extremal-forrelation" }, { "id": "promise-unbounded-R0-QE-promise-extremal-forrelation", "A": "R0", "B": "QE", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-extremal-forrelation, R0 is unbounded while QE=O(1). Thus R0 is not O(max(1,QE)^p) for any fixed finite p over partial Boolean functions.", "source": "GS26", "locator": "Theorem 4", "proof_summary": "Use GS26 Theorem 4. Exact quantum cost is constant in standard bit-oracle conventions, while R=tildeOmega(m^(1/4)). The logarithmic loss does not affect the unboundedness consequence used here.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-extremal-forrelation" }, { "id": "promise-unbounded-R0-Q-promise-extremal-forrelation", "A": "R0", "B": "Q", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-extremal-forrelation, R0 is unbounded while Q=O(1). Thus R0 is not O(max(1,Q)^p) for any fixed finite p over partial Boolean functions.", "source": "GS26", "locator": "Theorem 4", "proof_summary": "Use GS26 Theorem 4. Exact quantum cost is constant in standard bit-oracle conventions, while R=tildeOmega(m^(1/4)). The logarithmic loss does not affect the unboundedness consequence used here.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-extremal-forrelation" }, { "id": "promise-unbounded-R0-deg-promise-extremal-forrelation", "A": "R0", "B": "deg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-extremal-forrelation, R0 is unbounded while deg=O(1). Thus R0 is not O(max(1,deg)^p) for any fixed finite p over partial Boolean functions.", "source": "GS26", "locator": "Theorem 4", "proof_summary": "Use GS26 Theorem 4. Exact quantum cost is constant in standard bit-oracle conventions, while R=tildeOmega(m^(1/4)). The logarithmic loss does not affect the unboundedness consequence used here.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-extremal-forrelation" }, { "id": "promise-unbounded-R0-adeg-promise-extremal-forrelation", "A": "R0", "B": "adeg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-extremal-forrelation, R0 is unbounded while adeg=O(1). Thus R0 is not O(max(1,adeg)^p) for any fixed finite p over partial Boolean functions.", "source": "GS26", "locator": "Theorem 4", "proof_summary": "Use GS26 Theorem 4. Exact quantum cost is constant in standard bit-oracle conventions, while R=tildeOmega(m^(1/4)). The logarithmic loss does not affect the unboundedness consequence used here.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-extremal-forrelation" }, { "id": "promise-unbounded-D-QE-promise-extremal-forrelation", "A": "D", "B": "QE", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-extremal-forrelation, D is unbounded while QE=O(1). Thus D is not O(max(1,QE)^p) for any fixed finite p over partial Boolean functions.", "source": "GS26", "locator": "Theorem 4", "proof_summary": "Use GS26 Theorem 4. Exact quantum cost is constant in standard bit-oracle conventions, while R=tildeOmega(m^(1/4)). The logarithmic loss does not affect the unboundedness consequence used here.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-extremal-forrelation" }, { "id": "promise-unbounded-D-Q-promise-extremal-forrelation", "A": "D", "B": "Q", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-extremal-forrelation, D is unbounded while Q=O(1). Thus D is not O(max(1,Q)^p) for any fixed finite p over partial Boolean functions.", "source": "GS26", "locator": "Theorem 4", "proof_summary": "Use GS26 Theorem 4. Exact quantum cost is constant in standard bit-oracle conventions, while R=tildeOmega(m^(1/4)). The logarithmic loss does not affect the unboundedness consequence used here.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-extremal-forrelation" }, { "id": "promise-unbounded-D-deg-promise-extremal-forrelation", "A": "D", "B": "deg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-extremal-forrelation, D is unbounded while deg=O(1). Thus D is not O(max(1,deg)^p) for any fixed finite p over partial Boolean functions.", "source": "GS26", "locator": "Theorem 4", "proof_summary": "Use GS26 Theorem 4. Exact quantum cost is constant in standard bit-oracle conventions, while R=tildeOmega(m^(1/4)). The logarithmic loss does not affect the unboundedness consequence used here.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-extremal-forrelation" }, { "id": "promise-unbounded-D-adeg-promise-extremal-forrelation", "A": "D", "B": "adeg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-extremal-forrelation, D is unbounded while adeg=O(1). Thus D is not O(max(1,adeg)^p) for any fixed finite p over partial Boolean functions.", "source": "GS26", "locator": "Theorem 4", "proof_summary": "Use GS26 Theorem 4. Exact quantum cost is constant in standard bit-oracle conventions, while R=tildeOmega(m^(1/4)). The logarithmic loss does not affect the unboundedness consequence used here.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-extremal-forrelation" }, { "id": "promise-unbounded-D-R-promise-restricted-DJ", "A": "D", "B": "R", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-restricted-DJ, D is unbounded while R=O(1). Thus D is not O(max(1,R)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "The phase-query algorithm prepares the uniform state, applies phases (-1)^x_i and tests overlap with the uniform state. Its amplitude is 1 at zero and 0 on balanced inputs, hence QE=1. A weak certificate for zero needs m/2+1 zero positions, so C,D,R0=Theta(m). Random sampling gives R,RC=O(1); at zero exactly two disjoint half-size sensitive blocks can be packed, so bs=2. The promise has no opposite-output adjacent pair for m>=4. The unrestricted polynomial 2 sum_i x_i/m is exact of degree one.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument. The one-query circuit here is explicitly derived; DJ92 is the historical problem citation, not an audited original one-query attribution.", "witness_id": "promise-restricted-DJ", "proof_sources": [ "DJ92" ] }, { "id": "promise-unbounded-D-RC-promise-restricted-DJ", "A": "D", "B": "RC", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-restricted-DJ, D is unbounded while RC=O(1). Thus D is not O(max(1,RC)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "The phase-query algorithm prepares the uniform state, applies phases (-1)^x_i and tests overlap with the uniform state. Its amplitude is 1 at zero and 0 on balanced inputs, hence QE=1. A weak certificate for zero needs m/2+1 zero positions, so C,D,R0=Theta(m). Random sampling gives R,RC=O(1); at zero exactly two disjoint half-size sensitive blocks can be packed, so bs=2. The promise has no opposite-output adjacent pair for m>=4. The unrestricted polynomial 2 sum_i x_i/m is exact of degree one.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument. The one-query circuit here is explicitly derived; DJ92 is the historical problem citation, not an audited original one-query attribution.", "witness_id": "promise-restricted-DJ", "proof_sources": [ "DJ92" ] }, { "id": "promise-unbounded-D-bs-promise-restricted-DJ", "A": "D", "B": "bs", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-restricted-DJ, D is unbounded while bs=O(1). Thus D is not O(max(1,bs)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "The phase-query algorithm prepares the uniform state, applies phases (-1)^x_i and tests overlap with the uniform state. Its amplitude is 1 at zero and 0 on balanced inputs, hence QE=1. A weak certificate for zero needs m/2+1 zero positions, so C,D,R0=Theta(m). Random sampling gives R,RC=O(1); at zero exactly two disjoint half-size sensitive blocks can be packed, so bs=2. The promise has no opposite-output adjacent pair for m>=4. The unrestricted polynomial 2 sum_i x_i/m is exact of degree one.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument. The one-query circuit here is explicitly derived; DJ92 is the historical problem citation, not an audited original one-query attribution.", "witness_id": "promise-restricted-DJ", "proof_sources": [ "DJ92" ] }, { "id": "promise-unbounded-D-s-promise-restricted-DJ", "A": "D", "B": "s", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-restricted-DJ, D is unbounded while s=O(1). Thus D is not O(max(1,s)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "The phase-query algorithm prepares the uniform state, applies phases (-1)^x_i and tests overlap with the uniform state. Its amplitude is 1 at zero and 0 on balanced inputs, hence QE=1. A weak certificate for zero needs m/2+1 zero positions, so C,D,R0=Theta(m). Random sampling gives R,RC=O(1); at zero exactly two disjoint half-size sensitive blocks can be packed, so bs=2. The promise has no opposite-output adjacent pair for m>=4. The unrestricted polynomial 2 sum_i x_i/m is exact of degree one.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument. The one-query circuit here is explicitly derived; DJ92 is the historical problem citation, not an audited original one-query attribution.", "witness_id": "promise-restricted-DJ", "proof_sources": [ "DJ92" ] }, { "id": "promise-unbounded-D-lambda-promise-restricted-DJ", "A": "D", "B": "lambda", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-restricted-DJ, D is unbounded while lambda=O(1). Thus D is not O(max(1,lambda)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "The phase-query algorithm prepares the uniform state, applies phases (-1)^x_i and tests overlap with the uniform state. Its amplitude is 1 at zero and 0 on balanced inputs, hence QE=1. A weak certificate for zero needs m/2+1 zero positions, so C,D,R0=Theta(m). Random sampling gives R,RC=O(1); at zero exactly two disjoint half-size sensitive blocks can be packed, so bs=2. The promise has no opposite-output adjacent pair for m>=4. The unrestricted polynomial 2 sum_i x_i/m is exact of degree one.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument. The one-query circuit here is explicitly derived; DJ92 is the historical problem citation, not an audited original one-query attribution.", "witness_id": "promise-restricted-DJ", "proof_sources": [ "DJ92" ] }, { "id": "promise-unbounded-D-Q-promise-restricted-DJ", "A": "D", "B": "Q", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-restricted-DJ, D is unbounded while Q=O(1). Thus D is not O(max(1,Q)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "The phase-query algorithm prepares the uniform state, applies phases (-1)^x_i and tests overlap with the uniform state. Its amplitude is 1 at zero and 0 on balanced inputs, hence QE=1. A weak certificate for zero needs m/2+1 zero positions, so C,D,R0=Theta(m). Random sampling gives R,RC=O(1); at zero exactly two disjoint half-size sensitive blocks can be packed, so bs=2. The promise has no opposite-output adjacent pair for m>=4. The unrestricted polynomial 2 sum_i x_i/m is exact of degree one.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument. The one-query circuit here is explicitly derived; DJ92 is the historical problem citation, not an audited original one-query attribution.", "witness_id": "promise-restricted-DJ", "proof_sources": [ "DJ92" ] }, { "id": "promise-unbounded-D-QE-promise-restricted-DJ", "A": "D", "B": "QE", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-restricted-DJ, D is unbounded while QE=O(1). Thus D is not O(max(1,QE)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "The phase-query algorithm prepares the uniform state, applies phases (-1)^x_i and tests overlap with the uniform state. Its amplitude is 1 at zero and 0 on balanced inputs, hence QE=1. A weak certificate for zero needs m/2+1 zero positions, so C,D,R0=Theta(m). Random sampling gives R,RC=O(1); at zero exactly two disjoint half-size sensitive blocks can be packed, so bs=2. The promise has no opposite-output adjacent pair for m>=4. The unrestricted polynomial 2 sum_i x_i/m is exact of degree one.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument. The one-query circuit here is explicitly derived; DJ92 is the historical problem citation, not an audited original one-query attribution.", "witness_id": "promise-restricted-DJ", "proof_sources": [ "DJ92" ] }, { "id": "promise-unbounded-D-deg-promise-restricted-DJ", "A": "D", "B": "deg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-restricted-DJ, D is unbounded while deg=O(1). Thus D is not O(max(1,deg)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "The phase-query algorithm prepares the uniform state, applies phases (-1)^x_i and tests overlap with the uniform state. Its amplitude is 1 at zero and 0 on balanced inputs, hence QE=1. A weak certificate for zero needs m/2+1 zero positions, so C,D,R0=Theta(m). Random sampling gives R,RC=O(1); at zero exactly two disjoint half-size sensitive blocks can be packed, so bs=2. The promise has no opposite-output adjacent pair for m>=4. The unrestricted polynomial 2 sum_i x_i/m is exact of degree one.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument. The one-query circuit here is explicitly derived; DJ92 is the historical problem citation, not an audited original one-query attribution.", "witness_id": "promise-restricted-DJ", "proof_sources": [ "DJ92" ] }, { "id": "promise-unbounded-D-adeg-promise-restricted-DJ", "A": "D", "B": "adeg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-restricted-DJ, D is unbounded while adeg=O(1). Thus D is not O(max(1,adeg)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "The phase-query algorithm prepares the uniform state, applies phases (-1)^x_i and tests overlap with the uniform state. Its amplitude is 1 at zero and 0 on balanced inputs, hence QE=1. A weak certificate for zero needs m/2+1 zero positions, so C,D,R0=Theta(m). Random sampling gives R,RC=O(1); at zero exactly two disjoint half-size sensitive blocks can be packed, so bs=2. The promise has no opposite-output adjacent pair for m>=4. The unrestricted polynomial 2 sum_i x_i/m is exact of degree one.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument. The one-query circuit here is explicitly derived; DJ92 is the historical problem citation, not an audited original one-query attribution.", "witness_id": "promise-restricted-DJ", "proof_sources": [ "DJ92" ] }, { "id": "promise-unbounded-R0-R-promise-restricted-DJ", "A": "R0", "B": "R", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-restricted-DJ, R0 is unbounded while R=O(1). Thus R0 is not O(max(1,R)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "The phase-query algorithm prepares the uniform state, applies phases (-1)^x_i and tests overlap with the uniform state. Its amplitude is 1 at zero and 0 on balanced inputs, hence QE=1. A weak certificate for zero needs m/2+1 zero positions, so C,D,R0=Theta(m). Random sampling gives R,RC=O(1); at zero exactly two disjoint half-size sensitive blocks can be packed, so bs=2. The promise has no opposite-output adjacent pair for m>=4. The unrestricted polynomial 2 sum_i x_i/m is exact of degree one.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument. The one-query circuit here is explicitly derived; DJ92 is the historical problem citation, not an audited original one-query attribution.", "witness_id": "promise-restricted-DJ", "proof_sources": [ "DJ92" ] }, { "id": "promise-unbounded-R0-RC-promise-restricted-DJ", "A": "R0", "B": "RC", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-restricted-DJ, R0 is unbounded while RC=O(1). Thus R0 is not O(max(1,RC)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "The phase-query algorithm prepares the uniform state, applies phases (-1)^x_i and tests overlap with the uniform state. Its amplitude is 1 at zero and 0 on balanced inputs, hence QE=1. A weak certificate for zero needs m/2+1 zero positions, so C,D,R0=Theta(m). Random sampling gives R,RC=O(1); at zero exactly two disjoint half-size sensitive blocks can be packed, so bs=2. The promise has no opposite-output adjacent pair for m>=4. The unrestricted polynomial 2 sum_i x_i/m is exact of degree one.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument. The one-query circuit here is explicitly derived; DJ92 is the historical problem citation, not an audited original one-query attribution.", "witness_id": "promise-restricted-DJ", "proof_sources": [ "DJ92" ] }, { "id": "promise-unbounded-R0-bs-promise-restricted-DJ", "A": "R0", "B": "bs", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-restricted-DJ, R0 is unbounded while bs=O(1). Thus R0 is not O(max(1,bs)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "The phase-query algorithm prepares the uniform state, applies phases (-1)^x_i and tests overlap with the uniform state. Its amplitude is 1 at zero and 0 on balanced inputs, hence QE=1. A weak certificate for zero needs m/2+1 zero positions, so C,D,R0=Theta(m). Random sampling gives R,RC=O(1); at zero exactly two disjoint half-size sensitive blocks can be packed, so bs=2. The promise has no opposite-output adjacent pair for m>=4. The unrestricted polynomial 2 sum_i x_i/m is exact of degree one.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument. The one-query circuit here is explicitly derived; DJ92 is the historical problem citation, not an audited original one-query attribution.", "witness_id": "promise-restricted-DJ", "proof_sources": [ "DJ92" ] }, { "id": "promise-unbounded-R0-s-promise-restricted-DJ", "A": "R0", "B": "s", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-restricted-DJ, R0 is unbounded while s=O(1). Thus R0 is not O(max(1,s)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "The phase-query algorithm prepares the uniform state, applies phases (-1)^x_i and tests overlap with the uniform state. Its amplitude is 1 at zero and 0 on balanced inputs, hence QE=1. A weak certificate for zero needs m/2+1 zero positions, so C,D,R0=Theta(m). Random sampling gives R,RC=O(1); at zero exactly two disjoint half-size sensitive blocks can be packed, so bs=2. The promise has no opposite-output adjacent pair for m>=4. The unrestricted polynomial 2 sum_i x_i/m is exact of degree one.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument. The one-query circuit here is explicitly derived; DJ92 is the historical problem citation, not an audited original one-query attribution.", "witness_id": "promise-restricted-DJ", "proof_sources": [ "DJ92" ] }, { "id": "promise-unbounded-R0-lambda-promise-restricted-DJ", "A": "R0", "B": "lambda", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-restricted-DJ, R0 is unbounded while lambda=O(1). Thus R0 is not O(max(1,lambda)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "The phase-query algorithm prepares the uniform state, applies phases (-1)^x_i and tests overlap with the uniform state. Its amplitude is 1 at zero and 0 on balanced inputs, hence QE=1. A weak certificate for zero needs m/2+1 zero positions, so C,D,R0=Theta(m). Random sampling gives R,RC=O(1); at zero exactly two disjoint half-size sensitive blocks can be packed, so bs=2. The promise has no opposite-output adjacent pair for m>=4. The unrestricted polynomial 2 sum_i x_i/m is exact of degree one.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument. The one-query circuit here is explicitly derived; DJ92 is the historical problem citation, not an audited original one-query attribution.", "witness_id": "promise-restricted-DJ", "proof_sources": [ "DJ92" ] }, { "id": "promise-unbounded-R0-Q-promise-restricted-DJ", "A": "R0", "B": "Q", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-restricted-DJ, R0 is unbounded while Q=O(1). Thus R0 is not O(max(1,Q)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "The phase-query algorithm prepares the uniform state, applies phases (-1)^x_i and tests overlap with the uniform state. Its amplitude is 1 at zero and 0 on balanced inputs, hence QE=1. A weak certificate for zero needs m/2+1 zero positions, so C,D,R0=Theta(m). Random sampling gives R,RC=O(1); at zero exactly two disjoint half-size sensitive blocks can be packed, so bs=2. The promise has no opposite-output adjacent pair for m>=4. The unrestricted polynomial 2 sum_i x_i/m is exact of degree one.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument. The one-query circuit here is explicitly derived; DJ92 is the historical problem citation, not an audited original one-query attribution.", "witness_id": "promise-restricted-DJ", "proof_sources": [ "DJ92" ] }, { "id": "promise-unbounded-R0-QE-promise-restricted-DJ", "A": "R0", "B": "QE", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-restricted-DJ, R0 is unbounded while QE=O(1). Thus R0 is not O(max(1,QE)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "The phase-query algorithm prepares the uniform state, applies phases (-1)^x_i and tests overlap with the uniform state. Its amplitude is 1 at zero and 0 on balanced inputs, hence QE=1. A weak certificate for zero needs m/2+1 zero positions, so C,D,R0=Theta(m). Random sampling gives R,RC=O(1); at zero exactly two disjoint half-size sensitive blocks can be packed, so bs=2. The promise has no opposite-output adjacent pair for m>=4. The unrestricted polynomial 2 sum_i x_i/m is exact of degree one.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument. The one-query circuit here is explicitly derived; DJ92 is the historical problem citation, not an audited original one-query attribution.", "witness_id": "promise-restricted-DJ", "proof_sources": [ "DJ92" ] }, { "id": "promise-unbounded-R0-deg-promise-restricted-DJ", "A": "R0", "B": "deg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-restricted-DJ, R0 is unbounded while deg=O(1). Thus R0 is not O(max(1,deg)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "The phase-query algorithm prepares the uniform state, applies phases (-1)^x_i and tests overlap with the uniform state. Its amplitude is 1 at zero and 0 on balanced inputs, hence QE=1. A weak certificate for zero needs m/2+1 zero positions, so C,D,R0=Theta(m). Random sampling gives R,RC=O(1); at zero exactly two disjoint half-size sensitive blocks can be packed, so bs=2. The promise has no opposite-output adjacent pair for m>=4. The unrestricted polynomial 2 sum_i x_i/m is exact of degree one.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument. The one-query circuit here is explicitly derived; DJ92 is the historical problem citation, not an audited original one-query attribution.", "witness_id": "promise-restricted-DJ", "proof_sources": [ "DJ92" ] }, { "id": "promise-unbounded-R0-adeg-promise-restricted-DJ", "A": "R0", "B": "adeg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-restricted-DJ, R0 is unbounded while adeg=O(1). Thus R0 is not O(max(1,adeg)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "The phase-query algorithm prepares the uniform state, applies phases (-1)^x_i and tests overlap with the uniform state. Its amplitude is 1 at zero and 0 on balanced inputs, hence QE=1. A weak certificate for zero needs m/2+1 zero positions, so C,D,R0=Theta(m). Random sampling gives R,RC=O(1); at zero exactly two disjoint half-size sensitive blocks can be packed, so bs=2. The promise has no opposite-output adjacent pair for m>=4. The unrestricted polynomial 2 sum_i x_i/m is exact of degree one.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument. The one-query circuit here is explicitly derived; DJ92 is the historical problem citation, not an audited original one-query attribution.", "witness_id": "promise-restricted-DJ", "proof_sources": [ "DJ92" ] }, { "id": "promise-unbounded-C-R-promise-restricted-DJ", "A": "C", "B": "R", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-restricted-DJ, C is unbounded while R=O(1). Thus C is not O(max(1,R)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "The phase-query algorithm prepares the uniform state, applies phases (-1)^x_i and tests overlap with the uniform state. Its amplitude is 1 at zero and 0 on balanced inputs, hence QE=1. A weak certificate for zero needs m/2+1 zero positions, so C,D,R0=Theta(m). Random sampling gives R,RC=O(1); at zero exactly two disjoint half-size sensitive blocks can be packed, so bs=2. The promise has no opposite-output adjacent pair for m>=4. The unrestricted polynomial 2 sum_i x_i/m is exact of degree one.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument. The one-query circuit here is explicitly derived; DJ92 is the historical problem citation, not an audited original one-query attribution.", "witness_id": "promise-restricted-DJ", "proof_sources": [ "DJ92" ] }, { "id": "promise-unbounded-C-RC-promise-restricted-DJ", "A": "C", "B": "RC", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-restricted-DJ, C is unbounded while RC=O(1). Thus C is not O(max(1,RC)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "The phase-query algorithm prepares the uniform state, applies phases (-1)^x_i and tests overlap with the uniform state. Its amplitude is 1 at zero and 0 on balanced inputs, hence QE=1. A weak certificate for zero needs m/2+1 zero positions, so C,D,R0=Theta(m). Random sampling gives R,RC=O(1); at zero exactly two disjoint half-size sensitive blocks can be packed, so bs=2. The promise has no opposite-output adjacent pair for m>=4. The unrestricted polynomial 2 sum_i x_i/m is exact of degree one.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument. The one-query circuit here is explicitly derived; DJ92 is the historical problem citation, not an audited original one-query attribution.", "witness_id": "promise-restricted-DJ", "proof_sources": [ "DJ92" ] }, { "id": "promise-unbounded-C-bs-promise-restricted-DJ", "A": "C", "B": "bs", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-restricted-DJ, C is unbounded while bs=O(1). Thus C is not O(max(1,bs)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "The phase-query algorithm prepares the uniform state, applies phases (-1)^x_i and tests overlap with the uniform state. Its amplitude is 1 at zero and 0 on balanced inputs, hence QE=1. A weak certificate for zero needs m/2+1 zero positions, so C,D,R0=Theta(m). Random sampling gives R,RC=O(1); at zero exactly two disjoint half-size sensitive blocks can be packed, so bs=2. The promise has no opposite-output adjacent pair for m>=4. The unrestricted polynomial 2 sum_i x_i/m is exact of degree one.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument. The one-query circuit here is explicitly derived; DJ92 is the historical problem citation, not an audited original one-query attribution.", "witness_id": "promise-restricted-DJ", "proof_sources": [ "DJ92" ] }, { "id": "promise-unbounded-C-s-promise-restricted-DJ", "A": "C", "B": "s", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-restricted-DJ, C is unbounded while s=O(1). Thus C is not O(max(1,s)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "The phase-query algorithm prepares the uniform state, applies phases (-1)^x_i and tests overlap with the uniform state. Its amplitude is 1 at zero and 0 on balanced inputs, hence QE=1. A weak certificate for zero needs m/2+1 zero positions, so C,D,R0=Theta(m). Random sampling gives R,RC=O(1); at zero exactly two disjoint half-size sensitive blocks can be packed, so bs=2. The promise has no opposite-output adjacent pair for m>=4. The unrestricted polynomial 2 sum_i x_i/m is exact of degree one.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument. The one-query circuit here is explicitly derived; DJ92 is the historical problem citation, not an audited original one-query attribution.", "witness_id": "promise-restricted-DJ", "proof_sources": [ "DJ92" ] }, { "id": "promise-unbounded-C-lambda-promise-restricted-DJ", "A": "C", "B": "lambda", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-restricted-DJ, C is unbounded while lambda=O(1). Thus C is not O(max(1,lambda)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "The phase-query algorithm prepares the uniform state, applies phases (-1)^x_i and tests overlap with the uniform state. Its amplitude is 1 at zero and 0 on balanced inputs, hence QE=1. A weak certificate for zero needs m/2+1 zero positions, so C,D,R0=Theta(m). Random sampling gives R,RC=O(1); at zero exactly two disjoint half-size sensitive blocks can be packed, so bs=2. The promise has no opposite-output adjacent pair for m>=4. The unrestricted polynomial 2 sum_i x_i/m is exact of degree one.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument. The one-query circuit here is explicitly derived; DJ92 is the historical problem citation, not an audited original one-query attribution.", "witness_id": "promise-restricted-DJ", "proof_sources": [ "DJ92" ] }, { "id": "promise-unbounded-C-Q-promise-restricted-DJ", "A": "C", "B": "Q", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-restricted-DJ, C is unbounded while Q=O(1). Thus C is not O(max(1,Q)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "The phase-query algorithm prepares the uniform state, applies phases (-1)^x_i and tests overlap with the uniform state. Its amplitude is 1 at zero and 0 on balanced inputs, hence QE=1. A weak certificate for zero needs m/2+1 zero positions, so C,D,R0=Theta(m). Random sampling gives R,RC=O(1); at zero exactly two disjoint half-size sensitive blocks can be packed, so bs=2. The promise has no opposite-output adjacent pair for m>=4. The unrestricted polynomial 2 sum_i x_i/m is exact of degree one.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument. The one-query circuit here is explicitly derived; DJ92 is the historical problem citation, not an audited original one-query attribution.", "witness_id": "promise-restricted-DJ", "proof_sources": [ "DJ92" ] }, { "id": "promise-unbounded-C-QE-promise-restricted-DJ", "A": "C", "B": "QE", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-restricted-DJ, C is unbounded while QE=O(1). Thus C is not O(max(1,QE)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "The phase-query algorithm prepares the uniform state, applies phases (-1)^x_i and tests overlap with the uniform state. Its amplitude is 1 at zero and 0 on balanced inputs, hence QE=1. A weak certificate for zero needs m/2+1 zero positions, so C,D,R0=Theta(m). Random sampling gives R,RC=O(1); at zero exactly two disjoint half-size sensitive blocks can be packed, so bs=2. The promise has no opposite-output adjacent pair for m>=4. The unrestricted polynomial 2 sum_i x_i/m is exact of degree one.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument. The one-query circuit here is explicitly derived; DJ92 is the historical problem citation, not an audited original one-query attribution.", "witness_id": "promise-restricted-DJ", "proof_sources": [ "DJ92" ] }, { "id": "promise-unbounded-C-deg-promise-restricted-DJ", "A": "C", "B": "deg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-restricted-DJ, C is unbounded while deg=O(1). Thus C is not O(max(1,deg)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "The phase-query algorithm prepares the uniform state, applies phases (-1)^x_i and tests overlap with the uniform state. Its amplitude is 1 at zero and 0 on balanced inputs, hence QE=1. A weak certificate for zero needs m/2+1 zero positions, so C,D,R0=Theta(m). Random sampling gives R,RC=O(1); at zero exactly two disjoint half-size sensitive blocks can be packed, so bs=2. The promise has no opposite-output adjacent pair for m>=4. The unrestricted polynomial 2 sum_i x_i/m is exact of degree one.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument. The one-query circuit here is explicitly derived; DJ92 is the historical problem citation, not an audited original one-query attribution.", "witness_id": "promise-restricted-DJ", "proof_sources": [ "DJ92" ] }, { "id": "promise-unbounded-C-adeg-promise-restricted-DJ", "A": "C", "B": "adeg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-restricted-DJ, C is unbounded while adeg=O(1). Thus C is not O(max(1,adeg)^p) for any fixed finite p over partial Boolean functions.", "source": "definitions", "locator": "Repository derivation", "proof_summary": "The phase-query algorithm prepares the uniform state, applies phases (-1)^x_i and tests overlap with the uniform state. Its amplitude is 1 at zero and 0 on balanced inputs, hence QE=1. A weak certificate for zero needs m/2+1 zero positions, so C,D,R0=Theta(m). Random sampling gives R,RC=O(1); at zero exactly two disjoint half-size sensitive blocks can be packed, so bs=2. The promise has no opposite-output adjacent pair for m>=4. The unrestricted polynomial 2 sum_i x_i/m is exact of degree one.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument. The one-query circuit here is explicitly derived; DJ92 is the historical problem citation, not an audited original one-query attribution.", "witness_id": "promise-restricted-DJ", "proof_sources": [ "DJ92" ] }, { "id": "promise-unbounded-D-C-promise-unique-mark-color", "A": "D", "B": "C", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unique-mark-color, D is unbounded while C=O(1). Thus D is not O(max(1,C)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 9 and its proof", "proof_summary": "A one-bit weak certificate reads the marked position, so C=RC=fbs=bs=1. AKPVZ18 Theorem 9 uses the complete bipartite relation between the two halves to prove classical adversary Omega(m), hence R=Omega(m), and R0,D are at least as large. Scanning all coordinates gives matching upper bounds. The sum of the second-half input bits is an exact unrestricted degree-one polynomial. No two promised inputs are adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unique-mark-color" }, { "id": "promise-unbounded-D-RC-promise-unique-mark-color", "A": "D", "B": "RC", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unique-mark-color, D is unbounded while RC=O(1). Thus D is not O(max(1,RC)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 9 and its proof", "proof_summary": "A one-bit weak certificate reads the marked position, so C=RC=fbs=bs=1. AKPVZ18 Theorem 9 uses the complete bipartite relation between the two halves to prove classical adversary Omega(m), hence R=Omega(m), and R0,D are at least as large. Scanning all coordinates gives matching upper bounds. The sum of the second-half input bits is an exact unrestricted degree-one polynomial. No two promised inputs are adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unique-mark-color" }, { "id": "promise-unbounded-D-bs-promise-unique-mark-color", "A": "D", "B": "bs", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unique-mark-color, D is unbounded while bs=O(1). Thus D is not O(max(1,bs)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 9 and its proof", "proof_summary": "A one-bit weak certificate reads the marked position, so C=RC=fbs=bs=1. AKPVZ18 Theorem 9 uses the complete bipartite relation between the two halves to prove classical adversary Omega(m), hence R=Omega(m), and R0,D are at least as large. Scanning all coordinates gives matching upper bounds. The sum of the second-half input bits is an exact unrestricted degree-one polynomial. No two promised inputs are adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unique-mark-color" }, { "id": "promise-unbounded-D-s-promise-unique-mark-color", "A": "D", "B": "s", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unique-mark-color, D is unbounded while s=O(1). Thus D is not O(max(1,s)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 9 and its proof", "proof_summary": "A one-bit weak certificate reads the marked position, so C=RC=fbs=bs=1. AKPVZ18 Theorem 9 uses the complete bipartite relation between the two halves to prove classical adversary Omega(m), hence R=Omega(m), and R0,D are at least as large. Scanning all coordinates gives matching upper bounds. The sum of the second-half input bits is an exact unrestricted degree-one polynomial. No two promised inputs are adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unique-mark-color" }, { "id": "promise-unbounded-D-lambda-promise-unique-mark-color", "A": "D", "B": "lambda", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unique-mark-color, D is unbounded while lambda=O(1). Thus D is not O(max(1,lambda)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 9 and its proof", "proof_summary": "A one-bit weak certificate reads the marked position, so C=RC=fbs=bs=1. AKPVZ18 Theorem 9 uses the complete bipartite relation between the two halves to prove classical adversary Omega(m), hence R=Omega(m), and R0,D are at least as large. Scanning all coordinates gives matching upper bounds. The sum of the second-half input bits is an exact unrestricted degree-one polynomial. No two promised inputs are adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unique-mark-color" }, { "id": "promise-unbounded-D-deg-promise-unique-mark-color", "A": "D", "B": "deg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unique-mark-color, D is unbounded while deg=O(1). Thus D is not O(max(1,deg)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 9 and its proof", "proof_summary": "A one-bit weak certificate reads the marked position, so C=RC=fbs=bs=1. AKPVZ18 Theorem 9 uses the complete bipartite relation between the two halves to prove classical adversary Omega(m), hence R=Omega(m), and R0,D are at least as large. Scanning all coordinates gives matching upper bounds. The sum of the second-half input bits is an exact unrestricted degree-one polynomial. No two promised inputs are adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unique-mark-color" }, { "id": "promise-unbounded-D-adeg-promise-unique-mark-color", "A": "D", "B": "adeg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unique-mark-color, D is unbounded while adeg=O(1). Thus D is not O(max(1,adeg)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 9 and its proof", "proof_summary": "A one-bit weak certificate reads the marked position, so C=RC=fbs=bs=1. AKPVZ18 Theorem 9 uses the complete bipartite relation between the two halves to prove classical adversary Omega(m), hence R=Omega(m), and R0,D are at least as large. Scanning all coordinates gives matching upper bounds. The sum of the second-half input bits is an exact unrestricted degree-one polynomial. No two promised inputs are adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unique-mark-color" }, { "id": "promise-unbounded-R0-C-promise-unique-mark-color", "A": "R0", "B": "C", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unique-mark-color, R0 is unbounded while C=O(1). Thus R0 is not O(max(1,C)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 9 and its proof", "proof_summary": "A one-bit weak certificate reads the marked position, so C=RC=fbs=bs=1. AKPVZ18 Theorem 9 uses the complete bipartite relation between the two halves to prove classical adversary Omega(m), hence R=Omega(m), and R0,D are at least as large. Scanning all coordinates gives matching upper bounds. The sum of the second-half input bits is an exact unrestricted degree-one polynomial. No two promised inputs are adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unique-mark-color" }, { "id": "promise-unbounded-R0-RC-promise-unique-mark-color", "A": "R0", "B": "RC", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unique-mark-color, R0 is unbounded while RC=O(1). Thus R0 is not O(max(1,RC)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 9 and its proof", "proof_summary": "A one-bit weak certificate reads the marked position, so C=RC=fbs=bs=1. AKPVZ18 Theorem 9 uses the complete bipartite relation between the two halves to prove classical adversary Omega(m), hence R=Omega(m), and R0,D are at least as large. Scanning all coordinates gives matching upper bounds. The sum of the second-half input bits is an exact unrestricted degree-one polynomial. No two promised inputs are adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unique-mark-color" }, { "id": "promise-unbounded-R0-bs-promise-unique-mark-color", "A": "R0", "B": "bs", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unique-mark-color, R0 is unbounded while bs=O(1). Thus R0 is not O(max(1,bs)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 9 and its proof", "proof_summary": "A one-bit weak certificate reads the marked position, so C=RC=fbs=bs=1. AKPVZ18 Theorem 9 uses the complete bipartite relation between the two halves to prove classical adversary Omega(m), hence R=Omega(m), and R0,D are at least as large. Scanning all coordinates gives matching upper bounds. The sum of the second-half input bits is an exact unrestricted degree-one polynomial. No two promised inputs are adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unique-mark-color" }, { "id": "promise-unbounded-R0-s-promise-unique-mark-color", "A": "R0", "B": "s", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unique-mark-color, R0 is unbounded while s=O(1). Thus R0 is not O(max(1,s)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 9 and its proof", "proof_summary": "A one-bit weak certificate reads the marked position, so C=RC=fbs=bs=1. AKPVZ18 Theorem 9 uses the complete bipartite relation between the two halves to prove classical adversary Omega(m), hence R=Omega(m), and R0,D are at least as large. Scanning all coordinates gives matching upper bounds. The sum of the second-half input bits is an exact unrestricted degree-one polynomial. No two promised inputs are adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unique-mark-color" }, { "id": "promise-unbounded-R0-lambda-promise-unique-mark-color", "A": "R0", "B": "lambda", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unique-mark-color, R0 is unbounded while lambda=O(1). Thus R0 is not O(max(1,lambda)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 9 and its proof", "proof_summary": "A one-bit weak certificate reads the marked position, so C=RC=fbs=bs=1. AKPVZ18 Theorem 9 uses the complete bipartite relation between the two halves to prove classical adversary Omega(m), hence R=Omega(m), and R0,D are at least as large. Scanning all coordinates gives matching upper bounds. The sum of the second-half input bits is an exact unrestricted degree-one polynomial. No two promised inputs are adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unique-mark-color" }, { "id": "promise-unbounded-R0-deg-promise-unique-mark-color", "A": "R0", "B": "deg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unique-mark-color, R0 is unbounded while deg=O(1). Thus R0 is not O(max(1,deg)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 9 and its proof", "proof_summary": "A one-bit weak certificate reads the marked position, so C=RC=fbs=bs=1. AKPVZ18 Theorem 9 uses the complete bipartite relation between the two halves to prove classical adversary Omega(m), hence R=Omega(m), and R0,D are at least as large. Scanning all coordinates gives matching upper bounds. The sum of the second-half input bits is an exact unrestricted degree-one polynomial. No two promised inputs are adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unique-mark-color" }, { "id": "promise-unbounded-R0-adeg-promise-unique-mark-color", "A": "R0", "B": "adeg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unique-mark-color, R0 is unbounded while adeg=O(1). Thus R0 is not O(max(1,adeg)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 9 and its proof", "proof_summary": "A one-bit weak certificate reads the marked position, so C=RC=fbs=bs=1. AKPVZ18 Theorem 9 uses the complete bipartite relation between the two halves to prove classical adversary Omega(m), hence R=Omega(m), and R0,D are at least as large. Scanning all coordinates gives matching upper bounds. The sum of the second-half input bits is an exact unrestricted degree-one polynomial. No two promised inputs are adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unique-mark-color" }, { "id": "promise-unbounded-R-C-promise-unique-mark-color", "A": "R", "B": "C", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unique-mark-color, R is unbounded while C=O(1). Thus R is not O(max(1,C)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 9 and its proof", "proof_summary": "A one-bit weak certificate reads the marked position, so C=RC=fbs=bs=1. AKPVZ18 Theorem 9 uses the complete bipartite relation between the two halves to prove classical adversary Omega(m), hence R=Omega(m), and R0,D are at least as large. Scanning all coordinates gives matching upper bounds. The sum of the second-half input bits is an exact unrestricted degree-one polynomial. No two promised inputs are adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unique-mark-color" }, { "id": "promise-unbounded-R-RC-promise-unique-mark-color", "A": "R", "B": "RC", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unique-mark-color, R is unbounded while RC=O(1). Thus R is not O(max(1,RC)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 9 and its proof", "proof_summary": "A one-bit weak certificate reads the marked position, so C=RC=fbs=bs=1. AKPVZ18 Theorem 9 uses the complete bipartite relation between the two halves to prove classical adversary Omega(m), hence R=Omega(m), and R0,D are at least as large. Scanning all coordinates gives matching upper bounds. The sum of the second-half input bits is an exact unrestricted degree-one polynomial. No two promised inputs are adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unique-mark-color" }, { "id": "promise-unbounded-R-bs-promise-unique-mark-color", "A": "R", "B": "bs", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unique-mark-color, R is unbounded while bs=O(1). Thus R is not O(max(1,bs)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 9 and its proof", "proof_summary": "A one-bit weak certificate reads the marked position, so C=RC=fbs=bs=1. AKPVZ18 Theorem 9 uses the complete bipartite relation between the two halves to prove classical adversary Omega(m), hence R=Omega(m), and R0,D are at least as large. Scanning all coordinates gives matching upper bounds. The sum of the second-half input bits is an exact unrestricted degree-one polynomial. No two promised inputs are adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unique-mark-color" }, { "id": "promise-unbounded-R-s-promise-unique-mark-color", "A": "R", "B": "s", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unique-mark-color, R is unbounded while s=O(1). Thus R is not O(max(1,s)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 9 and its proof", "proof_summary": "A one-bit weak certificate reads the marked position, so C=RC=fbs=bs=1. AKPVZ18 Theorem 9 uses the complete bipartite relation between the two halves to prove classical adversary Omega(m), hence R=Omega(m), and R0,D are at least as large. Scanning all coordinates gives matching upper bounds. The sum of the second-half input bits is an exact unrestricted degree-one polynomial. No two promised inputs are adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unique-mark-color" }, { "id": "promise-unbounded-R-lambda-promise-unique-mark-color", "A": "R", "B": "lambda", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unique-mark-color, R is unbounded while lambda=O(1). Thus R is not O(max(1,lambda)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 9 and its proof", "proof_summary": "A one-bit weak certificate reads the marked position, so C=RC=fbs=bs=1. AKPVZ18 Theorem 9 uses the complete bipartite relation between the two halves to prove classical adversary Omega(m), hence R=Omega(m), and R0,D are at least as large. Scanning all coordinates gives matching upper bounds. The sum of the second-half input bits is an exact unrestricted degree-one polynomial. No two promised inputs are adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unique-mark-color" }, { "id": "promise-unbounded-R-deg-promise-unique-mark-color", "A": "R", "B": "deg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unique-mark-color, R is unbounded while deg=O(1). Thus R is not O(max(1,deg)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 9 and its proof", "proof_summary": "A one-bit weak certificate reads the marked position, so C=RC=fbs=bs=1. AKPVZ18 Theorem 9 uses the complete bipartite relation between the two halves to prove classical adversary Omega(m), hence R=Omega(m), and R0,D are at least as large. Scanning all coordinates gives matching upper bounds. The sum of the second-half input bits is an exact unrestricted degree-one polynomial. No two promised inputs are adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unique-mark-color" }, { "id": "promise-unbounded-R-adeg-promise-unique-mark-color", "A": "R", "B": "adeg", "exponent": null, "kind": "no_polynomial_bound", "statement": "On promise-unique-mark-color, R is unbounded while adeg=O(1). Thus R is not O(max(1,adeg)^p) for any fixed finite p over partial Boolean functions.", "source": "AKPVZ18", "locator": "Theorem 9 and its proof", "proof_summary": "A one-bit weak certificate reads the marked position, so C=RC=fbs=bs=1. AKPVZ18 Theorem 9 uses the complete bipartite relation between the two halves to prove classical adversary Omega(m), hence R=Omega(m), and R0,D are at least as large. Scanning all coordinates gives matching upper bounds. The sum of the second-half input bits is an exact unrestricted degree-one polynomial. No two promised inputs are adjacent.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This consequence uses weak promise certificates and unrestricted promise degree. It does not extend to globally bounded degree without a separate argument.", "witness_id": "promise-unique-mark-color" }, { "id": "partial-cfbs-CAdv", "A": "cfbs_boolean", "B": "CAdv", "exponent": 1, "kind": "upper_bound", "statement": "cfbs(f)<=2 CAdv(f).", "source": "ABK21", "locator": "Final Lemma 26", "proof_summary": "Choose a total Boolean completion by a weighted nearest original promised input. Doubling the original target weights covers every sensitive block of that completion at the original target.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "constant_factor": "2" }, { "id": "partial-CAdv-cfbs", "A": "CAdv", "B": "cfbs_boolean", "exponent": 1, "kind": "upper_bound", "statement": "CAdv(f)<=cfbs(f).", "source": "ABK21", "locator": "Final Lemma 27, attributed to Krišjānis Prūsis", "proof_summary": "Choose a completion attaining critical fractional block sensitivity and feasible fractional certificates at promised inputs. A coordinatewise choice of an intermediate cube input converts their coverage constraints into the classical minimum-weight constraint.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "constant_factor": "1" }, { "id": "relation-Q-R", "A": "Q", "B": "R", "exponent": 1, "kind": "upper_bound", "statement": "Q(S)<=R(S).", "source": "definitions", "locator": "Definitions of relation solvers and bounded polynomial systems; BK24 Section 2", "proof_summary": "Simulate a classical randomized solver on a quantum computer.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "relation-R-psR", "A": "R", "B": "psR", "exponent": 1, "kind": "upper_bound", "statement": "R(S)<=psR(S).", "source": "definitions", "locator": "Definitions of relation solvers and bounded polynomial systems; BK24 Section 2", "proof_summary": "A canonical valid answer is in particular a valid answer.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "relation-Q-psQ", "A": "Q", "B": "psQ", "exponent": 1, "kind": "upper_bound", "statement": "Q(S)<=psQ(S).", "source": "definitions", "locator": "Definitions of relation solvers and bounded polynomial systems; BK24 Section 2", "proof_summary": "A canonical quantum answer is in particular valid.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "relation-psQ-psR", "A": "psQ", "B": "psR", "exponent": 1, "kind": "upper_bound", "statement": "psQ(S)<=psR(S).", "source": "definitions", "locator": "Definitions of relation solvers and bounded polynomial systems; BK24 Section 2", "proof_summary": "Simulate the classical canonical solver.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "relation-psR-D", "A": "psR", "B": "D", "exponent": 1, "kind": "upper_bound", "statement": "psR(S)<=D(S).", "source": "definitions", "locator": "Definitions of relation solvers and bounded polynomial systems; BK24 Section 2", "proof_summary": "A deterministic solver itself chooses a canonical output.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "relation-psQ-D", "A": "psQ", "B": "D", "exponent": 1, "kind": "upper_bound", "statement": "psQ(S)<=D(S).", "source": "definitions", "locator": "Definitions of relation solvers and bounded polynomial systems; BK24 Section 2", "proof_summary": "Simulate the deterministic solver.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "relation-adeg-conical", "A": "rel_adeg", "B": "rel_conical", "exponent": 1, "kind": "upper_bound", "statement": "rel_adeg(S)<=rel_conical(S).", "source": "definitions", "locator": "Definitions of relation solvers and bounded polynomial systems; BK24 Section 2", "proof_summary": "Expand each nonnegative literal conjunction into a multilinear polynomial. A width-d conical system is a globally bounded valid-output polynomial system of degree at most d.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "relation-adeg-Q", "A": "rel_adeg", "B": "Q", "exponent": 1, "kind": "upper_bound", "statement": "rel_adeg(S)<=2 Q(S).", "source": "definitions", "locator": "Definitions of relation solvers and bounded polynomial systems; BK24 Section 2", "proof_summary": "For every output label, use its quantum acceptance polynomial. All are nonnegative on the full cube, their sum is at most one, and the valid mass is large on the promise.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "relation-cbs-fcbs", "A": "rel_cbs", "B": "rel_fcbs", "exponent": 1, "kind": "upper_bound", "statement": "cbs(S)<=fcbs(S).", "source": "definitions", "locator": "Definitions of relation solvers and bounded polynomial systems; BK24 Section 2", "proof_summary": "For each valid selector its critical block packing is bounded by its critical fractional packing; minimize over selectors.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "multi-s-bs", "A": "s", "B": "bs", "exponent": 1, "kind": "upper_bound", "statement": "s(f)<=bs(f).", "source": "CDM23full", "locator": "Theorem 3.1 and Theorem 3.2(1), including the complete proofs", "proof_summary": "Singleton sensitive blocks are disjoint.", "scope": { "object_kind": "multi_output_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "proof_sources": [ "CDM25" ], "journal_locator": "CDM25 Theorem 3.1(ii)" }, { "id": "multi-bs-R", "A": "bs", "B": "R", "exponent": 1, "kind": "upper_bound", "statement": "bs(f)<=3 R(f).", "source": "CDM23full", "locator": "Theorem 3.1 and Theorem 3.2(1), including the complete proofs", "proof_summary": "Against any disjoint sensitive block, a bounded-error solver must distinguish the changed output; sum the probabilities of querying each block.", "scope": { "object_kind": "multi_output_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "proof_sources": [ "CDM25" ], "journal_locator": "CDM25 Theorem 3.1(ii)" }, { "id": "multi-D-C2", "A": "D", "B": "C", "exponent": 2, "kind": "upper_bound", "statement": "D(f)<=C(f)^2.", "source": "CDM23full", "locator": "Theorem 3.1 and Theorem 3.2(1), including the complete proofs", "proof_summary": "Iteratively query a certificate of a consistent alternative output; the multi-output certificate argument bounds the number of stages.", "scope": { "object_kind": "multi_output_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "proof_sources": [ "CDM25" ], "journal_locator": "CDM25 Theorem 3.1(iii)" }, { "id": "multi-D-R3", "A": "D", "B": "R", "exponent": 3, "kind": "upper_bound", "statement": "D(f)<=54 R(f)^3.", "source": "CDM23full", "locator": "Theorem 3.1 and Theorem 3.2(1), including the complete proofs", "proof_summary": "Combine D<=2 C bs, C<=s bs and s<=bs<=3R; all are proved for total multi-output functions in the full report.", "scope": { "object_kind": "multi_output_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "proof_sources": [ "CDM25" ], "journal_locator": "CDM25 Theorem 3.1(v)" }, { "id": "partial-quadratic-RC-Q", "A": "RC", "B": "Q", "exponent": 2, "kind": "separation", "statement": "On the zero-or-unit-vector promise, RC=Theta(m) and Q=Theta(sqrt(m)), giving a quadratic separation.", "source": "definitions", "locator": "promise-unit-OR; BHMT02 Theorem 4 and the star sensitivity graph", "proof_summary": "Unrestricted polynomial sum_i x_i is exact and has degree one. Zero requires inspecting all coordinates classically; each unit position is a disjoint sensitive singleton at zero. The sensitivity graph is a star with norm sqrt(m). Grover search gives Q=O(sqrt(m)); the star adversary gives the matching lower bound. For exact search, apply BHMT02 Theorem 4 with hypothesized marked fraction 1/m and verify the returned index. When there is one mark it is found with certainty; when there are none verification rejects. This also proves QE=O(sqrt(m)).", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "proof_sources": [ "BHMT02", "BBCMW01" ], "witness_id": "promise-unit-OR" }, { "id": "partial-quadratic-RC-QE", "A": "RC", "B": "QE", "exponent": 2, "kind": "separation", "statement": "On the zero-or-unit-vector promise, RC=Theta(m) and QE=Theta(sqrt(m)), giving a quadratic separation.", "source": "definitions", "locator": "promise-unit-OR; BHMT02 Theorem 4 and the star sensitivity graph", "proof_summary": "Unrestricted polynomial sum_i x_i is exact and has degree one. Zero requires inspecting all coordinates classically; each unit position is a disjoint sensitive singleton at zero. The sensitivity graph is a star with norm sqrt(m). Grover search gives Q=O(sqrt(m)); the star adversary gives the matching lower bound. For exact search, apply BHMT02 Theorem 4 with hypothesized marked fraction 1/m and verify the returned index. When there is one mark it is found with certainty; when there are none verification rejects. This also proves QE=O(sqrt(m)).", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "proof_sources": [ "BHMT02", "BBCMW01" ], "witness_id": "promise-unit-OR" }, { "id": "partial-quadratic-bs-Q", "A": "bs", "B": "Q", "exponent": 2, "kind": "separation", "statement": "On the zero-or-unit-vector promise, bs=Theta(m) and Q=Theta(sqrt(m)), giving a quadratic separation.", "source": "definitions", "locator": "promise-unit-OR; BHMT02 Theorem 4 and the star sensitivity graph", "proof_summary": "Unrestricted polynomial sum_i x_i is exact and has degree one. Zero requires inspecting all coordinates classically; each unit position is a disjoint sensitive singleton at zero. The sensitivity graph is a star with norm sqrt(m). Grover search gives Q=O(sqrt(m)); the star adversary gives the matching lower bound. For exact search, apply BHMT02 Theorem 4 with hypothesized marked fraction 1/m and verify the returned index. When there is one mark it is found with certainty; when there are none verification rejects. This also proves QE=O(sqrt(m)).", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "proof_sources": [ "BHMT02", "BBCMW01" ], "witness_id": "promise-unit-OR" }, { "id": "partial-quadratic-bs-QE", "A": "bs", "B": "QE", "exponent": 2, "kind": "separation", "statement": "On the zero-or-unit-vector promise, bs=Theta(m) and QE=Theta(sqrt(m)), giving a quadratic separation.", "source": "definitions", "locator": "promise-unit-OR; BHMT02 Theorem 4 and the star sensitivity graph", "proof_summary": "Unrestricted polynomial sum_i x_i is exact and has degree one. Zero requires inspecting all coordinates classically; each unit position is a disjoint sensitive singleton at zero. The sensitivity graph is a star with norm sqrt(m). Grover search gives Q=O(sqrt(m)); the star adversary gives the matching lower bound. For exact search, apply BHMT02 Theorem 4 with hypothesized marked fraction 1/m and verify the returned index. When there is one mark it is found with certainty; when there are none verification rejects. This also proves QE=O(sqrt(m)).", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "proof_sources": [ "BHMT02", "BBCMW01" ], "witness_id": "promise-unit-OR" }, { "id": "partial-quadratic-s-Q", "A": "s", "B": "Q", "exponent": 2, "kind": "separation", "statement": "On the zero-or-unit-vector promise, s=Theta(m) and Q=Theta(sqrt(m)), giving a quadratic separation.", "source": "definitions", "locator": "promise-unit-OR; BHMT02 Theorem 4 and the star sensitivity graph", "proof_summary": "Unrestricted polynomial sum_i x_i is exact and has degree one. Zero requires inspecting all coordinates classically; each unit position is a disjoint sensitive singleton at zero. The sensitivity graph is a star with norm sqrt(m). Grover search gives Q=O(sqrt(m)); the star adversary gives the matching lower bound. For exact search, apply BHMT02 Theorem 4 with hypothesized marked fraction 1/m and verify the returned index. When there is one mark it is found with certainty; when there are none verification rejects. This also proves QE=O(sqrt(m)).", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "proof_sources": [ "BHMT02", "BBCMW01" ], "witness_id": "promise-unit-OR" }, { "id": "partial-quadratic-s-QE", "A": "s", "B": "QE", "exponent": 2, "kind": "separation", "statement": "On the zero-or-unit-vector promise, s=Theta(m) and QE=Theta(sqrt(m)), giving a quadratic separation.", "source": "definitions", "locator": "promise-unit-OR; BHMT02 Theorem 4 and the star sensitivity graph", "proof_summary": "Unrestricted polynomial sum_i x_i is exact and has degree one. Zero requires inspecting all coordinates classically; each unit position is a disjoint sensitive singleton at zero. The sensitivity graph is a star with norm sqrt(m). Grover search gives Q=O(sqrt(m)); the star adversary gives the matching lower bound. For exact search, apply BHMT02 Theorem 4 with hypothesized marked fraction 1/m and verify the returned index. When there is one mark it is found with certainty; when there are none verification rejects. This also proves QE=O(sqrt(m)).", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "proof_sources": [ "BHMT02", "BBCMW01" ], "witness_id": "promise-unit-OR" }, { "id": "partial-quadratic-s-lambda", "A": "s", "B": "lambda", "exponent": 2, "kind": "separation", "statement": "On the zero-or-unit-vector promise, s=Theta(m) and lambda=Theta(sqrt(m)), giving a quadratic separation.", "source": "definitions", "locator": "promise-unit-OR; BHMT02 Theorem 4 and the star sensitivity graph", "proof_summary": "Unrestricted polynomial sum_i x_i is exact and has degree one. Zero requires inspecting all coordinates classically; each unit position is a disjoint sensitive singleton at zero. The sensitivity graph is a star with norm sqrt(m). Grover search gives Q=O(sqrt(m)); the star adversary gives the matching lower bound. For exact search, apply BHMT02 Theorem 4 with hypothesized marked fraction 1/m and verify the returned index. When there is one mark it is found with certainty; when there are none verification rejects. This also proves QE=O(sqrt(m)).", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "proof_sources": [ "BHMT02", "BBCMW01" ], "witness_id": "promise-unit-OR" }, { "id": "search-PH-rel_adeg-rel_Adv", "A": "rel_adeg", "B": "rel_Adv", "exponent": null, "kind": "no_polynomial_bound", "statement": "On total pigeonhole search, rel_adeg is at least tildeOmega(n^(1/3)) while rel_Adv=O(log n), excluding every finite polynomial upper bound.", "source": "BK24", "locator": "Definition 44, Lemma 45 and Theorem 46", "proof_summary": "Any two inputs with disjoint collision certificates differ in at least n/2 bits, so uniform weights give a logarithmic classical adversary. Symmetrize the output-polynomial system and reduce to the collision lower bound. Use rel_adeg<=2Q<=2R for numerator transfer and rel_Adv<=rel_CAdv for denominator transfer.", "scope": { "object_kind": "search_relation", "totality": "total", "classes": [ "query_TFNP" ], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "polylogarithmic", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "Collision certificates include both colliding indices and their shared values. Do not apply the decision-function inequality Adv=O(approximate-degree^2) in the reverse direction, or confuse relation polynomial systems with a single polynomial.", "depends_on": [ "relation-adeg-CAdv-unbounded", "relation-adeg-Q", "relation-Q-R", "relation-Adv-CAdv" ] }, { "id": "search-PH-Q-rel_Adv", "A": "Q", "B": "rel_Adv", "exponent": null, "kind": "no_polynomial_bound", "statement": "On total pigeonhole search, Q is at least tildeOmega(n^(1/3)) while rel_Adv=O(log n), excluding every finite polynomial upper bound.", "source": "BK24", "locator": "Definition 44, Lemma 45 and Theorem 46", "proof_summary": "Any two inputs with disjoint collision certificates differ in at least n/2 bits, so uniform weights give a logarithmic classical adversary. Symmetrize the output-polynomial system and reduce to the collision lower bound. Use rel_adeg<=2Q<=2R for numerator transfer and rel_Adv<=rel_CAdv for denominator transfer.", "scope": { "object_kind": "search_relation", "totality": "total", "classes": [ "query_TFNP" ], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "polylogarithmic", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "Collision certificates include both colliding indices and their shared values. Do not apply the decision-function inequality Adv=O(approximate-degree^2) in the reverse direction, or confuse relation polynomial systems with a single polynomial.", "depends_on": [ "relation-adeg-CAdv-unbounded", "relation-adeg-Q", "relation-Q-R", "relation-Adv-CAdv" ] }, { "id": "search-PH-Q-rel_CAdv", "A": "Q", "B": "rel_CAdv", "exponent": null, "kind": "no_polynomial_bound", "statement": "On total pigeonhole search, Q is at least tildeOmega(n^(1/3)) while rel_CAdv=O(log n), excluding every finite polynomial upper bound.", "source": "BK24", "locator": "Definition 44, Lemma 45 and Theorem 46", "proof_summary": "Any two inputs with disjoint collision certificates differ in at least n/2 bits, so uniform weights give a logarithmic classical adversary. Symmetrize the output-polynomial system and reduce to the collision lower bound. Use rel_adeg<=2Q<=2R for numerator transfer and rel_Adv<=rel_CAdv for denominator transfer.", "scope": { "object_kind": "search_relation", "totality": "total", "classes": [ "query_TFNP" ], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "polylogarithmic", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "Collision certificates include both colliding indices and their shared values. Do not apply the decision-function inequality Adv=O(approximate-degree^2) in the reverse direction, or confuse relation polynomial systems with a single polynomial.", "depends_on": [ "relation-adeg-CAdv-unbounded", "relation-adeg-Q", "relation-Q-R", "relation-Adv-CAdv" ] }, { "id": "search-PH-R-rel_Adv", "A": "R", "B": "rel_Adv", "exponent": null, "kind": "no_polynomial_bound", "statement": "On total pigeonhole search, R is at least tildeOmega(n^(1/3)) while rel_Adv=O(log n), excluding every finite polynomial upper bound.", "source": "BK24", "locator": "Definition 44, Lemma 45 and Theorem 46", "proof_summary": "Any two inputs with disjoint collision certificates differ in at least n/2 bits, so uniform weights give a logarithmic classical adversary. Symmetrize the output-polynomial system and reduce to the collision lower bound. Use rel_adeg<=2Q<=2R for numerator transfer and rel_Adv<=rel_CAdv for denominator transfer.", "scope": { "object_kind": "search_relation", "totality": "total", "classes": [ "query_TFNP" ], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "polylogarithmic", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "Collision certificates include both colliding indices and their shared values. Do not apply the decision-function inequality Adv=O(approximate-degree^2) in the reverse direction, or confuse relation polynomial systems with a single polynomial.", "depends_on": [ "relation-adeg-CAdv-unbounded", "relation-adeg-Q", "relation-Q-R", "relation-Adv-CAdv" ] }, { "id": "search-PH-R-rel_CAdv", "A": "R", "B": "rel_CAdv", "exponent": null, "kind": "no_polynomial_bound", "statement": "On total pigeonhole search, R is at least tildeOmega(n^(1/3)) while rel_CAdv=O(log n), excluding every finite polynomial upper bound.", "source": "BK24", "locator": "Definition 44, Lemma 45 and Theorem 46", "proof_summary": "Any two inputs with disjoint collision certificates differ in at least n/2 bits, so uniform weights give a logarithmic classical adversary. Symmetrize the output-polynomial system and reduce to the collision lower bound. Use rel_adeg<=2Q<=2R for numerator transfer and rel_Adv<=rel_CAdv for denominator transfer.", "scope": { "object_kind": "search_relation", "totality": "total", "classes": [ "query_TFNP" ], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "polylogarithmic", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "Collision certificates include both colliding indices and their shared values. Do not apply the decision-function inequality Adv=O(approximate-degree^2) in the reverse direction, or confuse relation polynomial systems with a single polynomial.", "depends_on": [ "relation-adeg-CAdv-unbounded", "relation-adeg-Q", "relation-Q-R", "relation-Adv-CAdv" ] }, { "id": "search-R-conical-unbounded", "kind": "no_polynomial_bound", "A": "R", "B": "rel_conical", "exponent": null, "statement": "There are total query-TFNP relations with rel_conical=O(polylog n) but R=tildeOmega(sqrt(n)).", "scope": { "object_kind": "search_relation", "totality": "total", "classes": [ "query_TFNP" ], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "see_source", "status": "primary_statement_and_proof_route_checked", "source": "BK24", "locator": "Corollaries 6 and 40", "proof_summary": "The low-degree witness polynomials remain conical juntas under the total-search transformation; the shifted repeated-search quantum lower bound is unchanged. A quantum lower bound also lower-bounds classical randomized query complexity.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "proof_route_checked_not_independent_full_proof", "parameters": { "input_bits": "poly(n)", "Q": "Omega_tilde(sqrt(n))", "rel_conical": "polylog(n)" }, "notes": "Here n is the construction parameter, not necessarily the input length. These are bounded polynomial systems and two-sided bounded conical systems.", "depends_on": [ "search-Q-conical", "relation-Q-R" ] }, { "id": "search-psR-R-unbounded", "A": "psR", "B": "R", "exponent": null, "kind": "no_polynomial_bound", "statement": "AOES has R=O(1) and psR=Omega(N), as psQ<=psR.", "source": "AGL26", "locator": "ICALP Theorem 1 and Definition 22", "proof_summary": "The input hides a string through XOR blocks; any other string is a valid answer, so blind random output succeeds. A canonical solver can be amplified and converted into solving an XOR instance, giving the linear lower bound.", "scope": { "object_kind": "search_relation", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "Arbitrary total relation, not a claim of a short-verification query-TFNP separation.", "depends_on": [ "search-psQ-R-unbounded", "relation-psQ-psR" ] }, { "id": "search-nonadaptive-derandomization", "kind": "upper_bound", "A": "Dna", "B": "psRna", "exponent": 1, "statement": "For every total search relation, Dna(S)<=3 psRna(S).", "scope": { "object_kind": "search_relation", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "source": "GGR13", "locator": "Full report Theorem 4.1(2)", "proof_summary": "A coordinate with query probability below one third cannot change the canonical answer. The heavy coordinates, of which there are at most 3q, therefore determine the canonical output. Nonadaptivity makes this heavy set independent of the input.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "proof_route_checked_not_independent_full_proof", "proof_sources": [ "CDM23full" ], "notes": "The complete proof of the multi-output bound is in the ECCC report; the proceedings omits some proofs. This cubic 2023 bound predates AGL26’s discussion of an older quartic classical bound.", "constant_factor": "3" }, { "id": "search-D-psR4-historical", "kind": "upper_bound", "A": "D", "B": "psR", "exponent": 4, "statement": "The original GGR13 theorem gives D(S)=O(psR(S)^4) for total search, with an additional output-range-sensitive bound.", "scope": { "object_kind": "search_relation", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "source": "GGR13", "locator": "Full report Theorem 4.1(3)", "proof_summary": "Bound the range of a canonical selector in terms of query cost, encode its outputs by bits, and deterministically compute those total Boolean bits using the classical randomized-to-deterministic simulation.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "proof_route_checked_not_independent_full_proof", "proof_sources": [ "CDM23full" ], "notes": "The complete proof of the multi-output bound is in the ECCC report; the proceedings omits some proofs. This cubic 2023 bound predates AGL26’s discussion of an older quartic classical bound.", "historical": true, "superseded_by": [ "search-D-psR3" ] }, { "id": "search-ApproxHamWt-psR", "kind": "no_polynomial_bound", "A": "psR", "B": "R", "exponent": null, "statement": "Total approximate Hamming-weight search has R=O(1) and psR=Omega(n).", "scope": { "object_kind": "search_relation", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "source": "GGR13", "locator": "Full report Corollary 4.2", "proof_summary": "Random sampling estimates normalized weight within constant additive error. Any valid selector must change between low and high weights. Follow a Hamming walk through coordinates preserving the canonical answer until this is impossible; the resulting input has linearly many sensitive coordinates, forcing linearly many randomized queries.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "proof_route_checked_not_independent_full_proof", "proof_sources": [ "CDM23full" ], "notes": "An arbitrary total relation; this is not a short-verification query-TFNP claim. GGR13 uses strict error =2" ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false }, { "id": "andor-log-size-RandOr", "A": "RandOr", "B": "RSize", "exponent": null, "kind": "upper_bound", "statement": "log RSize(f)/(2 log n)<=RandOr(f)<=4 log RSize(f).", "source": "CDMRS23", "locator": "Lemma 4.2", "proof_summary": "For the upper bound balance a leaf-labelled tree with AND/OR queries; for the lower bound expand subset queries and count leaves. The randomized version applies the transformations to every support tree.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [ "n>=2" ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false }, { "id": "rank-log-size-sandwich", "A": "DTRank", "B": "DTSize", "exponent": null, "kind": "upper_bound", "statement": "rank(f)<=log DTSize(f)<=rank(f)*log(e n/rank(f)) for nonconstant total Boolean f.", "source": "CDMRS23", "locator": "Proposition 2.11, citing Ehrenfeucht–Haussler 1989", "proof_summary": "A rank-r tree has at least 2^r leaves. Orient light/heavy branches using child ranks; at most r rank-increasing choices on any path allow the binomial leaf-count upper bound.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "notes": "Original size k counts all nodes. The leaf-count formulation uses k=2L-1. Minimize each applicable bound carefully; no product of separately optimized tree parameters is inferred.", "proof_sources": [ "EH89", "EH89" ], "original_locator": "EH89 Lemma 1(i), p.235; rank definition p.234; function rank p.241" }, { "id": "random-rank-log-size", "A": "rrank", "B": "RSize", "exponent": null, "kind": "upper_bound", "statement": "rrank(f)<=log RSize(f)<=rrank(f)*log(e n/rrank(f)) for nonconstant total Boolean f.", "source": "CDMRS23", "locator": "Proposition 2.11", "proof_summary": "Apply the deterministic rank/size conversion separately to all trees in the support of the randomized algorithm.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false }, { "id": "search-size-Hamming-exponential", "A": "psDTSize", "B": "RSize", "exponent": null, "kind": "no_polynomial_bound", "statement": "Total ApproxHamWt has RSize=O(1) but psDTSize=exp(Omega(n)).", "source": "CDM23", "locator": "Theorem 4.6, using Theorem 4.5", "proof_summary": "Every canonical approximate-weight selector yields, by thresholding its output, a total completion of ApproxMaj. A corruption bound forces exponential randomized leaf size for every such completion. Constant sampling gives constant randomized solver size.", "scope": { "object_kind": "search_relation", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "This theorem is absent from the inspected earlier ECCC report. Theorem 4.5 explicitly credits its corruption argument to Swagato Sanyal." }, { "id": "search-size-CNF-random", "A": "psDTSize", "B": "RSize", "exponent": null, "kind": "no_polynomial_bound", "statement": "With high probability, sufficiently dense random 3-CNF search has RSize=O(1) and psDTSize=exp(Omega(n^(1/4)/log n)).", "source": "CDM23", "locator": "Corollary 4.4", "proof_summary": "Combine the exponential deterministic-size lower bound for random unsatisfiable formulas with Theorem 4.3’s deterministic/canonical logarithmic-size simulation. A random clause sample solves the relation with constant depth.", "scope": { "object_kind": "search_relation", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "A random formula is fixed to obtain a total relation on assignments. This unlifted construction is distinct from the stronger lifted family of GIPS21." }, { "id": "search-andor-canonical-Hamming", "A": "psRandOr", "B": "RandOr", "exponent": null, "kind": "no_polynomial_bound", "statement": "Total ApproxHamWt has RandOr=O(1), psRandOr=Omega(n/log n).", "source": "CDM23", "locator": "Theorem 5.6, using Lemma 5.5 and Theorem 4.6", "proof_summary": "Use the syntactic equivalence of AND/OR depth and logarithmic leaf size to transfer the exponential canonical-size lower bound.", "scope": { "object_kind": "search_relation", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "proof_sources": [ "CDM25" ], "journal_locator": "CDM25 Theorem 6.10" }, { "id": "search-derandomization-DandPos", "A": "DandPos", "B": "psRandPos", "exponent": 3, "kind": "upper_bound", "statement": "For total S on n bits with output range [m], DandPos(S)=O(psRandPos(S)^3 log^4(n) min(log m,psRandPos(S))).", "source": "CDM23", "locator": "Theorem 5.4", "proof_summary": "Encode the canonical output selector in bits, invoke the corresponding total Boolean restricted-query derandomization theorem, and bound its effective output range by its query cost.", "scope": { "object_kind": "search_relation", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "explicit_input_and_output_parameter_factors", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "proof_sources": [ "CDMRS23", "CDM25" ], "journal_locator": "CDM25 Theorem 6.5" }, { "id": "search-derandomization-DandOr", "A": "DandOr", "B": "psRandOr", "exponent": 4, "kind": "upper_bound", "statement": "For total S on n bits with output range [m], DandOr(S)=O(psRandOr(S)^4 log^7(n) min(log m,psRandOr(S))).", "source": "CDM23", "locator": "Theorem 5.7", "proof_summary": "Encode the canonical output selector in bits, invoke the corresponding total Boolean restricted-query derandomization theorem, and bound its effective output range by its query cost.", "scope": { "object_kind": "search_relation", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "explicit_input_and_output_parameter_factors", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "proof_sources": [ "CDMRS23", "CDM25" ], "journal_locator": "CDM25 Theorem 6.12" }, { "id": "search-andor-canonical-log-size", "A": "psRandOr", "B": "psDTSize", "exponent": null, "kind": "upper_bound", "statement": "log psDTSize(S)/(2 log n)<=psRandOr(S)<=4 log psDTSize(S).", "source": "CDM23", "locator": "Lemma 5.5", "proof_summary": "The underlying tree transformations are syntactic and preserve output labels. Apply them to the selector attaining each canonical optimum separately.", "scope": { "object_kind": "search_relation", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "proof_sources": [ "CDMRS23", "CDM25" ], "journal_locator": "CDM25 Lemma 6.9" }, { "id": "average-strict-zero-error-fix-coins", "A": "R0avg", "B": "Davg", "exponent": 1, "kind": "identity", "statement": "For finite promise domains with nonuniform trees, R0avg_mu(S)=Davg_mu(S)=psR0avg_mu(S).", "source": "definitions", "locator": "Finite-domain averaging; docs/average-cost.md", "proof_summary": "The event that fixed coins give a correct terminating algorithm on every promised input has probability one, because the input domain is finite. Average the mu-cost over those coins and fix a seed with cost at most the randomized average. Prune repeated queries. Deterministic algorithms are a subclass of strict Las Vegas and strict canonical zero-error algorithms.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [ "fixed input distribution mu; correctness required on every promised input" ] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "constant_factor": "1", "notes": "This is a repository derivation for the explicitly strict model, not an attribution of this sharpened equality to GIPS21 or a claim of first discovery." }, { "id": "average-published-zero-error-simulation", "A": "Davg", "B": "R0avg", "exponent": 1, "kind": "upper_bound", "statement": "GIPS21 states deterministic simulation on average with cost O(R0avg_mu log^2 n +1), preserving correctness on the input domain.", "source": "GIPS21", "locator": "Theorem 26 and proof", "proof_summary": "Select a logarithmic family of deterministic algorithms from the randomized solver, interleave their queries, and sum the input-distribution tail bounds for the first fast successful simulation.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [ "fixed input distribution mu; correctness required on every promised input" ] }, "loss": "explicit_input_length_logs", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "notes": "Published weaker bound retained as stated. Under the strict finite-domain Las Vegas definition used here, average-strict-zero-error-fix-coins gives exact equality. The proof of Theorem 26 also discusses solvers succeeding with probability 2/3, so one must not identify abort-allowed and strict conventions without specifying the model." }, { "id": "average-Find1-published", "A": "Davg", "B": null, "exponent": null, "kind": "upper_bound", "statement": "For every distribution on dense Find1 inputs, GIPS21 gives a correct deterministic solver of average cost at most log n+1.", "source": "GIPS21", "locator": "Proposition 25", "proof_summary": "Fix a logarithmic set of positions that hits a 1 on all but 1/n of the input distribution, query those first, and scan the remainder on failure.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [ "fixed input distribution mu; correctness required on every promised input" ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false }, { "id": "average-Find1-constant", "A": "Davg", "B": null, "exponent": null, "kind": "upper_bound", "statement": "For dense Find1 (weight>=ceil(n/2)), Davg_mu<= (n+1)/(ceil(n/2)+1)<2 for every distribution mu, in the nonuniform model.", "source": "definitions", "locator": "Random-permutation averaging; docs/average-cost.md", "proof_summary": "A uniformly random permutation scans until the first 1; on a fixed weight-k string its expected stopping position is (n+1)/(k+1). Average over mu and then fix a permutation whose mean cost is at most that average. Every fixed permutation is correct on all dense inputs.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [ "fixed input distribution mu; correctness required on every promised input" ] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "notes": "Elementary repository observation, with no first-priority claim. It improves the bound recorded in GIPS21 Proposition 25 under the explicitly same average-cost convention." }, { "id": "average-Hamming-canonical", "A": "psRavg", "B": null, "exponent": null, "kind": "upper_bound", "statement": "For every input distribution mu and fixed epsilon>0, epsilon-Hamming-weight estimation has psRavg_mu=O(log n/epsilon^2).", "source": "GIPS21", "locator": "Theorem 27", "proof_summary": "Fix a sample that estimates accurately for most mu-inputs. Cross-check it against an independent random sample; on disagreement read all bits and compute a fixed valid selector. The expensive branch is rare under mu, while canonical correctness holds pointwise.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [ "fixed input distribution mu; correctness required on every promised input" ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false }, { "id": "average-approximation-canonization", "A": "psRavg", "B": "R", "exponent": 1, "kind": "upper_bound", "statement": "If a q-query randomized algorithm epsilon/4-approximates a function into a metric space, a pseudo-deterministic epsilon-approximator has average query cost O(q log n) for every fixed input distribution.", "source": "GIPS21", "locator": "Theorem 28", "proof_summary": "Generalize the fixed-sample and cross-check proof for Hamming weight; on the exceptional branch read the input and evaluate the canonical approximation.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [ "fixed input distribution mu; correctness required on every promised input" ] }, "loss": "explicit_input_length_log", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "notes": "The numerator and denominator solve different approximation radii of the same metric-valued target. This is a typed approximation transformation, not a same-relation scalar power inequality." }, { "id": "average-Hamming-zero-error-lower", "A": "R0avg", "B": null, "exponent": null, "kind": "upper_bound", "statement": "For a full-support input distribution and 0=psQ.", "scope": { "object_kind": "search_relation", "totality": "total", "classes": [ "query_TFNP" ], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "Derived from the published family, not a first-discovery attribution. This concerns relations and does not contradict the total Boolean decision-function bound D=O(R0^2)." }, { "id": "quantum-search-deterministic-node-size", "A": "Q", "B": "DTNodes", "exponent": "1/2", "kind": "upper_bound", "statement": "For a relation solved by a deterministic tree with s total nodes, Q(S)=O(sqrt(s)).", "source": "CMP22", "locator": "Corollary 9(2)", "proof_summary": "Construct an optimal edge weighting for the classical tree and a span-program/dual-adversary quantum algorithm computing its reached leaf. Since that leaf is a unique deterministic function of the input, reduce the quantum leaf-identification error before mapping it to a valid output.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "proof_sources": [ "CMP25" ], "journal_locator": "Corollary 1.7 (deterministic) or Corollary 5.5 (randomized); Q2/5 preserved" }, { "id": "quantum-search-random-node-size", "A": "rel_Qeps", "B": "RNodes", "exponent": "1/2", "kind": "upper_bound", "statement": "Q_{2/5}(S)=O(sqrt(RNodes(S))).", "source": "CMP22", "locator": "Corollary 27", "proof_summary": "Sample a deterministic tree from the classical randomized support. Compute its reached leaf with high constant success and return the leaf label; the combined success is at least (9/10)*(2/3)=3/5.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [ "epsilon=2/5" ] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "notes": "Preserve the error margin. Do not relabel this as the default error-1/3 quantum search measure without an additional reduction.", "proof_sources": [ "CMP25" ], "journal_locator": "Corollary 1.7 (deterministic) or Corollary 5.5 (randomized); Q2/5 preserved" }, { "id": "quantum-Boolean-random-node-size", "A": "Q", "B": "RNodes", "exponent": "1/2", "kind": "upper_bound", "statement": "For partial or total Boolean decision functions, Q(f)=O(sqrt(RNodes(f))).", "source": "CMP22", "locator": "Corollary 27 and Boolean majority error reduction", "proof_summary": "Apply the relation result at error 2/5, then majority-amplify its Boolean answer a constant number of times to error 1/3.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "quantum-Boolean-random-leaf-size", "A": "Q", "B": "RSize", "exponent": "1/2", "kind": "upper_bound", "statement": "For partial or total Boolean decision functions, Q(f)=O(sqrt(RSize(f))).", "source": "CMP22", "locator": "Corollary 27 and the node/leaf conversion", "proof_summary": "RNodes=2 RSize-1 for full binary trees, so convert the size bound by a constant factor and apply Boolean error reduction.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "quantum-search-random-leaf-size", "A": "rel_Qeps", "B": "RSize", "exponent": "1/2", "kind": "upper_bound", "statement": "Q_{2/5}(S)=O(sqrt(RSize(S))).", "source": "CMP22", "locator": "Corollary 27, converted from nodes to leaves", "proof_summary": "Use RNodes=2 RSize-1, retaining the error-2/5 conclusion.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [ "epsilon=2/5" ] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "proof_sources": [ "CMP25" ], "journal_locator": "Corollary 1.7 (deterministic) or Corollary 5.5 (randomized); Q2/5 preserved" }, { "id": "guessing-rank-identity", "A": "DTGuess", "B": "DTRank", "exponent": 1, "kind": "identity", "statement": "Minimum guessing complexity equals minimum Strahler rank, including on a promise domain.", "source": "CMP22", "locator": "Claim 12 and Definition 3", "proof_summary": "For a tree, unequal child ranks let the larger-rank child be the guessed edge; equal ranks force a red edge into a child of that same rank. This is exactly the Strahler recurrence. Minimize the identity over correct trees.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "constant_factor": "1" }, { "id": "bomb-quantum-square", "A": "Bbomb", "B": "Q", "exponent": 2, "kind": "theta_bound", "statement": "At fixed constant explosion/total-error budgets, Bbomb(f)=Theta(Q(f)^2), including partial Boolean functions.", "source": "LL16", "locator": "Theorem 4.1", "proof_summary": "For the upper bound simulate each ordinary query by a quantum-Zeno sequence of gentle controlled bomb queries. For the lower bound use the adversary method to bound progress per query in terms of explosion probability. Boolean error reduction changes fixed error constants by constant factors.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17" }, { "id": "bomb-quantum-explosion-parameter", "A": "BbombParams", "B": "Q_eps", "exponent": 2, "kind": "theta_bound", "statement": "For 0=n-1 has RandPos=O(1) but DandPos=Omega(log n). Therefore no pure finite power of RandPos bounds DandPos for total Boolean functions.", "source": "KLMY21survey", "locator": "Example 3.17; randomized construction in KLMY21 Example 6.3", "proof_summary": "Split coordinates uniformly into two complementary sets and query their two conjunctions. At most one zero guarantees one conjunction is one; at least two zeros are separated with probability at least one half, so a constant number of independent trials suffices. The deterministic logarithmic lower bound follows from the n+1 nonzero monomials and the monomial-count bound for a depth-d positive-conjunction tree.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "proof_sources": [ "KLMY21" ], "notes": "This explains why the log n factor in DandPos=O(RandPos^3 log^4 n) cannot simply be suppressed, even when asking only for a finite power relation." }, { "id": "average-elementary-Ravg-R", "A": "Ravg", "B": "R", "exponent": 1, "kind": "upper_bound", "statement": "Ravg_mu(S)<=R(S) in the stated strong average-cost models, with mu fixed.", "source": "definitions", "locator": "Definitions of worst-case and average cost", "proof_summary": "Use the same admissible algorithm: averaging cannot increase a worst-case cost bound, and a canonical or zero-error solver also meets the corresponding weaker output requirement. A deterministic solver is canonical.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [ "fixed input distribution mu; correctness required on every promised input" ] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "constant_factor": "1", "notes": "No average-error budget is introduced. For two averaged measures, both use the same fixed mu." }, { "id": "average-elementary-R0avg-R0", "A": "R0avg", "B": "R0", "exponent": 1, "kind": "upper_bound", "statement": "R0avg_mu(S)<=R0(S) in the stated strong average-cost models, with mu fixed.", "source": "definitions", "locator": "Definitions of worst-case and average cost", "proof_summary": "Use the same admissible algorithm: averaging cannot increase a worst-case cost bound, and a canonical or zero-error solver also meets the corresponding weaker output requirement. A deterministic solver is canonical.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [ "fixed input distribution mu; correctness required on every promised input" ] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "constant_factor": "1", "notes": "No average-error budget is introduced. For two averaged measures, both use the same fixed mu." }, { "id": "average-elementary-Davg-D", "A": "Davg", "B": "D", "exponent": 1, "kind": "upper_bound", "statement": "Davg_mu(S)<=D(S) in the stated strong average-cost models, with mu fixed.", "source": "definitions", "locator": "Definitions of worst-case and average cost", "proof_summary": "Use the same admissible algorithm: averaging cannot increase a worst-case cost bound, and a canonical or zero-error solver also meets the corresponding weaker output requirement. A deterministic solver is canonical.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [ "fixed input distribution mu; correctness required on every promised input" ] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "constant_factor": "1", "notes": "No average-error budget is introduced. For two averaged measures, both use the same fixed mu." }, { "id": "average-elementary-psRavg-psR", "A": "psRavg", "B": "psR", "exponent": 1, "kind": "upper_bound", "statement": "psRavg_mu(S)<=psR(S) in the stated strong average-cost models, with mu fixed.", "source": "definitions", "locator": "Definitions of worst-case and average cost", "proof_summary": "Use the same admissible algorithm: averaging cannot increase a worst-case cost bound, and a canonical or zero-error solver also meets the corresponding weaker output requirement. A deterministic solver is canonical.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [ "fixed input distribution mu; correctness required on every promised input" ] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "constant_factor": "1", "notes": "No average-error budget is introduced. For two averaged measures, both use the same fixed mu." }, { "id": "average-elementary-Ravg-psRavg", "A": "Ravg", "B": "psRavg", "exponent": 1, "kind": "upper_bound", "statement": "Ravg_mu(S)<=psRavg(S) in the stated strong average-cost models, with mu fixed.", "source": "definitions", "locator": "Definitions of worst-case and average cost", "proof_summary": "Use the same admissible algorithm: averaging cannot increase a worst-case cost bound, and a canonical or zero-error solver also meets the corresponding weaker output requirement. A deterministic solver is canonical.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [ "fixed input distribution mu; correctness required on every promised input" ] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "constant_factor": "1", "notes": "No average-error budget is introduced. For two averaged measures, both use the same fixed mu." }, { "id": "average-elementary-Ravg-R0avg", "A": "Ravg", "B": "R0avg", "exponent": 1, "kind": "upper_bound", "statement": "Ravg_mu(S)<=R0avg(S) in the stated strong average-cost models, with mu fixed.", "source": "definitions", "locator": "Definitions of worst-case and average cost", "proof_summary": "Use the same admissible algorithm: averaging cannot increase a worst-case cost bound, and a canonical or zero-error solver also meets the corresponding weaker output requirement. A deterministic solver is canonical.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [ "fixed input distribution mu; correctness required on every promised input" ] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "constant_factor": "1", "notes": "No average-error budget is introduced. For two averaged measures, both use the same fixed mu." }, { "id": "average-elementary-psRavg-Davg", "A": "psRavg", "B": "Davg", "exponent": 1, "kind": "upper_bound", "statement": "psRavg_mu(S)<=Davg(S) in the stated strong average-cost models, with mu fixed.", "source": "definitions", "locator": "Definitions of worst-case and average cost", "proof_summary": "Use the same admissible algorithm: averaging cannot increase a worst-case cost bound, and a canonical or zero-error solver also meets the corresponding weaker output requirement. A deterministic solver is canonical.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [ "fixed input distribution mu; correctness required on every promised input" ] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "constant_factor": "1", "notes": "No average-error budget is introduced. For two averaged measures, both use the same fixed mu." }, { "id": "average-Hamming-unbounded-Davg", "A": "Davg", "B": "Ravg", "exponent": null, "kind": "no_polynomial_bound", "statement": "For fixed 0=n-1 has exact sparsity n+1 and approximate sparsity O(log n).", "source": "Sparse25", "locator": "Remark 1.2 and Claim 4.15", "proof_summary": "For threshold f(x)=1 iff |x|>=n-1, the unique exact polynomial is sum_i product_{j!=i}x_j -(n-1)product_j x_j, with n+1 terms. A separating collection of O(log n) pairs of subsets separates each pair of coordinates in at least a 2/3 fraction of its members. Average the constant-size polynomials testing that all zeros lie on one side of both partitions. This is exact on positive inputs and at most 1/3 on negative inputs.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric", "monotone" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false }, { "id": "monomial-weight-no-pure-approx-power", "A": "monomialWeight", "B": "approxMonomialWeight", "exponent": null, "kind": "no_polynomial_bound", "statement": "No finite pure polynomial bound: threshold |x|>=n-1 has exact weight 2n-1 and approximate weight O(1).", "source": "Sparse25", "locator": "Section 4.3 and Claim 4.15", "proof_summary": "For threshold f(x)=1 iff |x|>=n-1, the unique exact polynomial is sum_i product_{j!=i}x_j -(n-1)product_j x_j, with n+1 terms. A separating collection of O(log n) pairs of subsets separates each pair of coordinates in at least a 2/3 fraction of its members. Average the constant-size polynomials testing that all zeros lie on one side of both partitions. This is exact on positive inputs and at most 1/3 on negative inputs. The average has bounded coefficient weight, whereas the displayed exact polynomial has weight 2n-1.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric", "monotone" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false }, { "id": "extremal-forrelation-nonadaptive-2026", "A": "Rna", "B": "QE", "exponent": null, "kind": "no_polynomial_bound", "statement": "Extremal Forrelation has QE=1 and Rna=Omega(2^((1/2-o(1))*n)); hence no finite Rna versus QE polynomial bound. This is not an adaptive R lower bound.", "source": "CCM26", "locator": "Theorems 1 and 3; Appendix D", "proof_summary": "Generalize the bent-function hidden-subspace construction and bound total variation between nonadaptive query transcripts by counting dependent query configurations. Optimize the moment parameter. The v1 partial-spread argument is replaced, not reused.", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [ "Nonadaptive classical algorithms; n is oracle address length, so Boolean input length is 2^(n+1)." ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false }, { "id": "multi-C-s-bs", "A": "C", "B": "bs", "exponent": null, "kind": "upper_bound", "statement": "C(f)<=s(f)*bs(f) for total multi-output functions.", "source": "CDM25", "locator": "Theorem 3.1(i)", "proof_summary": "Take a maximal family of disjoint minimal sensitive blocks. Their union is a certificate, and each block has size at most s(f), since flipping back any coordinate from the altered input changes the output.", "scope": { "object_kind": "multi_output_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "factors": [ { "measure": "s", "exponent": 1 }, { "measure": "bs", "exponent": 1 } ] }, { "id": "multi-D-C-bs", "A": "D", "B": "C", "exponent": null, "kind": "upper_bound", "statement": "D(f)<=2*C(f)*bs(f) for total multi-output functions.", "source": "CDM25", "locator": "Theorem 3.1(iv)", "proof_summary": "Query a consistent certificate for up to 2bs(f) stages. If the residual function is not constant, two consistent inputs with different outputs yield enough disjoint sensitive blocks to contradict the block-sensitivity bound.", "scope": { "object_kind": "multi_output_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "factors": [ { "measure": "C", "exponent": 1 }, { "measure": "bs", "exponent": 1 } ], "constant": 2 }, { "id": "search-deterministic-andor-log-size", "A": "DandOr", "B": "DTSize", "exponent": null, "kind": "upper_bound", "statement": "log DTSize(S)/(2 log n)<=DandOr(S)<=4 log DTSize(S), for total search; sizes count leaves.", "source": "CDM25", "locator": "Proposition 6.6 and extension paragraph", "proof_summary": "Expand each AND/OR query into a bit-query tree for one direction; balance a bit-query tree using AND/OR queries for the other. These syntactic transformations preserve output labels and apply to each tree in a randomized distribution.", "scope": { "object_kind": "search_relation", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false }, { "id": "search-randomized-andor-log-size", "A": "RandOr", "B": "RSize", "exponent": null, "kind": "upper_bound", "statement": "log RSize(S)/(2 log n)<=RandOr(S)<=4 log RSize(S), for total search; sizes count leaves.", "source": "CDM25", "locator": "Proposition 6.6 and extension paragraph", "proof_summary": "Expand each AND/OR query into a bit-query tree for one direction; balance a bit-query tree using AND/OR queries for the other. These syntactic transformations preserve output labels and apply to each tree in a randomized distribution.", "scope": { "object_kind": "search_relation", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false }, { "id": "search-DTSize-from-DandOr", "A": "DTSize", "B": "DandOr", "exponent": null, "kind": "upper_bound", "statement": "DTSize(S)<=2^(2*DandOr(S)*log n).", "source": "CDM25", "locator": "Proposition 6.6; Lemma 6.9 for pseudo-deterministic variant", "proof_summary": "Expand each positive AND or OR query to bit queries. For the pseudo-deterministic case choose the selector attaining its optimum; the syntactic expansion preserves that selector and its success probability.", "scope": { "object_kind": "search_relation", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "notes": "Exponential size conversion; not a finite scalar power bound." }, { "id": "search-RSize-from-RandOr", "A": "RSize", "B": "RandOr", "exponent": null, "kind": "upper_bound", "statement": "RSize(S)<=2^(2*RandOr(S)*log n).", "source": "CDM25", "locator": "Proposition 6.6; Lemma 6.9 for pseudo-deterministic variant", "proof_summary": "Expand each positive AND or OR query to bit queries. For the pseudo-deterministic case choose the selector attaining its optimum; the syntactic expansion preserves that selector and its success probability.", "scope": { "object_kind": "search_relation", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "notes": "Exponential size conversion; not a finite scalar power bound." }, { "id": "search-psDTSize-from-psRandOr", "A": "psDTSize", "B": "psRandOr", "exponent": null, "kind": "upper_bound", "statement": "psDTSize(S)<=2^(2*psRandOr(S)*log n).", "source": "CDM25", "locator": "Proposition 6.6; Lemma 6.9 for pseudo-deterministic variant", "proof_summary": "Expand each positive AND or OR query to bit queries. For the pseudo-deterministic case choose the selector attaining its optimum; the syntactic expansion preserves that selector and its success probability.", "scope": { "object_kind": "search_relation", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "notes": "Exponential size conversion; not a finite scalar power bound." }, { "id": "EH89-rank-node-count", "A": "DTRank", "B": "DTNodes", "exponent": null, "kind": "upper_bound", "statement": "For a reduced n-variable binary tree of rank r>=1 and k total nodes: 2^(r+1)-1<=k<=2*sum_{i=0}^r binom(n,i)-1<=2*(e*n/r)^r-1. Function optima inherit the corresponding inequalities.", "source": "EH89", "locator": "Lemma 1(i), p.235", "proof_summary": "The smallest rank-r tree contains a complete rank-r branching skeleton. The maximum leaf count obeys L(n,r)=L(n-1,r)+L(n-1,r-1), with L(n,0)=L(0,r)=1, hence the binomial sum. A full binary tree has k=2L-1.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "explicit_input_and_rank_parameter_factors", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "notes": "The displayed r,k statement is about a single tree; not an equality for independently minimizing trees." }, { "id": "Nis89-D-R2-cubic", "A": "D", "B": "NisR2expect", "exponent": 3, "kind": "upper_bound", "statement": "D(f)<=8 R_2(f)^3.", "source": "Nis89", "locator": "Theorem 4, p.334", "proof_summary": "Combine C<=bs^2, D<=2*C_1*R_2 and bs<=2R_2.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "constant": 8 }, { "id": "Nis89-bs-R2", "A": "bs", "B": "NisR2expect", "exponent": 1, "kind": "upper_bound", "statement": "bs(f)<=2 R_2(f).", "source": "Nis89", "locator": "Lemma 4.2, p.334", "proof_summary": "Couple runs at x and each disjoint sensitive-block neighbor. Correctness with error 1/4 forces probability at least 1/2 of querying each block; sum over disjoint blocks.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "constant": 2 }, { "id": "Nis89-D-R1-square", "A": "D", "B": "NisR1expect", "exponent": 2, "kind": "upper_bound", "statement": "D(f)<=2 R_1(f)^2.", "source": "Nis89", "locator": "Theorem 3, p.333", "proof_summary": "For one-bit identity, reject immediately with probability 1/2 and otherwise query. Expected cost is 1/2 and acceptance on input 1 is 1/2, so the printed D<=2R_1^2 gives 1<=1/2. The constant cannot hold under the literal source convention.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "source_statement_refuted_under_literal_cost_convention", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "constant": 2, "historical": true, "notes": "Literal normalization discrepancy established. Repository derivation Nis89-expected-one-sided-repair gives the sharp universal factor 4. Nis91 journal text remains uninspected; no claim about a published erratum." }, { "id": "Nis89-D-C1-R2", "A": "D", "B": "NisR2expect", "exponent": null, "kind": "upper_bound", "statement": "D(f)<=2*C1(f)*R_2(f).", "source": "Nis89", "locator": "Lemma 4.1, pp.333–334", "proof_summary": "Query a consistent 1-certificate each round. Coupling a 0-input with an input satisfying this certificate forces at least half a query of expected cost to be removed each round; after at most 2R_2 rounds the residual function is constant.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "factors": [ { "measure": "C1", "exponent": 1 }, { "measure": "NisR2expect", "exponent": 1 } ] }, { "id": "search-elementary-rel_deg-QE", "A": "rel_deg", "B": "QE", "exponent": 1, "kind": "upper_bound", "statement": "rel_deg(S)<=2 QE(S).", "source": "BK24", "locator": "Section 2, Definitions 10–12 and standard solver definitions", "proof_summary": "Use all exact quantum output-probability polynomials, of degree at most twice the query count. They form a bounded nonnegative output distribution on the full cube.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "search-elementary-rel_adeg-rel_deg", "A": "rel_adeg", "B": "rel_deg", "exponent": 1, "kind": "upper_bound", "statement": "rel_adeg(S)<=rel_deg(S).", "source": "BK24", "locator": "Section 2, Definitions 10–12 and standard solver definitions", "proof_summary": "An exact bounded system satisfies the approximate success constraint.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "search-elementary-rel_udeg-rel_adeg", "A": "rel_udeg", "B": "rel_adeg", "exponent": 1, "kind": "upper_bound", "statement": "rel_udeg(S)<=rel_adeg(S).", "source": "BK24", "locator": "Section 2, Definitions 10–12 and standard solver definitions", "proof_summary": "Drop the off-promise positivity and normalization constraints.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "search-elementary-rel_adeg-rel_udeg", "A": "rel_adeg", "B": "rel_udeg", "exponent": 1, "kind": "upper_bound", "statement": "rel_adeg(S)=rel_udeg(S) for total relations.", "source": "BK24", "locator": "Section 2, Definitions 10–12 and standard solver definitions", "proof_summary": "On a total domain there are no off-promise inputs; the two feasible systems coincide.", "scope": { "object_kind": "search_relation", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "search-elementary-rel_conical-R", "A": "rel_conical", "B": "R", "exponent": 1, "kind": "upper_bound", "statement": "rel_conical(S)<=R(S).", "source": "BK24", "locator": "Section 2, Definitions 10–12 and standard solver definitions", "proof_summary": "For every output sum the probabilities of leaves labeled with that output. Each leaf is a conjunction of at most R queried literals with a nonnegative coefficient. The full-cube normalization is at most one.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "search-elementary-Q-QE", "A": "Q", "B": "QE", "exponent": 1, "kind": "upper_bound", "statement": "Q(S)<=QE(S).", "source": "BK24", "locator": "Section 2, Definitions 10–12 and standard solver definitions", "proof_summary": "An exact quantum solver is also a bounded-error solver.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "search-elementary-QE-D", "A": "QE", "B": "D", "exponent": 1, "kind": "upper_bound", "statement": "QE(S)<=D(S).", "source": "BK24", "locator": "Section 2, Definitions 10–12 and standard solver definitions", "proof_summary": "Execute the deterministic solver coherently and measure its valid output.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "search-elementary-R-Rfail", "A": "R", "B": "Rfail", "exponent": 1, "kind": "upper_bound", "statement": "R(S)<=Rfail(S).", "source": "BK24", "locator": "Section 2, Definitions 10–12 and standard solver definitions", "proof_summary": "Replace detectable failure by any fixed label, counting it as error.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "search-elementary-Rfail-D", "A": "Rfail", "B": "D", "exponent": 1, "kind": "upper_bound", "statement": "Rfail(S)<=D(S).", "source": "BK24", "locator": "Section 2, Definitions 10–12 and standard solver definitions", "proof_summary": "A deterministic solver never fails.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "search-elementary-R0-D", "A": "R0", "B": "D", "exponent": 1, "kind": "upper_bound", "statement": "R0(S)<=D(S).", "source": "BK24", "locator": "Section 2, Definitions 10–12 and standard solver definitions", "proof_summary": "A deterministic solver is an always-correct randomized solver with the same expected cost.", "scope": { "object_kind": "search_relation", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "arbitrary_finite", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "relevant-certificate-exp", "A": "nRelevant", "B": "C", "exponent": null, "kind": "nonpolynomial_upper_bound", "statement": "nRelevant(f)<=4^C(f)/2.", "source": "Wel22", "locator": "Theorem 1.1, equation (1.4)", "proof_summary": "Use the restriction-decreasing coordinate potential built from certificate measures, and induct under restrictions.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false }, { "id": "relevant-monotone-degree", "A": "nRelevant", "B": "deg", "exponent": null, "kind": "nonpolynomial_upper_bound", "statement": "nRelevant(f)<=1.325*2^deg(f).", "source": "Wel22", "locator": "Theorem 1.1", "proof_summary": "Apply the coordinate-potential restriction argument with stronger monotone local-degree constraints.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "monotone" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false }, { "id": "relevant-monotone-sensitivity", "A": "nRelevant", "B": "s", "exponent": null, "kind": "nonpolynomial_upper_bound", "statement": "nRelevant(f)<=4^s(f)/2.", "source": "Wel22", "locator": "Theorem 1.1", "proof_summary": "Specialize the coordinate-measure potential to monotone sensitivity and bound the potential after restrictions.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "monotone" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false }, { "id": "relevant-monotone-D", "A": "nRelevant", "B": "D", "exponent": null, "kind": "nonpolynomial_upper_bound", "statement": "nRelevant(f)<=2^D(f)/4+2.", "source": "Wel22", "locator": "Theorem 1.1", "proof_summary": "Use the monotone decision-tree recurrence and restriction-decreasing local depth potential.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "monotone" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false }, { "id": "relevant-degree-CHS-historical", "A": "nRelevant", "B": "deg", "exponent": null, "kind": "nonpolynomial_upper_bound", "statement": "nRelevant(f)<=6.614*2^deg(f); later constant improved to 4.394 by Wel22.", "source": "CHS20", "locator": "Theorem 1.1 and Section 2", "proof_summary": "Weight each relevant coordinate by 2^(-local degree). Restrict a maxonomial hitting set and bound the increase in the normalized relevant-variable count; sum the resulting convergent series.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "historical": true, "superseded_by": [ "nRelevant-deg-exp" ] }, { "id": "relevant-degree-no-power", "A": "nRelevant", "B": "deg", "exponent": null, "kind": "no_polynomial_bound", "statement": "No fixed polynomial upper bound of nRelevant by degree: read-once complete depth-d decision trees on distinct internal-node variables depend on 2^d-1 variables and have degree d.", "source": "CHS20", "locator": "Introduction, p.1", "proof_summary": "Use an address-style complete depth-d tree with fresh variables at every internal node and leaf labels chosen so every node query is relevant. Its degree is at most d, and the deepest queries witness degree d.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false }, { "id": "permutation-degree-sensitivity", "A": "permDegree", "B": "permS", "exponent": 2, "kind": "upper_bound", "statement": "deg(f)<=2*s(f)^2 on S_n, with s the disjoint-transposition packing sensitivity.", "source": "DFLLV21", "locator": "Corollary 5.6", "proof_summary": "Use Young-tableau pseudo-characters to witness a nonzero top-degree component; embed a Boolean cube of dimension at least degree/2 whose neighbors map to disjoint transposition neighbors. Apply Huang’s cube sensitivity theorem.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric_group" ], "input_alphabet": "permutation", "output_alphabet": "bits", "cost": "forward_or_inverse_symbol_queries", "error": "as_in_measure_definitions", "assumptions": [ "Domain S_n; not permutation-invariant Boolean-cube functions." ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "constant": 2, "proof_sources": [ "DFLLV22author", "DFLLV26" ], "later_manuscript_check": "Main indexed statements checked against the 13 May 2022 author version; theorem numbering remains as separately indicated. Final journal text unchecked.", "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "permutation-bs-degree", "A": "permBS", "B": "permAdeg", "exponent": 2, "kind": "upper_bound", "statement": "bs(f)<=6*approx_degree(f)^2 on S_n.", "source": "DFLLV21", "locator": "Theorem 3.1; Lemma 3.6", "proof_summary": "Disjoint sensitive permutation blocks generate a Boolean subcube. Restrict the approximating polynomial to this cube and symmetrize; the standard approximate-degree/block-sensitivity argument applies.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric_group" ], "input_alphabet": "permutation", "output_alphabet": "bits", "cost": "forward_or_inverse_symbol_queries", "error": "as_in_measure_definitions", "assumptions": [ "Domain S_n; not permutation-invariant Boolean-cube functions." ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "constant": 6, "proof_sources": [ "DFLLV22author", "DFLLV26" ], "later_manuscript_check": "Main indexed statements checked against the 13 May 2022 author version; theorem numbering remains as separately indicated. Final journal text unchecked.", "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "permutation-s-edge", "A": "permS", "B": "permEdgeS", "exponent": 1, "kind": "upper_bound", "statement": "s(f)<=t(f); the packing sensitivity and the total transposition edge count are different measures.", "source": "DFLLV21", "locator": "Section 2.2, Notes on sensitivity", "proof_summary": "Every family of disjoint sensitive transpositions is a subset of all sensitive transpositions.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric_group" ], "input_alphabet": "permutation", "output_alphabet": "bits", "cost": "forward_or_inverse_symbol_queries", "error": "as_in_measure_definitions", "assumptions": [ "Domain S_n; not permutation-invariant Boolean-cube functions." ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "proof_sources": [ "DFLLV22author", "DFLLV26" ], "later_manuscript_check": "Main indexed statements checked against the 13 May 2022 author version; theorem numbering remains as separately indicated. Final journal text unchecked.", "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "completion-D-equals-D", "A": "completion_D", "B": "D", "exponent": 1, "kind": "upper_bound", "statement": "completion_D(f)=D(f) for every partial Boolean function.", "source": "HKMP26", "locator": "Observation 49, p.23", "proof_summary": "A deterministic decision tree for the partial function outputs some bit off-promise and hence defines a total completion of the same depth. Conversely every completion solver solves the partial function.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "D-equals-completion-D", "A": "D", "B": "completion_D", "exponent": 1, "kind": "upper_bound", "statement": "D(f)=completion_D(f).", "source": "HKMP26", "locator": "Observation 49, p.23", "proof_summary": "Restrict an optimal total-completion solver to the promised domain; use the reverse construction for equality.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "native-to-completion-R0", "A": "R0", "B": "completion_R0", "exponent": 1, "kind": "upper_bound", "statement": "R0(f)<=completion_R0(f).", "source": "HKMP26", "locator": "Definition 45; restriction of a total solver or witness", "proof_summary": "Every total completion agrees on the promise. Restrict its solver, certificates, sensitive blocks or polynomial to the promised task, then minimize. For degree this is the unrestricted promise degree, a relaxation of the total polynomial.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "native-to-completion-R", "A": "R", "B": "completion_R", "exponent": 1, "kind": "upper_bound", "statement": "R(f)<=completion_R(f).", "source": "HKMP26", "locator": "Definition 45; restriction of a total solver or witness", "proof_summary": "Every total completion agrees on the promise. Restrict its solver, certificates, sensitive blocks or polynomial to the promised task, then minimize. For degree this is the unrestricted promise degree, a relaxation of the total polynomial.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "native-to-completion-C", "A": "C", "B": "completion_C", "exponent": 1, "kind": "upper_bound", "statement": "C(f)<=completion_C(f).", "source": "HKMP26", "locator": "Definition 45; restriction of a total solver or witness", "proof_summary": "Every total completion agrees on the promise. Restrict its solver, certificates, sensitive blocks or polynomial to the promised task, then minimize. For degree this is the unrestricted promise degree, a relaxation of the total polynomial.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "native-to-completion-bs", "A": "bs", "B": "completion_bs", "exponent": 1, "kind": "upper_bound", "statement": "bs(f)<=completion_bs(f).", "source": "HKMP26", "locator": "Definition 45; restriction of a total solver or witness", "proof_summary": "Every total completion agrees on the promise. Restrict its solver, certificates, sensitive blocks or polynomial to the promised task, then minimize. For degree this is the unrestricted promise degree, a relaxation of the total polynomial.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "native-to-completion-s", "A": "s", "B": "completion_s", "exponent": 1, "kind": "upper_bound", "statement": "s(f)<=completion_s(f).", "source": "HKMP26", "locator": "Definition 45; restriction of a total solver or witness", "proof_summary": "Every total completion agrees on the promise. Restrict its solver, certificates, sensitive blocks or polynomial to the promised task, then minimize. For degree this is the unrestricted promise degree, a relaxation of the total polynomial.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "native-to-completion-QE", "A": "QE", "B": "completion_QE", "exponent": 1, "kind": "upper_bound", "statement": "QE(f)<=completion_QE(f).", "source": "HKMP26", "locator": "Definition 45; restriction of a total solver or witness", "proof_summary": "Every total completion agrees on the promise. Restrict its solver, certificates, sensitive blocks or polynomial to the promised task, then minimize. For degree this is the unrestricted promise degree, a relaxation of the total polynomial.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "native-to-completion-deg", "A": "deg", "B": "completion_deg", "exponent": 1, "kind": "upper_bound", "statement": "deg(f)<=completion_deg(f).", "source": "HKMP26", "locator": "Definition 45; restriction of a total solver or witness", "proof_summary": "Every total completion agrees on the promise. Restrict its solver, certificates, sensitive blocks or polynomial to the promised task, then minimize. For degree this is the unrestricted promise degree, a relaxation of the total polynomial.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "native-to-completion-Q", "A": "Q", "B": "completion_Q", "exponent": 1, "kind": "upper_bound", "statement": "Q(f)<=completion_Q(f).", "source": "HKMP26", "locator": "Definition 45; restriction of a total solver or witness", "proof_summary": "Every total completion agrees on the promise. Restrict its solver, certificates, sensitive blocks or polynomial to the promised task, then minimize. For degree this is the unrestricted promise degree, a relaxation of the total polynomial.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "native-to-completion-adeg", "A": "adeg", "B": "completion_adeg", "exponent": 1, "kind": "upper_bound", "statement": "adeg(f)<=completion_adeg(f).", "source": "HKMP26", "locator": "Definition 45; restriction of a total solver or witness", "proof_summary": "Every total completion agrees on the promise. Restrict its solver, certificates, sensitive blocks or polynomial to the promised task, then minimize. For degree this is the unrestricted promise degree, a relaxation of the total polynomial.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "promise-s-bs", "A": "promiseS", "B": "promiseBS", "exponent": 1, "kind": "upper_bound", "statement": "s prime<=bs prime.", "source": "HKMP26", "locator": "Observation 16", "proof_summary": "Single-coordinate sensitive exits are disjoint sensitive blocks.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "promise-bs-strongC", "A": "promiseBS", "B": "Cstrong", "exponent": 1, "kind": "upper_bound", "statement": "bs prime<=Cstrong.", "source": "HKMP26", "locator": "Observation 16", "proof_summary": "A strong certificate intersects each disjoint block that exits the promise or changes the output.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "critical-bs-exit", "A": "criticalBScompletion", "B": "promiseBS", "exponent": 1, "kind": "upper_bound", "statement": "cbs<=bs prime.", "source": "HKMP26", "locator": "Lemma 35", "proof_summary": "At promised inputs every block sensitive in a completion either changes the original output or exits the promise.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "bs-critical-completion", "A": "bs", "B": "criticalBScompletion", "exponent": 1, "kind": "upper_bound", "statement": "bs<=cbs.", "source": "HKMP26", "locator": "Lemma 35", "proof_summary": "Every completion retains all promised opposite-output sensitive blocks.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "critical-bs-bounded-adeg", "A": "criticalBScompletion", "B": "cubeBoundedAdeg", "exponent": 2, "kind": "upper_bound", "statement": "cbs=O(cubeBoundedAdeg^2).", "source": "HKMP26", "locator": "Lemma 20, citing ABK21", "proof_summary": "Use the completion-critical block-sensitivity versus globally bounded polynomial theorem; off-promise boundedness is essential.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "D-critical-certificate-bs", "A": "D", "B": "criticalCertificate", "exponent": null, "kind": "upper_bound", "statement": "D<=criticalCertificate*criticalBScompletion.", "source": "HKMP26", "locator": "Lemma 33(3)", "proof_summary": "Choose a completion attaining both nested optima. Query consistent positive certificates; if too many rounds leave conflicting promised outputs, their disjoint disagreement sets contradict the completion-critical block bound.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "factors": [ { "measure": "criticalCertificate", "exponent": 1 }, { "measure": "criticalBScompletion", "exponent": 1 } ] }, { "id": "promise-C-bs-exitS", "A": "C", "B": "bs", "exponent": null, "kind": "upper_bound", "statement": "C<=bs*promiseS.", "source": "HKMP26", "locator": "Lemma 32", "proof_summary": "A maximal family of disjoint minimal promised opposite-output blocks covers a weak certificate. At a minimal block endpoint, undoing any one coordinate either changes the output back or exits the promise, bounding block size by exit-sensitive sensitivity.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "factors": [ { "measure": "bs", "exponent": 1 }, { "measure": "promiseS", "exponent": 1 } ] }, { "id": "promise-D-weak-strong-C", "A": "D", "B": "C", "exponent": null, "kind": "upper_bound", "statement": "D<=C*Cstrong.", "source": "HKMP26", "locator": "Lemma 34", "proof_summary": "Query a consistent strong 1-certificate in each round. Against any surviving 0-input, each round reveals a new coordinate of its weak 0-certificate; after C rounds the answer is determined.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "factors": [ { "measure": "C", "exponent": 1 }, { "measure": "Cstrong", "exponent": 1 } ] }, { "id": "relevant-degree-43935", "A": "nRelevant", "B": "deg", "exponent": null, "kind": "nonpolynomial_upper_bound", "statement": "nRelevant(f)<=4.3935*2^deg(f).", "source": "Wel22", "locator": "Theorem 3.13, p.10; visually checked", "proof_summary": "Use exact-rational small-degree LP block-sensitivity bounds through degree 14, the general block-sensitivity improvement and the coordinate-potential recurrence to bound its limiting constant by 4.3935.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "constant": { "value": "4.3935", "normalization": "relevant coordinates" }, "notes": "Sharper than introductory 4.394, not a new later publication. The reported Maple LP computation was not independently reproduced." }, { "id": "Nis89-expected-one-sided-repair", "A": "D", "B": "NisR1expect", "exponent": 2, "kind": "upper_bound", "statement": "D(f)<=4*NisR1expect(f)^2; factor 4 is optimal for this exact expected-cost, success-1/2 convention.", "source": "Nis89", "locator": "Repository derivation using Section 1.2 and the certificate-query method of Lemma 4.1", "proof_summary": "Let T be worst-case expected cost. On any 1-input, accepting transcripts are certificates, so C1<=2T. Query a consistent 1-certificate each round. For any surviving 0-input, coupling it with an input satisfying each queried certificate forces probability at least 1/2 of probing a new disagreement block. Blocks from different rounds are disjoint, so at most floor(2T) rounds are possible. Thus D<=2T*C1<=4T^2. One-bit identity has T=1/2 (acceptance implies a query), attaining equality.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "constant": 4, "notes": "Not attributed as an original Nisan constant or a published correction. The default capped R_1 is a distinct measure." }, { "id": "perm-chain-permBS-permFBS", "A": "permBS", "B": "permFBS", "exponent": 1, "kind": "upper_bound", "statement": "bs<=fbs.", "source": "DFLLV21", "locator": "Theorem 3.1; Section 4.2 parameters; Lemma 3.5 for C", "proof_summary": "The unit-weight integral packing is feasible for the fractional LP.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric_group" ], "input_alphabet": "permutation", "output_alphabet": "bits", "cost": "forward_or_inverse_symbol_queries", "error": "as_in_measure_definitions", "assumptions": [ "Domain S_n; not permutation-invariant Boolean-cube functions." ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "proof_sources": [ "DFLLV22author", "DFLLV26" ], "later_manuscript_check": "Main indexed statements checked against the 13 May 2022 author version; theorem numbering remains as separately indicated. Final journal text unchecked.", "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "perm-chain-permFBS-permC", "A": "permFBS", "B": "permC", "exponent": 1, "kind": "upper_bound", "statement": "fbs<=C.", "source": "DFLLV21", "locator": "Theorem 3.1; Section 4.2 parameters; Lemma 3.5 for C", "proof_summary": "A certificate must intersect every opposite-output disagreement block; use the certificate as a feasible dual cover.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric_group" ], "input_alphabet": "permutation", "output_alphabet": "bits", "cost": "forward_or_inverse_symbol_queries", "error": "as_in_measure_definitions", "assumptions": [ "Domain S_n; not permutation-invariant Boolean-cube functions." ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "proof_sources": [ "DFLLV22author", "DFLLV26" ], "later_manuscript_check": "Main indexed statements checked against the 13 May 2022 author version; theorem numbering remains as separately indicated. Final journal text unchecked.", "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "perm-chain-permC-permUC", "A": "permC", "B": "permUC", "exponent": 1, "kind": "upper_bound", "statement": "C<=U.", "source": "DFLLV21", "locator": "Theorem 3.1; Section 4.2 parameters; Lemma 3.5 for C", "proof_summary": "Every unambiguous partition certificate is an ordinary certificate.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric_group" ], "input_alphabet": "permutation", "output_alphabet": "bits", "cost": "forward_or_inverse_symbol_queries", "error": "as_in_measure_definitions", "assumptions": [ "Domain S_n; not permutation-invariant Boolean-cube functions." ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "proof_sources": [ "DFLLV22author", "DFLLV26" ], "later_manuscript_check": "Main indexed statements checked against the 13 May 2022 author version; theorem numbering remains as separately indicated. Final journal text unchecked.", "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "perm-chain-permUC-permD", "A": "permUC", "B": "permD", "exponent": 1, "kind": "upper_bound", "statement": "U<=D.", "source": "DFLLV21", "locator": "Theorem 3.1; Section 4.2 parameters; Lemma 3.5 for C", "proof_summary": "Each decision-tree leaf supplies a compatible permutation-entry certificate. The leaf sets partition S_n.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric_group" ], "input_alphabet": "permutation", "output_alphabet": "bits", "cost": "forward_or_inverse_symbol_queries", "error": "as_in_measure_definitions", "assumptions": [ "Domain S_n; not permutation-invariant Boolean-cube functions." ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "proof_sources": [ "DFLLV22author", "DFLLV26" ], "later_manuscript_check": "Main indexed statements checked against the 13 May 2022 author version; theorem numbering remains as separately indicated. Final journal text unchecked.", "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "perm-chain-permBS-permR", "A": "permBS", "B": "permR", "exponent": 1, "kind": "upper_bound", "statement": "bs<=3R.", "source": "DFLLV21", "locator": "Theorem 3.1; Section 4.2 parameters; Lemma 3.5 for C", "proof_summary": "Choose disjoint sensitive neighbors and apply the classical distinguishing/coupling lower bound at error 1/3.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric_group" ], "input_alphabet": "permutation", "output_alphabet": "bits", "cost": "forward_or_inverse_symbol_queries", "error": "as_in_measure_definitions", "assumptions": [ "Domain S_n; not permutation-invariant Boolean-cube functions." ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "proof_sources": [ "DFLLV22author", "DFLLV26" ], "later_manuscript_check": "Main indexed statements checked against the 13 May 2022 author version; theorem numbering remains as separately indicated. Final journal text unchecked.", "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "perm-chain-permR-permRfail23", "A": "permR", "B": "permRfail23", "exponent": 1, "kind": "upper_bound", "statement": "R<=R_0.", "source": "DFLLV21", "locator": "Theorem 3.1; Section 4.2 parameters; Lemma 3.5 for C", "proof_summary": "On detectable failure, output a uniformly random bit. Success is at least 1/3+(2/3)/2=2/3 with the same depth cap; this uses Boolean output.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric_group" ], "input_alphabet": "permutation", "output_alphabet": "bits", "cost": "forward_or_inverse_symbol_queries", "error": "as_in_measure_definitions", "assumptions": [ "Domain S_n; not permutation-invariant Boolean-cube functions." ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "proof_sources": [ "DFLLV22author", "DFLLV26" ], "later_manuscript_check": "Main indexed statements checked against the 13 May 2022 author version; theorem numbering remains as separately indicated. Final journal text unchecked.", "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "perm-chain-permRfail23-permD", "A": "permRfail23", "B": "permD", "exponent": 1, "kind": "upper_bound", "statement": "R_0<=D.", "source": "DFLLV21", "locator": "Theorem 3.1; Section 4.2 parameters; Lemma 3.5 for C", "proof_summary": "The deterministic tree never fails.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric_group" ], "input_alphabet": "permutation", "output_alphabet": "bits", "cost": "forward_or_inverse_symbol_queries", "error": "as_in_measure_definitions", "assumptions": [ "Domain S_n; not permutation-invariant Boolean-cube functions." ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "proof_sources": [ "DFLLV22author", "DFLLV26" ], "later_manuscript_check": "Main indexed statements checked against the 13 May 2022 author version; theorem numbering remains as separately indicated. Final journal text unchecked.", "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "perm-chain-permAdeg-permQ", "A": "permAdeg", "B": "permQ", "exponent": 1, "kind": "upper_bound", "statement": "approx_degree<=2Q.", "source": "DFLLV21", "locator": "Theorem 3.1; Section 4.2 parameters; Lemma 3.5 for C", "proof_summary": "Quantum acceptance probabilities are polynomials of degree at most twice the symbol-query count in the domain indicators.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric_group" ], "input_alphabet": "permutation", "output_alphabet": "bits", "cost": "forward_or_inverse_symbol_queries", "error": "as_in_measure_definitions", "assumptions": [ "Domain S_n; not permutation-invariant Boolean-cube functions." ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "proof_sources": [ "DFLLV22author", "DFLLV26" ], "later_manuscript_check": "Main indexed statements checked against the 13 May 2022 author version; theorem numbering remains as separately indicated. Final journal text unchecked.", "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "perm-chain-permQ-permQE", "A": "permQ", "B": "permQE", "exponent": 1, "kind": "upper_bound", "statement": "Q<=Q_E.", "source": "DFLLV21", "locator": "Theorem 3.1; Section 4.2 parameters; Lemma 3.5 for C", "proof_summary": "An exact solver is bounded-error.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric_group" ], "input_alphabet": "permutation", "output_alphabet": "bits", "cost": "forward_or_inverse_symbol_queries", "error": "as_in_measure_definitions", "assumptions": [ "Domain S_n; not permutation-invariant Boolean-cube functions." ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "proof_sources": [ "DFLLV22author", "DFLLV26" ], "later_manuscript_check": "Main indexed statements checked against the 13 May 2022 author version; theorem numbering remains as separately indicated. Final journal text unchecked.", "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "perm-chain-permQE-permD", "A": "permQE", "B": "permD", "exponent": 1, "kind": "upper_bound", "statement": "Q_E<=D.", "source": "DFLLV21", "locator": "Theorem 3.1; Section 4.2 parameters; Lemma 3.5 for C", "proof_summary": "Execute each deterministic symbol query coherently.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric_group" ], "input_alphabet": "permutation", "output_alphabet": "bits", "cost": "forward_or_inverse_symbol_queries", "error": "as_in_measure_definitions", "assumptions": [ "Domain S_n; not permutation-invariant Boolean-cube functions." ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "proof_sources": [ "DFLLV22author", "DFLLV26" ], "later_manuscript_check": "Main indexed statements checked against the 13 May 2022 author version; theorem numbering remains as separately indicated. Final journal text unchecked.", "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "perm-chain-permD-permBS", "A": "permD", "B": "permBS", "exponent": 4, "kind": "upper_bound", "statement": "D<=18*bs^4.", "source": "DFLLV21", "locator": "Theorem 3.1; Section 4.2 parameters; Lemma 3.5 for C", "proof_summary": "Specialize the general certificate-query simulation with Delta=2, conflict bound M=n and beta>=1/3 for perfect matchings representing permutations.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric_group" ], "input_alphabet": "permutation", "output_alphabet": "bits", "cost": "forward_or_inverse_symbol_queries", "error": "as_in_measure_definitions", "assumptions": [ "Domain S_n; not permutation-invariant Boolean-cube functions." ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "proof_sources": [ "DFLLV22author", "DFLLV26" ], "later_manuscript_check": "Main indexed statements checked against the 13 May 2022 author version; theorem numbering remains as separately indicated. Final journal text unchecked.", "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "perm-chain-permC-permBS", "A": "permC", "B": "permBS", "exponent": 2, "kind": "upper_bound", "statement": "C<=3*bs^2.", "source": "DFLLV21", "locator": "Theorem 3.1; Section 4.2 parameters; Lemma 3.5 for C", "proof_summary": "Apply Lemma 3.5 with beta>=1/3: maximal disjoint minimal disagreement sets form a certificate.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric_group" ], "input_alphabet": "permutation", "output_alphabet": "bits", "cost": "forward_or_inverse_symbol_queries", "error": "as_in_measure_definitions", "assumptions": [ "Domain S_n; not permutation-invariant Boolean-cube functions." ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "proof_sources": [ "DFLLV22author", "DFLLV26" ], "later_manuscript_check": "Main indexed statements checked against the 13 May 2022 author version; theorem numbering remains as separately indicated. Final journal text unchecked.", "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "perm-chain-permS-permBS", "A": "permS", "B": "permBS", "exponent": 1, "kind": "upper_bound", "statement": "s<=bs.", "source": "DFLLV21", "locator": "Theorem 3.1; Section 4.2 parameters; Lemma 3.5 for C", "proof_summary": "Disjoint sensitive transpositions are a special case of disjoint sensitive moved-point supports.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric_group" ], "input_alphabet": "permutation", "output_alphabet": "bits", "cost": "forward_or_inverse_symbol_queries", "error": "as_in_measure_definitions", "assumptions": [ "Domain S_n; not permutation-invariant Boolean-cube functions." ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "proof_sources": [ "DFLLV22author", "DFLLV26" ], "later_manuscript_check": "Main indexed statements checked against the 13 May 2022 author version; theorem numbering remains as separately indicated. Final journal text unchecked.", "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "perm-chain-permAdeg-permDegree", "A": "permAdeg", "B": "permDegree", "exponent": 1, "kind": "upper_bound", "statement": "approx_degree<=degree.", "source": "DFLLV21", "locator": "Section 2.2 and Theorem 3.1", "proof_summary": "An exact polynomial approximates with zero error.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric_group" ], "input_alphabet": "permutation", "output_alphabet": "bits", "cost": "forward_or_inverse_symbol_queries", "error": "as_in_measure_definitions", "assumptions": [ "Domain S_n; not permutation-invariant Boolean-cube functions." ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "proof_sources": [ "DFLLV22author", "DFLLV26" ], "later_manuscript_check": "Main indexed statements checked against the 13 May 2022 author version; theorem numbering remains as separately indicated. Final journal text unchecked.", "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "perm-chain-permDegree-permD", "A": "permDegree", "B": "permD", "exponent": 1, "kind": "upper_bound", "statement": "degree<=D.", "source": "DFLLV21", "locator": "Section 2.2 and Theorem 3.1", "proof_summary": "Sum the products of entry indicators along accepting leaves of an optimal tree; each product has degree at most the depth.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric_group" ], "input_alphabet": "permutation", "output_alphabet": "bits", "cost": "forward_or_inverse_symbol_queries", "error": "as_in_measure_definitions", "assumptions": [ "Domain S_n; not permutation-invariant Boolean-cube functions." ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "proof_sources": [ "DFLLV22author", "DFLLV26" ], "later_manuscript_check": "Main indexed statements checked against the 13 May 2022 author version; theorem numbering remains as separately indicated. Final journal text unchecked.", "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "perm-chain-permQ-permR", "A": "permQ", "B": "permR", "exponent": 1, "kind": "upper_bound", "statement": "Q<=R.", "source": "DFLLV21", "locator": "Section 2.2 and Theorem 3.1", "proof_summary": "Simulate a capped classical randomized symbol-query algorithm with a quantum query algorithm, including its random choices.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric_group" ], "input_alphabet": "permutation", "output_alphabet": "bits", "cost": "forward_or_inverse_symbol_queries", "error": "as_in_measure_definitions", "assumptions": [ "Domain S_n; not permutation-invariant Boolean-cube functions." ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "proof_sources": [ "DFLLV22author", "DFLLV26" ], "later_manuscript_check": "Main indexed statements checked against the 13 May 2022 author version; theorem numbering remains as separately indicated. Final journal text unchecked.", "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "critical-bs-completion-bs", "A": "criticalBScompletion", "B": "completion_bs", "exponent": 1, "kind": "upper_bound", "statement": "criticalBScompletion<=completion_bs.", "source": "HKMP26", "locator": "Definitions 18–19 and 45", "proof_summary": "For every completion, its maximum block sensitivity over promised points is at most its maximum over the whole cube. Minimize the former and evaluate it at a completion minimizing the latter.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "weak-C-critical-certificate", "A": "C", "B": "criticalCertificate", "exponent": 1, "kind": "upper_bound", "statement": "C<=criticalCertificate.", "source": "HKMP26", "locator": "Definitions 18–19 and 45", "proof_summary": "Any full-cube certificate in a block-optimal total completion is a valid weak promise certificate. Maximize over promised inputs and then minimize over block-optimal completions.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "perm-edge-no-power-permD", "A": "permEdgeS", "B": "permD", "exponent": null, "kind": "no_polynomial_bound", "statement": "No finite power bounds permEdgeS by permD: the single-entry indicator has edge sensitivity n-1 and permD=O(1).", "source": "definitions", "locator": "Repository example using DFLLV21 Section 2.2 definitions", "proof_summary": "For f(pi)=[pi(1)=1], one symbol query solves the function and degree is one. Its n-1 sensitive transpositions at a positive input all share position 1, so the disjoint packing is one. A one-entry certificate and its n disjoint output cases give certificate, fractional and unambiguous complexities at most one. Quantum and randomized costs are also at most one.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric_group" ], "input_alphabet": "permutation", "output_alphabet": "bits", "cost": "forward_or_inverse_symbol_queries", "error": "as_in_measure_definitions", "assumptions": [ "Domain S_n; not permutation-invariant Boolean-cube functions." ] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "proof_sources": [ "DFLLV21" ], "witness_id": "permutation-single-entry" }, { "id": "perm-edge-no-power-permRfail23", "A": "permEdgeS", "B": "permRfail23", "exponent": null, "kind": "no_polynomial_bound", "statement": "No finite power bounds permEdgeS by permRfail23: the single-entry indicator has edge sensitivity n-1 and permRfail23=O(1).", "source": "definitions", "locator": "Repository example using DFLLV21 Section 2.2 definitions", "proof_summary": "For f(pi)=[pi(1)=1], one symbol query solves the function and degree is one. Its n-1 sensitive transpositions at a positive input all share position 1, so the disjoint packing is one. A one-entry certificate and its n disjoint output cases give certificate, fractional and unambiguous complexities at most one. Quantum and randomized costs are also at most one.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric_group" ], "input_alphabet": "permutation", "output_alphabet": "bits", "cost": "forward_or_inverse_symbol_queries", "error": "as_in_measure_definitions", "assumptions": [ "Domain S_n; not permutation-invariant Boolean-cube functions." ] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "proof_sources": [ "DFLLV21" ], "witness_id": "permutation-single-entry" }, { "id": "perm-edge-no-power-permR", "A": "permEdgeS", "B": "permR", "exponent": null, "kind": "no_polynomial_bound", "statement": "No finite power bounds permEdgeS by permR: the single-entry indicator has edge sensitivity n-1 and permR=O(1).", "source": "definitions", "locator": "Repository example using DFLLV21 Section 2.2 definitions", "proof_summary": "For f(pi)=[pi(1)=1], one symbol query solves the function and degree is one. Its n-1 sensitive transpositions at a positive input all share position 1, so the disjoint packing is one. A one-entry certificate and its n disjoint output cases give certificate, fractional and unambiguous complexities at most one. Quantum and randomized costs are also at most one.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric_group" ], "input_alphabet": "permutation", "output_alphabet": "bits", "cost": "forward_or_inverse_symbol_queries", "error": "as_in_measure_definitions", "assumptions": [ "Domain S_n; not permutation-invariant Boolean-cube functions." ] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "proof_sources": [ "DFLLV21" ], "witness_id": "permutation-single-entry" }, { "id": "perm-edge-no-power-permC", "A": "permEdgeS", "B": "permC", "exponent": null, "kind": "no_polynomial_bound", "statement": "No finite power bounds permEdgeS by permC: the single-entry indicator has edge sensitivity n-1 and permC=O(1).", "source": "definitions", "locator": "Repository example using DFLLV21 Section 2.2 definitions", "proof_summary": "For f(pi)=[pi(1)=1], one symbol query solves the function and degree is one. Its n-1 sensitive transpositions at a positive input all share position 1, so the disjoint packing is one. A one-entry certificate and its n disjoint output cases give certificate, fractional and unambiguous complexities at most one. Quantum and randomized costs are also at most one.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric_group" ], "input_alphabet": "permutation", "output_alphabet": "bits", "cost": "forward_or_inverse_symbol_queries", "error": "as_in_measure_definitions", "assumptions": [ "Domain S_n; not permutation-invariant Boolean-cube functions." ] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "proof_sources": [ "DFLLV21" ], "witness_id": "permutation-single-entry" }, { "id": "perm-edge-no-power-permUC", "A": "permEdgeS", "B": "permUC", "exponent": null, "kind": "no_polynomial_bound", "statement": "No finite power bounds permEdgeS by permUC: the single-entry indicator has edge sensitivity n-1 and permUC=O(1).", "source": "definitions", "locator": "Repository example using DFLLV21 Section 2.2 definitions", "proof_summary": "For f(pi)=[pi(1)=1], one symbol query solves the function and degree is one. Its n-1 sensitive transpositions at a positive input all share position 1, so the disjoint packing is one. A one-entry certificate and its n disjoint output cases give certificate, fractional and unambiguous complexities at most one. Quantum and randomized costs are also at most one.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric_group" ], "input_alphabet": "permutation", "output_alphabet": "bits", "cost": "forward_or_inverse_symbol_queries", "error": "as_in_measure_definitions", "assumptions": [ "Domain S_n; not permutation-invariant Boolean-cube functions." ] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "proof_sources": [ "DFLLV21" ], "witness_id": "permutation-single-entry" }, { "id": "perm-edge-no-power-permFBS", "A": "permEdgeS", "B": "permFBS", "exponent": null, "kind": "no_polynomial_bound", "statement": "No finite power bounds permEdgeS by permFBS: the single-entry indicator has edge sensitivity n-1 and permFBS=O(1).", "source": "definitions", "locator": "Repository example using DFLLV21 Section 2.2 definitions", "proof_summary": "For f(pi)=[pi(1)=1], one symbol query solves the function and degree is one. Its n-1 sensitive transpositions at a positive input all share position 1, so the disjoint packing is one. A one-entry certificate and its n disjoint output cases give certificate, fractional and unambiguous complexities at most one. Quantum and randomized costs are also at most one.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric_group" ], "input_alphabet": "permutation", "output_alphabet": "bits", "cost": "forward_or_inverse_symbol_queries", "error": "as_in_measure_definitions", "assumptions": [ "Domain S_n; not permutation-invariant Boolean-cube functions." ] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "proof_sources": [ "DFLLV21" ], "witness_id": "permutation-single-entry" }, { "id": "perm-edge-no-power-permBS", "A": "permEdgeS", "B": "permBS", "exponent": null, "kind": "no_polynomial_bound", "statement": "No finite power bounds permEdgeS by permBS: the single-entry indicator has edge sensitivity n-1 and permBS=O(1).", "source": "definitions", "locator": "Repository example using DFLLV21 Section 2.2 definitions", "proof_summary": "For f(pi)=[pi(1)=1], one symbol query solves the function and degree is one. Its n-1 sensitive transpositions at a positive input all share position 1, so the disjoint packing is one. A one-entry certificate and its n disjoint output cases give certificate, fractional and unambiguous complexities at most one. Quantum and randomized costs are also at most one.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric_group" ], "input_alphabet": "permutation", "output_alphabet": "bits", "cost": "forward_or_inverse_symbol_queries", "error": "as_in_measure_definitions", "assumptions": [ "Domain S_n; not permutation-invariant Boolean-cube functions." ] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "proof_sources": [ "DFLLV21" ], "witness_id": "permutation-single-entry" }, { "id": "perm-edge-no-power-permS", "A": "permEdgeS", "B": "permS", "exponent": null, "kind": "no_polynomial_bound", "statement": "No finite power bounds permEdgeS by permS: the single-entry indicator has edge sensitivity n-1 and permS=O(1).", "source": "definitions", "locator": "Repository example using DFLLV21 Section 2.2 definitions", "proof_summary": "For f(pi)=[pi(1)=1], one symbol query solves the function and degree is one. Its n-1 sensitive transpositions at a positive input all share position 1, so the disjoint packing is one. A one-entry certificate and its n disjoint output cases give certificate, fractional and unambiguous complexities at most one. Quantum and randomized costs are also at most one.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric_group" ], "input_alphabet": "permutation", "output_alphabet": "bits", "cost": "forward_or_inverse_symbol_queries", "error": "as_in_measure_definitions", "assumptions": [ "Domain S_n; not permutation-invariant Boolean-cube functions." ] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "proof_sources": [ "DFLLV21" ], "witness_id": "permutation-single-entry" }, { "id": "perm-edge-no-power-permDegree", "A": "permEdgeS", "B": "permDegree", "exponent": null, "kind": "no_polynomial_bound", "statement": "No finite power bounds permEdgeS by permDegree: the single-entry indicator has edge sensitivity n-1 and permDegree=O(1).", "source": "definitions", "locator": "Repository example using DFLLV21 Section 2.2 definitions", "proof_summary": "For f(pi)=[pi(1)=1], one symbol query solves the function and degree is one. Its n-1 sensitive transpositions at a positive input all share position 1, so the disjoint packing is one. A one-entry certificate and its n disjoint output cases give certificate, fractional and unambiguous complexities at most one. Quantum and randomized costs are also at most one.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric_group" ], "input_alphabet": "permutation", "output_alphabet": "bits", "cost": "forward_or_inverse_symbol_queries", "error": "as_in_measure_definitions", "assumptions": [ "Domain S_n; not permutation-invariant Boolean-cube functions." ] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "proof_sources": [ "DFLLV21" ], "witness_id": "permutation-single-entry" }, { "id": "perm-edge-no-power-permAdeg", "A": "permEdgeS", "B": "permAdeg", "exponent": null, "kind": "no_polynomial_bound", "statement": "No finite power bounds permEdgeS by permAdeg: the single-entry indicator has edge sensitivity n-1 and permAdeg=O(1).", "source": "definitions", "locator": "Repository example using DFLLV21 Section 2.2 definitions", "proof_summary": "For f(pi)=[pi(1)=1], one symbol query solves the function and degree is one. Its n-1 sensitive transpositions at a positive input all share position 1, so the disjoint packing is one. A one-entry certificate and its n disjoint output cases give certificate, fractional and unambiguous complexities at most one. Quantum and randomized costs are also at most one.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric_group" ], "input_alphabet": "permutation", "output_alphabet": "bits", "cost": "forward_or_inverse_symbol_queries", "error": "as_in_measure_definitions", "assumptions": [ "Domain S_n; not permutation-invariant Boolean-cube functions." ] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "proof_sources": [ "DFLLV21" ], "witness_id": "permutation-single-entry" }, { "id": "perm-edge-no-power-permQE", "A": "permEdgeS", "B": "permQE", "exponent": null, "kind": "no_polynomial_bound", "statement": "No finite power bounds permEdgeS by permQE: the single-entry indicator has edge sensitivity n-1 and permQE=O(1).", "source": "definitions", "locator": "Repository example using DFLLV21 Section 2.2 definitions", "proof_summary": "For f(pi)=[pi(1)=1], one symbol query solves the function and degree is one. Its n-1 sensitive transpositions at a positive input all share position 1, so the disjoint packing is one. A one-entry certificate and its n disjoint output cases give certificate, fractional and unambiguous complexities at most one. Quantum and randomized costs are also at most one.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric_group" ], "input_alphabet": "permutation", "output_alphabet": "bits", "cost": "forward_or_inverse_symbol_queries", "error": "as_in_measure_definitions", "assumptions": [ "Domain S_n; not permutation-invariant Boolean-cube functions." ] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "proof_sources": [ "DFLLV21" ], "witness_id": "permutation-single-entry" }, { "id": "perm-edge-no-power-permQ", "A": "permEdgeS", "B": "permQ", "exponent": null, "kind": "no_polynomial_bound", "statement": "No finite power bounds permEdgeS by permQ: the single-entry indicator has edge sensitivity n-1 and permQ=O(1).", "source": "definitions", "locator": "Repository example using DFLLV21 Section 2.2 definitions", "proof_summary": "For f(pi)=[pi(1)=1], one symbol query solves the function and degree is one. Its n-1 sensitive transpositions at a positive input all share position 1, so the disjoint packing is one. A one-entry certificate and its n disjoint output cases give certificate, fractional and unambiguous complexities at most one. Quantum and randomized costs are also at most one.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric_group" ], "input_alphabet": "permutation", "output_alphabet": "bits", "cost": "forward_or_inverse_symbol_queries", "error": "as_in_measure_definitions", "assumptions": [ "Domain S_n; not permutation-invariant Boolean-cube functions." ] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "proof_sources": [ "DFLLV21" ], "witness_id": "permutation-single-entry" }, { "id": "relevant-depth-exact-upper", "A": "nRelevant", "B": "D", "exponent": null, "kind": "nonpolynomial_upper_bound", "statement": "nRelevant(f)<=2^D(f)-1, attained by a complete decision tree with a distinct variable at every internal node.", "source": "Wel22", "locator": "Section 3.6; total-core comparison chains", "proof_summary": "A depth-d binary tree has at most 2^d-1 internal nodes, and every relevant variable must label some internal node. The recursively addressed tree on fresh variables attains the bound.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false }, { "id": "core-at-most-relevant-D", "A": "D", "B": "nRelevant", "exponent": 1, "kind": "upper_bound", "statement": "D(f)<=O(nRelevant(f)) for total Boolean functions.", "source": "Wel22", "locator": "Section 3.6; total-core comparison chains", "proof_summary": "Query all relevant variables. This gives D<=nRelevant; the indexed total-core comparison bounds the indicated measure by O(D).", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "relevant-no-power-D", "A": "nRelevant", "B": "D", "exponent": null, "kind": "no_polynomial_bound", "statement": "No finite c bounds nRelevant(f) by O(D(f)^c) over all total Boolean functions.", "source": "Wel22", "locator": "Section 3.6; total-core comparison chains", "proof_summary": "On the complete fresh-variable depth-d decision tree, nRelevant=2^d-1 while every core measure is O(d). Exponential growth exceeds every fixed power.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "witness_id": "relevant-complete-address-tree", "proof_sources": [ "CHS20" ] }, { "id": "core-at-most-relevant-R0", "A": "R0", "B": "nRelevant", "exponent": 1, "kind": "upper_bound", "statement": "R0(f)<=O(nRelevant(f)) for total Boolean functions.", "source": "Wel22", "locator": "Section 3.6; total-core comparison chains", "proof_summary": "Query all relevant variables. This gives D<=nRelevant; the indexed total-core comparison bounds the indicated measure by O(D).", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "relevant-no-power-R0", "A": "nRelevant", "B": "R0", "exponent": null, "kind": "no_polynomial_bound", "statement": "No finite c bounds nRelevant(f) by O(R0(f)^c) over all total Boolean functions.", "source": "Wel22", "locator": "Section 3.6; total-core comparison chains", "proof_summary": "On the complete fresh-variable depth-d decision tree, nRelevant=2^d-1 while every core measure is O(d). Exponential growth exceeds every fixed power.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "witness_id": "relevant-complete-address-tree", "proof_sources": [ "CHS20" ] }, { "id": "core-at-most-relevant-R", "A": "R", "B": "nRelevant", "exponent": 1, "kind": "upper_bound", "statement": "R(f)<=O(nRelevant(f)) for total Boolean functions.", "source": "Wel22", "locator": "Section 3.6; total-core comparison chains", "proof_summary": "Query all relevant variables. This gives D<=nRelevant; the indexed total-core comparison bounds the indicated measure by O(D).", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "relevant-no-power-R", "A": "nRelevant", "B": "R", "exponent": null, "kind": "no_polynomial_bound", "statement": "No finite c bounds nRelevant(f) by O(R(f)^c) over all total Boolean functions.", "source": "Wel22", "locator": "Section 3.6; total-core comparison chains", "proof_summary": "On the complete fresh-variable depth-d decision tree, nRelevant=2^d-1 while every core measure is O(d). Exponential growth exceeds every fixed power.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "witness_id": "relevant-complete-address-tree", "proof_sources": [ "CHS20" ] }, { "id": "core-at-most-relevant-C", "A": "C", "B": "nRelevant", "exponent": 1, "kind": "upper_bound", "statement": "C(f)<=O(nRelevant(f)) for total Boolean functions.", "source": "Wel22", "locator": "Section 3.6; total-core comparison chains", "proof_summary": "Query all relevant variables. This gives D<=nRelevant; the indexed total-core comparison bounds the indicated measure by O(D).", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "relevant-no-power-C", "A": "nRelevant", "B": "C", "exponent": null, "kind": "no_polynomial_bound", "statement": "No finite c bounds nRelevant(f) by O(C(f)^c) over all total Boolean functions.", "source": "Wel22", "locator": "Section 3.6; total-core comparison chains", "proof_summary": "On the complete fresh-variable depth-d decision tree, nRelevant=2^d-1 while every core measure is O(d). Exponential growth exceeds every fixed power.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "witness_id": "relevant-complete-address-tree", "proof_sources": [ "CHS20" ] }, { "id": "core-at-most-relevant-RC", "A": "RC", "B": "nRelevant", "exponent": 1, "kind": "upper_bound", "statement": "RC(f)<=O(nRelevant(f)) for total Boolean functions.", "source": "Wel22", "locator": "Section 3.6; total-core comparison chains", "proof_summary": "Query all relevant variables. This gives D<=nRelevant; the indexed total-core comparison bounds the indicated measure by O(D).", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "relevant-no-power-RC", "A": "nRelevant", "B": "RC", "exponent": null, "kind": "no_polynomial_bound", "statement": "No finite c bounds nRelevant(f) by O(RC(f)^c) over all total Boolean functions.", "source": "Wel22", "locator": "Section 3.6; total-core comparison chains", "proof_summary": "On the complete fresh-variable depth-d decision tree, nRelevant=2^d-1 while every core measure is O(d). Exponential growth exceeds every fixed power.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "witness_id": "relevant-complete-address-tree", "proof_sources": [ "CHS20" ] }, { "id": "core-at-most-relevant-bs", "A": "bs", "B": "nRelevant", "exponent": 1, "kind": "upper_bound", "statement": "bs(f)<=O(nRelevant(f)) for total Boolean functions.", "source": "Wel22", "locator": "Section 3.6; total-core comparison chains", "proof_summary": "Query all relevant variables. This gives D<=nRelevant; the indexed total-core comparison bounds the indicated measure by O(D).", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "relevant-no-power-bs", "A": "nRelevant", "B": "bs", "exponent": null, "kind": "no_polynomial_bound", "statement": "No finite c bounds nRelevant(f) by O(bs(f)^c) over all total Boolean functions.", "source": "Wel22", "locator": "Section 3.6; total-core comparison chains", "proof_summary": "On the complete fresh-variable depth-d decision tree, nRelevant=2^d-1 while every core measure is O(d). Exponential growth exceeds every fixed power.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "witness_id": "relevant-complete-address-tree", "proof_sources": [ "CHS20" ] }, { "id": "core-at-most-relevant-s", "A": "s", "B": "nRelevant", "exponent": 1, "kind": "upper_bound", "statement": "s(f)<=O(nRelevant(f)) for total Boolean functions.", "source": "Wel22", "locator": "Section 3.6; total-core comparison chains", "proof_summary": "Query all relevant variables. This gives D<=nRelevant; the indexed total-core comparison bounds the indicated measure by O(D).", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "relevant-no-power-s", "A": "nRelevant", "B": "s", "exponent": null, "kind": "no_polynomial_bound", "statement": "No finite c bounds nRelevant(f) by O(s(f)^c) over all total Boolean functions.", "source": "Wel22", "locator": "Section 3.6; total-core comparison chains", "proof_summary": "On the complete fresh-variable depth-d decision tree, nRelevant=2^d-1 while every core measure is O(d). Exponential growth exceeds every fixed power.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "witness_id": "relevant-complete-address-tree", "proof_sources": [ "CHS20" ] }, { "id": "core-at-most-relevant-lambda", "A": "lambda", "B": "nRelevant", "exponent": 1, "kind": "upper_bound", "statement": "lambda(f)<=O(nRelevant(f)) for total Boolean functions.", "source": "Wel22", "locator": "Section 3.6; total-core comparison chains", "proof_summary": "Query all relevant variables. This gives D<=nRelevant; the indexed total-core comparison bounds the indicated measure by O(D).", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "relevant-no-power-lambda", "A": "nRelevant", "B": "lambda", "exponent": null, "kind": "no_polynomial_bound", "statement": "No finite c bounds nRelevant(f) by O(lambda(f)^c) over all total Boolean functions.", "source": "Wel22", "locator": "Section 3.6; total-core comparison chains", "proof_summary": "On the complete fresh-variable depth-d decision tree, nRelevant=2^d-1 while every core measure is O(d). Exponential growth exceeds every fixed power.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "witness_id": "relevant-complete-address-tree", "proof_sources": [ "CHS20" ] }, { "id": "core-at-most-relevant-QE", "A": "QE", "B": "nRelevant", "exponent": 1, "kind": "upper_bound", "statement": "QE(f)<=O(nRelevant(f)) for total Boolean functions.", "source": "Wel22", "locator": "Section 3.6; total-core comparison chains", "proof_summary": "Query all relevant variables. This gives D<=nRelevant; the indexed total-core comparison bounds the indicated measure by O(D).", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "relevant-no-power-QE", "A": "nRelevant", "B": "QE", "exponent": null, "kind": "no_polynomial_bound", "statement": "No finite c bounds nRelevant(f) by O(QE(f)^c) over all total Boolean functions.", "source": "Wel22", "locator": "Section 3.6; total-core comparison chains", "proof_summary": "On the complete fresh-variable depth-d decision tree, nRelevant=2^d-1 while every core measure is O(d). Exponential growth exceeds every fixed power.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "witness_id": "relevant-complete-address-tree", "proof_sources": [ "CHS20" ] }, { "id": "core-at-most-relevant-deg", "A": "deg", "B": "nRelevant", "exponent": 1, "kind": "upper_bound", "statement": "deg(f)<=O(nRelevant(f)) for total Boolean functions.", "source": "Wel22", "locator": "Section 3.6; total-core comparison chains", "proof_summary": "Query all relevant variables. This gives D<=nRelevant; the indexed total-core comparison bounds the indicated measure by O(D).", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "core-at-most-relevant-Q", "A": "Q", "B": "nRelevant", "exponent": 1, "kind": "upper_bound", "statement": "Q(f)<=O(nRelevant(f)) for total Boolean functions.", "source": "Wel22", "locator": "Section 3.6; total-core comparison chains", "proof_summary": "Query all relevant variables. This gives D<=nRelevant; the indexed total-core comparison bounds the indicated measure by O(D).", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "relevant-no-power-Q", "A": "nRelevant", "B": "Q", "exponent": null, "kind": "no_polynomial_bound", "statement": "No finite c bounds nRelevant(f) by O(Q(f)^c) over all total Boolean functions.", "source": "Wel22", "locator": "Section 3.6; total-core comparison chains", "proof_summary": "On the complete fresh-variable depth-d decision tree, nRelevant=2^d-1 while every core measure is O(d). Exponential growth exceeds every fixed power.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "witness_id": "relevant-complete-address-tree", "proof_sources": [ "CHS20" ] }, { "id": "core-at-most-relevant-adeg", "A": "adeg", "B": "nRelevant", "exponent": 1, "kind": "upper_bound", "statement": "adeg(f)<=O(nRelevant(f)) for total Boolean functions.", "source": "Wel22", "locator": "Section 3.6; total-core comparison chains", "proof_summary": "Query all relevant variables. This gives D<=nRelevant; the indexed total-core comparison bounds the indicated measure by O(D).", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "relevant-no-power-adeg", "A": "nRelevant", "B": "adeg", "exponent": null, "kind": "no_polynomial_bound", "statement": "No finite c bounds nRelevant(f) by O(adeg(f)^c) over all total Boolean functions.", "source": "Wel22", "locator": "Section 3.6; total-core comparison chains", "proof_summary": "On the complete fresh-variable depth-d decision tree, nRelevant=2^d-1 while every core measure is O(d). Exponential growth exceeds every fixed power.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "witness_id": "relevant-complete-address-tree", "proof_sources": [ "CHS20" ] }, { "id": "relevant-hybrid-degree-sensitivity", "A": "nRelevant", "B": "deg", "exponent": null, "kind": "nonpolynomial_upper_bound", "statement": "nRelevant=O(sqrt(2^degree*4^sensitivity)).", "source": "Wel22", "locator": "Theorem 3.18, equations (3.25)–(3.26)", "proof_summary": "The author outlines applying the restriction method to convex combinations of the coordinate-degree, coordinate-sensitivity and coordinate-certificate measures. The paper explicitly omits the technical proof for brevity.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "statement_checked_proof_pending", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "notes": "Mixed-measure bound; not a scalar exponent in the B measure. Second inequality retains the source tildeO notation; hidden factors not independently extracted." }, { "id": "relevant-hybrid-certificate-sensitivity", "A": "nRelevant", "B": "C", "exponent": null, "kind": "nonpolynomial_upper_bound", "statement": "nRelevant=tildeO(sqrt(4^certificate*4^sensitivity)).", "source": "Wel22", "locator": "Theorem 3.18, equations (3.25)–(3.26)", "proof_summary": "The author outlines applying the restriction method to convex combinations of the coordinate-degree, coordinate-sensitivity and coordinate-certificate measures. The paper explicitly omits the technical proof for brevity.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "statement_checked_proof_pending", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "notes": "Mixed-measure bound; not a scalar exponent in the B measure. Second inequality retains the source tildeO notation; hidden factors not independently extracted." }, { "id": "completion-relevance-Dna", "A": "completion_nRelevant", "B": "Dna", "exponent": 1, "kind": "upper_bound", "statement": "completion_nRelevant(f)=Dna(f) for partial Boolean decision functions.", "source": "HKMP26", "locator": "Repository derivation from Definition 45 and fixed-set nonadaptive queries", "proof_summary": "A nonadaptive algorithm queries a fixed set S that separates every promised opposite-output pair. Define a completion depending only on S, assigning arbitrary values to unseen S-patterns. Conversely, querying the relevant coordinates of any completion solves f nonadaptively.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "proof_sources": [ "Wel22", "GGR13" ] }, { "id": "Dna-completion-relevance", "A": "Dna", "B": "completion_nRelevant", "exponent": 1, "kind": "upper_bound", "statement": "Dna(f)=completion_nRelevant(f).", "source": "HKMP26", "locator": "Repository derivation from Definition 45 and fixed-set nonadaptive queries", "proof_summary": "The same projection-separation argument gives the reverse inequality.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "proof_sources": [ "Wel22", "GGR13" ] }, { "id": "completion-relevance-depth-upper", "A": "completion_nRelevant", "B": "D", "exponent": null, "kind": "nonpolynomial_upper_bound", "statement": "completion_nRelevant(f)<=2^D(f)-1.", "source": "HKMP26", "locator": "Repository derivation from Definition 45 and fixed-set nonadaptive queries", "proof_summary": "An optimal partial decision tree induces a total completion of the same depth. Its number of relevant variables is at most its 2^D-1 internal nodes.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "proof_sources": [ "Wel22", "GGR13" ] }, { "id": "completion-relevance-depth-lower", "A": "D", "B": "completion_nRelevant", "exponent": 1, "kind": "upper_bound", "statement": "D(f)<=completion_nRelevant(f).", "source": "HKMP26", "locator": "Repository derivation from Definition 45 and fixed-set nonadaptive queries", "proof_summary": "Query the relevant coordinates of an optimal completion, or use the equality with nonadaptive complexity.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "proof_sources": [ "Wel22", "GGR13" ] }, { "id": "completion-fbs-FC", "A": "completion_fbs", "B": "completion_FC", "exponent": 1, "kind": "upper_bound", "statement": "completion_fbs=completion_FC.", "source": "HKMP26", "locator": "Definition 45; LP duality in the total-function definitions", "proof_summary": "The total-function LP-dual identity fbs=FC holds for every completion. Taking minima over the identical completion set preserves equality.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "completion-FC-fbs", "A": "completion_FC", "B": "completion_fbs", "exponent": 1, "kind": "upper_bound", "statement": "completion_FC=completion_fbs.", "source": "HKMP26", "locator": "Definition 45; LP duality", "proof_summary": "The same pointwise identity holds before taking minima.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true }, { "id": "domain-chain-domainAdeg-domainDegree", "A": "domainAdeg", "B": "domainDegree", "exponent": 1, "kind": "upper_bound", "statement": "approx_degree<=degree.", "source": "DFLLV22author", "locator": "Theorem 3.1", "proof_summary": "An exact representation is a zero-error approximant.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "composable_query_domain" ], "input_alphabet": "query_domain", "output_alphabet": "bits", "cost": "one_unique_element_answer_per_query", "error": "as_in_measure_definitions", "assumptions": [ "n-element set inputs; query family covers the universe; queries intersect every input in exactly one element." ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "proof_sources": [ "DFLLV26" ], "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "domain-chain-domainDegree-domainD", "A": "domainDegree", "B": "domainD", "exponent": 1, "kind": "upper_bound", "statement": "degree<=D.", "source": "DFLLV22author", "locator": "Theorem 3.1", "proof_summary": "Sum the products of membership indicators on accepting decision-tree paths.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "composable_query_domain" ], "input_alphabet": "query_domain", "output_alphabet": "bits", "cost": "one_unique_element_answer_per_query", "error": "as_in_measure_definitions", "assumptions": [ "n-element set inputs; query family covers the universe; queries intersect every input in exactly one element." ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "proof_sources": [ "DFLLV26" ], "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "domain-chain-domainBS-domainFBS", "A": "domainBS", "B": "domainFBS", "exponent": 1, "kind": "upper_bound", "statement": "bs<=fbs.", "source": "DFLLV22author", "locator": "Theorem 3.1", "proof_summary": "Integral disjoint blocks give a feasible fractional packing.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "composable_query_domain" ], "input_alphabet": "query_domain", "output_alphabet": "bits", "cost": "one_unique_element_answer_per_query", "error": "as_in_measure_definitions", "assumptions": [ "n-element set inputs; query family covers the universe; queries intersect every input in exactly one element." ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "proof_sources": [ "DFLLV26" ], "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "domain-chain-domainFBS-domainC", "A": "domainFBS", "B": "domainC", "exponent": 1, "kind": "upper_bound", "statement": "fbs<=C.", "source": "DFLLV22author", "locator": "Theorem 3.1", "proof_summary": "A certificate supplies a feasible dual cover for every opposite-output disagreement set.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "composable_query_domain" ], "input_alphabet": "query_domain", "output_alphabet": "bits", "cost": "one_unique_element_answer_per_query", "error": "as_in_measure_definitions", "assumptions": [ "n-element set inputs; query family covers the universe; queries intersect every input in exactly one element." ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "proof_sources": [ "DFLLV26" ], "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "domain-chain-domainC-domainUC", "A": "domainC", "B": "domainUC", "exponent": 1, "kind": "upper_bound", "statement": "C<=U.", "source": "DFLLV22author", "locator": "Theorem 3.1", "proof_summary": "Every partition certificate is a certificate.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "composable_query_domain" ], "input_alphabet": "query_domain", "output_alphabet": "bits", "cost": "one_unique_element_answer_per_query", "error": "as_in_measure_definitions", "assumptions": [ "n-element set inputs; query family covers the universe; queries intersect every input in exactly one element." ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "proof_sources": [ "DFLLV26" ], "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "domain-chain-domainUC-domainD", "A": "domainUC", "B": "domainD", "exponent": 1, "kind": "upper_bound", "statement": "U<=D.", "source": "DFLLV22author", "locator": "Theorem 3.1", "proof_summary": "Tree leaves give unambiguous certificates.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "composable_query_domain" ], "input_alphabet": "query_domain", "output_alphabet": "bits", "cost": "one_unique_element_answer_per_query", "error": "as_in_measure_definitions", "assumptions": [ "n-element set inputs; query family covers the universe; queries intersect every input in exactly one element." ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "proof_sources": [ "DFLLV26" ], "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "domain-chain-domainBS-domainR", "A": "domainBS", "B": "domainR", "exponent": 1, "kind": "upper_bound", "statement": "bs<=3R.", "source": "DFLLV22author", "locator": "Theorem 3.1", "proof_summary": "Disjoint sensitive alternatives must be distinguished with constant success.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "composable_query_domain" ], "input_alphabet": "query_domain", "output_alphabet": "bits", "cost": "one_unique_element_answer_per_query", "error": "as_in_measure_definitions", "assumptions": [ "n-element set inputs; query family covers the universe; queries intersect every input in exactly one element." ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "proof_sources": [ "DFLLV26" ], "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "domain-chain-domainR-domainRfail23", "A": "domainR", "B": "domainRfail23", "exponent": 1, "kind": "upper_bound", "statement": "R<=R_0.", "source": "DFLLV22author", "locator": "Theorem 3.1", "proof_summary": "Guess a fresh fair bit on failure; success at least 1/3+(2/3)/2=2/3, with no extra queries.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "composable_query_domain" ], "input_alphabet": "query_domain", "output_alphabet": "bits", "cost": "one_unique_element_answer_per_query", "error": "as_in_measure_definitions", "assumptions": [ "n-element set inputs; query family covers the universe; queries intersect every input in exactly one element." ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "proof_sources": [ "DFLLV26" ], "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "domain-chain-domainRfail23-domainD", "A": "domainRfail23", "B": "domainD", "exponent": 1, "kind": "upper_bound", "statement": "R_0<=D.", "source": "DFLLV22author", "locator": "Theorem 3.1", "proof_summary": "A deterministic tree never fails.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "composable_query_domain" ], "input_alphabet": "query_domain", "output_alphabet": "bits", "cost": "one_unique_element_answer_per_query", "error": "as_in_measure_definitions", "assumptions": [ "n-element set inputs; query family covers the universe; queries intersect every input in exactly one element." ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "proof_sources": [ "DFLLV26" ], "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "domain-chain-domainAdeg-domainQ", "A": "domainAdeg", "B": "domainQ", "exponent": 1, "kind": "upper_bound", "statement": "approx_degree<=2Q.", "source": "DFLLV22author", "locator": "Theorem 3.1", "proof_summary": "Acceptance probabilities are domain-indicator polynomials of degree at most twice the query count.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "composable_query_domain" ], "input_alphabet": "query_domain", "output_alphabet": "bits", "cost": "one_unique_element_answer_per_query", "error": "as_in_measure_definitions", "assumptions": [ "n-element set inputs; query family covers the universe; queries intersect every input in exactly one element." ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "proof_sources": [ "DFLLV26" ], "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "domain-chain-domainQ-domainQE", "A": "domainQ", "B": "domainQE", "exponent": 1, "kind": "upper_bound", "statement": "Q<=Q_E.", "source": "DFLLV22author", "locator": "Theorem 3.1", "proof_summary": "Exact algorithms are bounded-error algorithms.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "composable_query_domain" ], "input_alphabet": "query_domain", "output_alphabet": "bits", "cost": "one_unique_element_answer_per_query", "error": "as_in_measure_definitions", "assumptions": [ "n-element set inputs; query family covers the universe; queries intersect every input in exactly one element." ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "proof_sources": [ "DFLLV26" ], "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "domain-chain-domainQE-domainD", "A": "domainQE", "B": "domainD", "exponent": 1, "kind": "upper_bound", "statement": "Q_E<=D.", "source": "DFLLV22author", "locator": "Theorem 3.1", "proof_summary": "Coherently simulate the deterministic symbol-query tree.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "composable_query_domain" ], "input_alphabet": "query_domain", "output_alphabet": "bits", "cost": "one_unique_element_answer_per_query", "error": "as_in_measure_definitions", "assumptions": [ "n-element set inputs; query family covers the universe; queries intersect every input in exactly one element." ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "proof_sources": [ "DFLLV26" ], "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "domain-chain-domainS-domainBS", "A": "domainS", "B": "domainBS", "exponent": 1, "kind": "upper_bound", "statement": "s<=bs.", "source": "DFLLV22author", "locator": "Theorem 3.1", "proof_summary": "Chunk-size disjoint changes are a subclass of disjoint changes.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "composable_query_domain" ], "input_alphabet": "query_domain", "output_alphabet": "bits", "cost": "one_unique_element_answer_per_query", "error": "as_in_measure_definitions", "assumptions": [ "n-element set inputs; query family covers the universe; queries intersect every input in exactly one element." ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "proof_sources": [ "DFLLV26" ], "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "domain-chain-domainBS-domainAdeg", "A": "domainBS", "B": "domainAdeg", "exponent": 2, "kind": "upper_bound", "statement": "bs<=6*approx_degree^2.", "source": "DFLLV22author", "locator": "Theorem 3.1", "proof_summary": "Composability embeds the disjoint-block alternatives in a Boolean cube; apply the symmetrized polynomial degree lower bound.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "composable_query_domain" ], "input_alphabet": "query_domain", "output_alphabet": "bits", "cost": "one_unique_element_answer_per_query", "error": "as_in_measure_definitions", "assumptions": [ "n-element set inputs; query family covers the universe; queries intersect every input in exactly one element." ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "proof_sources": [ "DFLLV26" ], "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "domain-parameterized-domainD-domainBS", "A": "domainD", "B": "domainBS", "exponent": 4, "kind": "upper_bound", "statement": "D<=beta^(-2)*max(Delta,n/M)*bs^4.", "source": "DFLLV22author", "locator": "Theorem 3.1", "proof_summary": "Apply the parameterized certificate-query and block-shortening arguments of Theorem 3.1 and Lemma 3.5. Domain parameters remain explicit.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "composable_query_domain" ], "input_alphabet": "query_domain", "output_alphabet": "bits", "cost": "one_unique_element_answer_per_query", "error": "as_in_measure_definitions", "assumptions": [ "n-element set inputs; query family covers the universe; queries intersect every input in exactly one element." ] }, "loss": "domain_parameters", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "notes": "M>0 required for n/M; no pure-power bound is asserted uniformly over domains.", "proof_sources": [ "DFLLV26" ], "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "domain-parameterized-domainC-domainBS", "A": "domainC", "B": "domainBS", "exponent": 2, "kind": "upper_bound", "statement": "C<=beta^(-1)*bs^2.", "source": "DFLLV22author", "locator": "Theorem 3.1", "proof_summary": "Apply the parameterized certificate-query and block-shortening arguments of Theorem 3.1 and Lemma 3.5. Domain parameters remain explicit.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "composable_query_domain" ], "input_alphabet": "query_domain", "output_alphabet": "bits", "cost": "one_unique_element_answer_per_query", "error": "as_in_measure_definitions", "assumptions": [ "n-element set inputs; query family covers the universe; queries intersect every input in exactly one element." ] }, "loss": "domain_parameters", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "notes": "M>0 required for n/M; no pure-power bound is asserted uniformly over domains.", "proof_sources": [ "DFLLV26" ], "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "forward-permutation-no-power-domainDegree", "A": "domainD", "B": "domainDegree", "exponent": null, "kind": "no_polynomial_bound", "statement": "Forward-only deterministic complexity is n/2 while domainDegree=1 for f(pi)=[pi^(-1)(1)<=n/2].", "source": "definitions", "locator": "Repository example using Sections 2.1,2.2 and 4.3", "proof_summary": "One inverse query finds the position of 1, and sum_{i<=n/2} [pi(i)=1] has degree one. Query an entire half for the forward upper bound. For fewer than n/2 queries an adversary answers with distinct values other than 1, leaving possible positions in both halves.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "forward_only_permutation_queries" ], "input_alphabet": "permutation", "output_alphabet": "bits", "cost": "forward_or_inverse_symbol_queries", "error": "as_in_measure_definitions", "assumptions": [ "A uses only forward pi(i) queries; permD permits forward and inverse queries. n is even." ] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "proof_sources": [ "DFLLV22author", "DFLLV26" ], "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "forward-permutation-no-power-permD", "A": "domainD", "B": "permD", "exponent": null, "kind": "no_polynomial_bound", "statement": "Forward-only deterministic complexity is n/2 while permD=1 for f(pi)=[pi^(-1)(1)<=n/2].", "source": "definitions", "locator": "Repository example using Sections 2.1,2.2 and 4.3", "proof_summary": "One inverse query finds the position of 1, and sum_{i<=n/2} [pi(i)=1] has degree one. Query an entire half for the forward upper bound. For fewer than n/2 queries an adversary answers with distinct values other than 1, leaving possible positions in both halves.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "forward_only_permutation_queries" ], "input_alphabet": "permutation", "output_alphabet": "bits", "cost": "forward_or_inverse_symbol_queries", "error": "as_in_measure_definitions", "assumptions": [ "A uses only forward pi(i) queries; permD permits forward and inverse queries. n is even." ] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "proof_sources": [ "DFLLV22author", "DFLLV26" ], "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "completion-containment-completion_C0-completion_C", "A": "completion_C0", "B": "completion_C", "exponent": 1, "kind": "upper_bound", "statement": "completion_C0<= completion_C.", "source": "HKMP26", "locator": "Definition 45; LP duality in the total-function definitions", "proof_summary": "Evaluate the componentwise total-function inequality at a completion optimal for the right-hand measure, then minimize the left.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "notes": "Do not interchange min_F and max_b: minimizing completions can differ." }, { "id": "completion-containment-completion_C1-completion_C", "A": "completion_C1", "B": "completion_C", "exponent": 1, "kind": "upper_bound", "statement": "completion_C1<= completion_C.", "source": "HKMP26", "locator": "Definition 45; LP duality in the total-function definitions", "proof_summary": "Evaluate the componentwise total-function inequality at a completion optimal for the right-hand measure, then minimize the left.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "notes": "Do not interchange min_F and max_b: minimizing completions can differ." }, { "id": "completion-containment-completion_UC0-completion_UC", "A": "completion_UC0", "B": "completion_UC", "exponent": 1, "kind": "upper_bound", "statement": "completion_UC0<= completion_UC.", "source": "HKMP26", "locator": "Definition 45; LP duality in the total-function definitions", "proof_summary": "Evaluate the componentwise total-function inequality at a completion optimal for the right-hand measure, then minimize the left.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "notes": "Do not interchange min_F and max_b: minimizing completions can differ." }, { "id": "completion-containment-completion_UC1-completion_UC", "A": "completion_UC1", "B": "completion_UC", "exponent": 1, "kind": "upper_bound", "statement": "completion_UC1<= completion_UC.", "source": "HKMP26", "locator": "Definition 45; LP duality in the total-function definitions", "proof_summary": "Evaluate the componentwise total-function inequality at a completion optimal for the right-hand measure, then minimize the left.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "notes": "Do not interchange min_F and max_b: minimizing completions can differ." }, { "id": "completion-containment-completion_C-completion_UC", "A": "completion_C", "B": "completion_UC", "exponent": 1, "kind": "upper_bound", "statement": "completion_C<= completion_UC.", "source": "HKMP26", "locator": "Definition 45; LP duality in the total-function definitions", "proof_summary": "Evaluate the componentwise total-function inequality at a completion optimal for the right-hand measure, then minimize the left.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "notes": "Do not interchange min_F and max_b: minimizing completions can differ." }, { "id": "BD16-domainD-domainQ", "A": "domainD", "B": "domainQ", "exponent": 18, "kind": "upper_bound", "statement": "D(f)=O(Q(f)^18) on a single multislice, uniformly over multiplicities and alphabet size.", "source": "BD16", "locator": "Theorems 5 and 12", "proof_summary": "Either certificates are large and Corollary 9 supplies the lower bound, or Lemma 10 restricts to small certificates using a bounded set of alphabet symbols. Hard disjoint certificate arrangements reduce to permutation inversion, and the adversary method gives the remaining quantum lower bound.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "single_multislice_forward_queries" ], "input_alphabet": "finite_symbols", "output_alphabet": "bits", "cost": "one_unique_element_answer_per_query", "error": "as_in_measure_definitions", "assumptions": [ "Domain is one complete coordinate-permutation orbit in [M]^n; queries reveal forward coordinate values only." ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "proof_sources": [ "DFLLV26" ], "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained.", "notes": "An indexed original-source bound, not a globally certified best exponent. For binary full slices compare the later HKMP26 improvements." }, { "id": "BD16-domainD-domainR", "A": "domainD", "B": "domainR", "exponent": 9, "kind": "upper_bound", "statement": "D(f)=O(R(f)^9) on a single multislice, uniformly over multiplicities and alphabet size.", "source": "BD16", "locator": "Theorem 12", "proof_summary": "Use the same large-certificate versus restricted-symbol hard-instance dichotomy; the classical distinguishing lower bound yields the ninth-power relationship.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "single_multislice_forward_queries" ], "input_alphabet": "finite_symbols", "output_alphabet": "bits", "cost": "one_unique_element_answer_per_query", "error": "as_in_measure_definitions", "assumptions": [ "Domain is one complete coordinate-permutation orbit in [M]^n; queries reveal forward coordinate values only." ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "proof_sources": [ "DFLLV26" ], "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained.", "notes": "An indexed original-source bound, not a globally certified best exponent. For binary full slices compare the later HKMP26 improvements." }, { "id": "BD16-domainC-domainBS", "A": "domainC", "B": "domainBS", "exponent": 2, "kind": "upper_bound", "statement": "C(f)<=3*bs(f)^2 on a single multislice.", "source": "BD16", "locator": "Theorem 8 and Definition 7", "proof_summary": "Theorem 8 gives C<=3*bs*s_2. A maximal disjoint collection of minimal sensitive blocks forms a certificate, and cycle decomposition bounds block size by 3*s_2. Since s_2<=bs, the square follows.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "single_multislice_forward_queries" ], "input_alphabet": "finite_symbols", "output_alphabet": "bits", "cost": "one_unique_element_answer_per_query", "error": "as_in_measure_definitions", "assumptions": [ "Domain is one complete coordinate-permutation orbit in [M]^n; queries reveal forward coordinate values only." ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "proof_sources": [ "DFLLV26" ], "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained.", "notes": "An indexed original-source bound, not a globally certified best exponent. For binary full slices compare the later HKMP26 improvements." }, { "id": "side-C0-C", "kind": "upper_bound", "A": "C0", "B": "C", "exponent": 1, "statement": "C0(f)<=O(C(f)).", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "AIK26 preliminary definitions, article p.3; repository elementary derivation", "proof_summary": "The maximum over inputs of one output is at most the maximum over both outputs.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "AIK26", "BdW02" ], "scalar_inference_allowed": true }, { "id": "side-C0-UC0", "kind": "upper_bound", "A": "C0", "B": "UC0", "exponent": 1, "statement": "C0(f)<=O(UC0(f)).", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "AIK26 preliminary definitions, article p.3; repository elementary derivation", "proof_summary": "Each term of an unambiguous representation is a certificate; drop the uniqueness requirement.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "AIK26", "BdW02" ], "scalar_inference_allowed": true }, { "id": "side-UC0-UC", "kind": "upper_bound", "A": "UC0", "B": "UC", "exponent": 1, "statement": "UC0(f)<=O(UC(f)).", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "AIK26 preliminary definitions, article p.3; repository elementary derivation", "proof_summary": "UC is the maximum of the separately optimized two output unambiguous certificate costs.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "AIK26", "BdW02" ], "scalar_inference_allowed": true }, { "id": "side-UC0-D", "kind": "upper_bound", "A": "UC0", "B": "D", "exponent": 1, "statement": "UC0(f)<=O(D(f)).", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "AIK26 preliminary definitions, article p.3; repository elementary derivation", "proof_summary": "The leaves of a deterministic tree labeled by the chosen output partition that output class into disjoint subcubes of codimension at most the tree depth.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "AIK26", "BdW02" ], "scalar_inference_allowed": true }, { "id": "side-C1-C", "kind": "upper_bound", "A": "C1", "B": "C", "exponent": 1, "statement": "C1(f)<=O(C(f)).", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "AIK26 preliminary definitions, article p.3; repository elementary derivation", "proof_summary": "The maximum over inputs of one output is at most the maximum over both outputs.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "AIK26", "BdW02" ], "scalar_inference_allowed": true }, { "id": "side-C1-UC1", "kind": "upper_bound", "A": "C1", "B": "UC1", "exponent": 1, "statement": "C1(f)<=O(UC1(f)).", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "AIK26 preliminary definitions, article p.3; repository elementary derivation", "proof_summary": "Each term of an unambiguous representation is a certificate; drop the uniqueness requirement.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "AIK26", "BdW02" ], "scalar_inference_allowed": true }, { "id": "side-UC1-UC", "kind": "upper_bound", "A": "UC1", "B": "UC", "exponent": 1, "statement": "UC1(f)<=O(UC(f)).", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "AIK26 preliminary definitions, article p.3; repository elementary derivation", "proof_summary": "UC is the maximum of the separately optimized two output unambiguous certificate costs.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "AIK26", "BdW02" ], "scalar_inference_allowed": true }, { "id": "side-UC1-D", "kind": "upper_bound", "A": "UC1", "B": "D", "exponent": 1, "statement": "UC1(f)<=O(D(f)).", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "AIK26 preliminary definitions, article p.3; repository elementary derivation", "proof_summary": "The leaves of a deterministic tree labeled by the chosen output partition that output class into disjoint subcubes of codimension at most the tree depth.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "AIK26", "BdW02" ], "scalar_inference_allowed": true }, { "id": "side-UC-D", "kind": "upper_bound", "A": "UC", "B": "D", "exponent": 1, "statement": "UC(f)<=O(D(f)).", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "AIK26 preliminary definitions, article p.3; repository elementary derivation", "proof_summary": "Apply the leaf partition argument to each output; take their maximum.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "AIK26", "BdW02" ], "scalar_inference_allowed": true }, { "id": "side-R-R1", "kind": "upper_bound", "A": "R", "B": "R1", "exponent": 1, "statement": "R(f)<=O(R1(f)).", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "AIK26 preliminary definitions, article p.3; repository elementary derivation", "proof_summary": "A no-false-positive, positive-success-at-least-2/3 algorithm has two-sided error at most 1/3.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "AIK26", "BdW02" ], "scalar_inference_allowed": true }, { "id": "side-R1-D", "kind": "upper_bound", "A": "R1", "B": "D", "exponent": 1, "statement": "R1(f)<=O(D(f)).", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "AIK26 preliminary definitions, article p.3; repository elementary derivation", "proof_summary": "A deterministic tree has no false positives and succeeds with probability one.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "AIK26", "BdW02" ], "scalar_inference_allowed": true }, { "id": "side-R1-R0", "kind": "upper_bound", "A": "R1", "B": "R0", "exponent": 1, "statement": "R1(f)<=O(R0(f)).", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "AIK26 preliminary definitions, article p.3; repository elementary derivation", "proof_summary": "Run an expected zero-error algorithm to three times its worst-input expectation. If it has not halted, output zero. Markov bounds failure on positive inputs by 1/3; negative inputs never produce one. Integer ceilings and the zero-cost constant case do not change the big-O bound.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "AIK26", "BdW02" ], "scalar_inference_allowed": true }, { "id": "side-C1-R1", "kind": "upper_bound", "A": "C1", "B": "R1", "exponent": 1, "statement": "C1(f)<=O(R1(f)).", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "AIK26 preliminary definitions, article p.3; repository elementary derivation", "proof_summary": "For a positive input, some accepting query transcript has positive probability. Its queried literals exclude every negative input: otherwise the same positive-probability event of random choices would produce the same accepting transcript there, contradicting no false positives. The transcript is a positive certificate of width at most the query cap.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "AIK26", "BdW02" ], "scalar_inference_allowed": true }, { "id": "side-deg-UC0", "kind": "upper_bound", "A": "deg", "B": "UC0", "exponent": 1, "statement": "deg(f)<=O(UC0(f)).", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "AIK26 preliminary definitions, article p.3; repository elementary derivation", "proof_summary": "Sum the disjoint zero-certificate indicator monomials; this is exactly 1-f on every Boolean input. Each indicator has degree at most UC0; complementing does not raise degree.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "AIK26", "BdW02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-D-C1-OR", "kind": "no_polynomial_bound", "A": "D", "B": "C1", "exponent": null, "statement": "On OR_n, C1=1 and D tends to infinity; no finite power of max(1,C1) bounds D.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On OR_n, C1=1, while D diverges. Approximate degree is Omega(sqrt(n)) by the indexed block-sensitivity lower bound, since bs=n. Polynomial and query simulations, and s=bs=n with lambda>=sqrt(s), give divergence of all the remaining listed core/query measures; R1 dominates R and Q0 dominates Q up to constant amplification.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-R0-C1-OR", "kind": "no_polynomial_bound", "A": "R0", "B": "C1", "exponent": null, "statement": "On OR_n, C1=1 and R0 tends to infinity; no finite power of max(1,C1) bounds R0.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On OR_n, C1=1, while R0 diverges. Approximate degree is Omega(sqrt(n)) by the indexed block-sensitivity lower bound, since bs=n. Polynomial and query simulations, and s=bs=n with lambda>=sqrt(s), give divergence of all the remaining listed core/query measures; R1 dominates R and Q0 dominates Q up to constant amplification.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-R-C1-OR", "kind": "no_polynomial_bound", "A": "R", "B": "C1", "exponent": null, "statement": "On OR_n, C1=1 and R tends to infinity; no finite power of max(1,C1) bounds R.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On OR_n, C1=1, while R diverges. Approximate degree is Omega(sqrt(n)) by the indexed block-sensitivity lower bound, since bs=n. Polynomial and query simulations, and s=bs=n with lambda>=sqrt(s), give divergence of all the remaining listed core/query measures; R1 dominates R and Q0 dominates Q up to constant amplification.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-C-C1-OR", "kind": "no_polynomial_bound", "A": "C", "B": "C1", "exponent": null, "statement": "On OR_n, C1=1 and C tends to infinity; no finite power of max(1,C1) bounds C.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On OR_n, C1=1, while C diverges. Approximate degree is Omega(sqrt(n)) by the indexed block-sensitivity lower bound, since bs=n. Polynomial and query simulations, and s=bs=n with lambda>=sqrt(s), give divergence of all the remaining listed core/query measures; R1 dominates R and Q0 dominates Q up to constant amplification.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-RC-C1-OR", "kind": "no_polynomial_bound", "A": "RC", "B": "C1", "exponent": null, "statement": "On OR_n, C1=1 and RC tends to infinity; no finite power of max(1,C1) bounds RC.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On OR_n, C1=1, while RC diverges. At the unique extreme input, the n singleton sensitive blocks give fractional block sensitivity n. LP duality and the indexed RC equivalence give RC=Theta(n); QC=Theta(sqrt(RC)) gives QC=Theta(sqrt(n)).", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-bs-C1-OR", "kind": "no_polynomial_bound", "A": "bs", "B": "C1", "exponent": null, "statement": "On OR_n, C1=1 and bs tends to infinity; no finite power of max(1,C1) bounds bs.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On OR_n, C1=1, while bs diverges. Approximate degree is Omega(sqrt(n)) by the indexed block-sensitivity lower bound, since bs=n. Polynomial and query simulations, and s=bs=n with lambda>=sqrt(s), give divergence of all the remaining listed core/query measures; R1 dominates R and Q0 dominates Q up to constant amplification.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-s-C1-OR", "kind": "no_polynomial_bound", "A": "s", "B": "C1", "exponent": null, "statement": "On OR_n, C1=1 and s tends to infinity; no finite power of max(1,C1) bounds s.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On OR_n, C1=1, while s diverges. Approximate degree is Omega(sqrt(n)) by the indexed block-sensitivity lower bound, since bs=n. Polynomial and query simulations, and s=bs=n with lambda>=sqrt(s), give divergence of all the remaining listed core/query measures; R1 dominates R and Q0 dominates Q up to constant amplification.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-lambda-C1-OR", "kind": "no_polynomial_bound", "A": "lambda", "B": "C1", "exponent": null, "statement": "On OR_n, C1=1 and lambda tends to infinity; no finite power of max(1,C1) bounds lambda.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On OR_n, C1=1, while lambda diverges. Approximate degree is Omega(sqrt(n)) by the indexed block-sensitivity lower bound, since bs=n. Polynomial and query simulations, and s=bs=n with lambda>=sqrt(s), give divergence of all the remaining listed core/query measures; R1 dominates R and Q0 dominates Q up to constant amplification.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-QE-C1-OR", "kind": "no_polynomial_bound", "A": "QE", "B": "C1", "exponent": null, "statement": "On OR_n, C1=1 and QE tends to infinity; no finite power of max(1,C1) bounds QE.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On OR_n, C1=1, while QE diverges. Approximate degree is Omega(sqrt(n)) by the indexed block-sensitivity lower bound, since bs=n. Polynomial and query simulations, and s=bs=n with lambda>=sqrt(s), give divergence of all the remaining listed core/query measures; R1 dominates R and Q0 dominates Q up to constant amplification.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-deg-C1-OR", "kind": "no_polynomial_bound", "A": "deg", "B": "C1", "exponent": null, "statement": "On OR_n, C1=1 and deg tends to infinity; no finite power of max(1,C1) bounds deg.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On OR_n, C1=1, while deg diverges. Exact degree is n, by the nonzero full multilinear coefficient. Either output unambiguous representation gives an exact polynomial of degree at most its width.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-Q-C1-OR", "kind": "no_polynomial_bound", "A": "Q", "B": "C1", "exponent": null, "statement": "On OR_n, C1=1 and Q tends to infinity; no finite power of max(1,C1) bounds Q.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On OR_n, C1=1, while Q diverges. Approximate degree is Omega(sqrt(n)) by the indexed block-sensitivity lower bound, since bs=n. Polynomial and query simulations, and s=bs=n with lambda>=sqrt(s), give divergence of all the remaining listed core/query measures; R1 dominates R and Q0 dominates Q up to constant amplification.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-adeg-C1-OR", "kind": "no_polynomial_bound", "A": "adeg", "B": "C1", "exponent": null, "statement": "On OR_n, C1=1 and adeg tends to infinity; no finite power of max(1,C1) bounds adeg.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On OR_n, C1=1, while adeg diverges. Approximate degree is Omega(sqrt(n)) by the indexed block-sensitivity lower bound, since bs=n. Polynomial and query simulations, and s=bs=n with lambda>=sqrt(s), give divergence of all the remaining listed core/query measures; R1 dominates R and Q0 dominates Q up to constant amplification.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-nRelevant-C1-OR", "kind": "no_polynomial_bound", "A": "nRelevant", "B": "C1", "exponent": null, "statement": "On OR_n, C1=1 and nRelevant tends to infinity; no finite power of max(1,C1) bounds nRelevant.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On OR_n, C1=1, while nRelevant diverges. All n variables are relevant.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-C0-C1-OR", "kind": "no_polynomial_bound", "A": "C0", "B": "C1", "exponent": null, "statement": "On OR_n, C1=1 and C0 tends to infinity; no finite power of max(1,C1) bounds C0.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On OR_n, C1=1, while C0 diverges. The opposite-output input requires fixing all n bits.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-UC-C1-OR", "kind": "no_polynomial_bound", "A": "UC", "B": "C1", "exponent": null, "statement": "On OR_n, C1=1 and UC tends to infinity; no finite power of max(1,C1) bounds UC.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On OR_n, C1=1, while UC diverges. Exact degree is n, by the nonzero full multilinear coefficient. Either output unambiguous representation gives an exact polynomial of degree at most its width.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-UC0-C1-OR", "kind": "no_polynomial_bound", "A": "UC0", "B": "C1", "exponent": null, "statement": "On OR_n, C1=1 and UC0 tends to infinity; no finite power of max(1,C1) bounds UC0.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On OR_n, C1=1, while UC0 diverges. Exact degree is n, by the nonzero full multilinear coefficient. Either output unambiguous representation gives an exact polynomial of degree at most its width.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-UC1-C1-OR", "kind": "no_polynomial_bound", "A": "UC1", "B": "C1", "exponent": null, "statement": "On OR_n, C1=1 and UC1 tends to infinity; no finite power of max(1,C1) bounds UC1.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On OR_n, C1=1, while UC1 diverges. Exact degree is n, by the nonzero full multilinear coefficient. Either output unambiguous representation gives an exact polynomial of degree at most its width.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-fbs-C1-OR", "kind": "no_polynomial_bound", "A": "fbs", "B": "C1", "exponent": null, "statement": "On OR_n, C1=1 and fbs tends to infinity; no finite power of max(1,C1) bounds fbs.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On OR_n, C1=1, while fbs diverges. At the unique extreme input, the n singleton sensitive blocks give fractional block sensitivity n. LP duality and the indexed RC equivalence give RC=Theta(n); QC=Theta(sqrt(RC)) gives QC=Theta(sqrt(n)).", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-FC-C1-OR", "kind": "no_polynomial_bound", "A": "FC", "B": "C1", "exponent": null, "statement": "On OR_n, C1=1 and FC tends to infinity; no finite power of max(1,C1) bounds FC.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On OR_n, C1=1, while FC diverges. At the unique extreme input, the n singleton sensitive blocks give fractional block sensitivity n. LP duality and the indexed RC equivalence give RC=Theta(n); QC=Theta(sqrt(RC)) gives QC=Theta(sqrt(n)).", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-QC-C1-OR", "kind": "no_polynomial_bound", "A": "QC", "B": "C1", "exponent": null, "statement": "On OR_n, C1=1 and QC tends to infinity; no finite power of max(1,C1) bounds QC.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On OR_n, C1=1, while QC diverges. At the unique extreme input, the n singleton sensitive blocks give fractional block sensitivity n. LP duality and the indexed RC equivalence give RC=Theta(n); QC=Theta(sqrt(RC)) gives QC=Theta(sqrt(n)).", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-Q0-C1-OR", "kind": "no_polynomial_bound", "A": "Q0", "B": "C1", "exponent": null, "statement": "On OR_n, C1=1 and Q0 tends to infinity; no finite power of max(1,C1) bounds Q0.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On OR_n, C1=1, while Q0 diverges. Approximate degree is Omega(sqrt(n)) by the indexed block-sensitivity lower bound, since bs=n. Polynomial and query simulations, and s=bs=n with lambda>=sqrt(s), give divergence of all the remaining listed core/query measures; R1 dominates R and Q0 dominates Q up to constant amplification.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-R1-C1-OR", "kind": "no_polynomial_bound", "A": "R1", "B": "C1", "exponent": null, "statement": "On OR_n, C1=1 and R1 tends to infinity; no finite power of max(1,C1) bounds R1.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On OR_n, C1=1, while R1 diverges. Approximate degree is Omega(sqrt(n)) by the indexed block-sensitivity lower bound, since bs=n. Polynomial and query simulations, and s=bs=n with lambda>=sqrt(s), give divergence of all the remaining listed core/query measures; R1 dominates R and Q0 dominates Q up to constant amplification.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-D-C0-AND", "kind": "no_polynomial_bound", "A": "D", "B": "C0", "exponent": null, "statement": "On AND_n, C0=1 and D tends to infinity; no finite power of max(1,C0) bounds D.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On AND_n, C0=1, while D diverges. Approximate degree is Omega(sqrt(n)) by the indexed block-sensitivity lower bound, since bs=n. Polynomial and query simulations, and s=bs=n with lambda>=sqrt(s), give divergence of all the remaining listed core/query measures; R1 dominates R and Q0 dominates Q up to constant amplification.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-R0-C0-AND", "kind": "no_polynomial_bound", "A": "R0", "B": "C0", "exponent": null, "statement": "On AND_n, C0=1 and R0 tends to infinity; no finite power of max(1,C0) bounds R0.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On AND_n, C0=1, while R0 diverges. Approximate degree is Omega(sqrt(n)) by the indexed block-sensitivity lower bound, since bs=n. Polynomial and query simulations, and s=bs=n with lambda>=sqrt(s), give divergence of all the remaining listed core/query measures; R1 dominates R and Q0 dominates Q up to constant amplification.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-R-C0-AND", "kind": "no_polynomial_bound", "A": "R", "B": "C0", "exponent": null, "statement": "On AND_n, C0=1 and R tends to infinity; no finite power of max(1,C0) bounds R.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On AND_n, C0=1, while R diverges. Approximate degree is Omega(sqrt(n)) by the indexed block-sensitivity lower bound, since bs=n. Polynomial and query simulations, and s=bs=n with lambda>=sqrt(s), give divergence of all the remaining listed core/query measures; R1 dominates R and Q0 dominates Q up to constant amplification.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-C-C0-AND", "kind": "no_polynomial_bound", "A": "C", "B": "C0", "exponent": null, "statement": "On AND_n, C0=1 and C tends to infinity; no finite power of max(1,C0) bounds C.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On AND_n, C0=1, while C diverges. Approximate degree is Omega(sqrt(n)) by the indexed block-sensitivity lower bound, since bs=n. Polynomial and query simulations, and s=bs=n with lambda>=sqrt(s), give divergence of all the remaining listed core/query measures; R1 dominates R and Q0 dominates Q up to constant amplification.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-RC-C0-AND", "kind": "no_polynomial_bound", "A": "RC", "B": "C0", "exponent": null, "statement": "On AND_n, C0=1 and RC tends to infinity; no finite power of max(1,C0) bounds RC.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On AND_n, C0=1, while RC diverges. At the unique extreme input, the n singleton sensitive blocks give fractional block sensitivity n. LP duality and the indexed RC equivalence give RC=Theta(n); QC=Theta(sqrt(RC)) gives QC=Theta(sqrt(n)).", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-bs-C0-AND", "kind": "no_polynomial_bound", "A": "bs", "B": "C0", "exponent": null, "statement": "On AND_n, C0=1 and bs tends to infinity; no finite power of max(1,C0) bounds bs.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On AND_n, C0=1, while bs diverges. Approximate degree is Omega(sqrt(n)) by the indexed block-sensitivity lower bound, since bs=n. Polynomial and query simulations, and s=bs=n with lambda>=sqrt(s), give divergence of all the remaining listed core/query measures; R1 dominates R and Q0 dominates Q up to constant amplification.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-s-C0-AND", "kind": "no_polynomial_bound", "A": "s", "B": "C0", "exponent": null, "statement": "On AND_n, C0=1 and s tends to infinity; no finite power of max(1,C0) bounds s.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On AND_n, C0=1, while s diverges. Approximate degree is Omega(sqrt(n)) by the indexed block-sensitivity lower bound, since bs=n. Polynomial and query simulations, and s=bs=n with lambda>=sqrt(s), give divergence of all the remaining listed core/query measures; R1 dominates R and Q0 dominates Q up to constant amplification.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-lambda-C0-AND", "kind": "no_polynomial_bound", "A": "lambda", "B": "C0", "exponent": null, "statement": "On AND_n, C0=1 and lambda tends to infinity; no finite power of max(1,C0) bounds lambda.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On AND_n, C0=1, while lambda diverges. Approximate degree is Omega(sqrt(n)) by the indexed block-sensitivity lower bound, since bs=n. Polynomial and query simulations, and s=bs=n with lambda>=sqrt(s), give divergence of all the remaining listed core/query measures; R1 dominates R and Q0 dominates Q up to constant amplification.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-QE-C0-AND", "kind": "no_polynomial_bound", "A": "QE", "B": "C0", "exponent": null, "statement": "On AND_n, C0=1 and QE tends to infinity; no finite power of max(1,C0) bounds QE.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On AND_n, C0=1, while QE diverges. Approximate degree is Omega(sqrt(n)) by the indexed block-sensitivity lower bound, since bs=n. Polynomial and query simulations, and s=bs=n with lambda>=sqrt(s), give divergence of all the remaining listed core/query measures; R1 dominates R and Q0 dominates Q up to constant amplification.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-deg-C0-AND", "kind": "no_polynomial_bound", "A": "deg", "B": "C0", "exponent": null, "statement": "On AND_n, C0=1 and deg tends to infinity; no finite power of max(1,C0) bounds deg.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On AND_n, C0=1, while deg diverges. Exact degree is n, by the nonzero full multilinear coefficient. Either output unambiguous representation gives an exact polynomial of degree at most its width.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-Q-C0-AND", "kind": "no_polynomial_bound", "A": "Q", "B": "C0", "exponent": null, "statement": "On AND_n, C0=1 and Q tends to infinity; no finite power of max(1,C0) bounds Q.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On AND_n, C0=1, while Q diverges. Approximate degree is Omega(sqrt(n)) by the indexed block-sensitivity lower bound, since bs=n. Polynomial and query simulations, and s=bs=n with lambda>=sqrt(s), give divergence of all the remaining listed core/query measures; R1 dominates R and Q0 dominates Q up to constant amplification.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-adeg-C0-AND", "kind": "no_polynomial_bound", "A": "adeg", "B": "C0", "exponent": null, "statement": "On AND_n, C0=1 and adeg tends to infinity; no finite power of max(1,C0) bounds adeg.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On AND_n, C0=1, while adeg diverges. Approximate degree is Omega(sqrt(n)) by the indexed block-sensitivity lower bound, since bs=n. Polynomial and query simulations, and s=bs=n with lambda>=sqrt(s), give divergence of all the remaining listed core/query measures; R1 dominates R and Q0 dominates Q up to constant amplification.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-nRelevant-C0-AND", "kind": "no_polynomial_bound", "A": "nRelevant", "B": "C0", "exponent": null, "statement": "On AND_n, C0=1 and nRelevant tends to infinity; no finite power of max(1,C0) bounds nRelevant.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On AND_n, C0=1, while nRelevant diverges. All n variables are relevant.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-C1-C0-AND", "kind": "no_polynomial_bound", "A": "C1", "B": "C0", "exponent": null, "statement": "On AND_n, C0=1 and C1 tends to infinity; no finite power of max(1,C0) bounds C1.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On AND_n, C0=1, while C1 diverges. The opposite-output input requires fixing all n bits.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-UC-C0-AND", "kind": "no_polynomial_bound", "A": "UC", "B": "C0", "exponent": null, "statement": "On AND_n, C0=1 and UC tends to infinity; no finite power of max(1,C0) bounds UC.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On AND_n, C0=1, while UC diverges. Exact degree is n, by the nonzero full multilinear coefficient. Either output unambiguous representation gives an exact polynomial of degree at most its width.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-UC0-C0-AND", "kind": "no_polynomial_bound", "A": "UC0", "B": "C0", "exponent": null, "statement": "On AND_n, C0=1 and UC0 tends to infinity; no finite power of max(1,C0) bounds UC0.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On AND_n, C0=1, while UC0 diverges. Exact degree is n, by the nonzero full multilinear coefficient. Either output unambiguous representation gives an exact polynomial of degree at most its width.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-UC1-C0-AND", "kind": "no_polynomial_bound", "A": "UC1", "B": "C0", "exponent": null, "statement": "On AND_n, C0=1 and UC1 tends to infinity; no finite power of max(1,C0) bounds UC1.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On AND_n, C0=1, while UC1 diverges. Exact degree is n, by the nonzero full multilinear coefficient. Either output unambiguous representation gives an exact polynomial of degree at most its width.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-fbs-C0-AND", "kind": "no_polynomial_bound", "A": "fbs", "B": "C0", "exponent": null, "statement": "On AND_n, C0=1 and fbs tends to infinity; no finite power of max(1,C0) bounds fbs.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On AND_n, C0=1, while fbs diverges. At the unique extreme input, the n singleton sensitive blocks give fractional block sensitivity n. LP duality and the indexed RC equivalence give RC=Theta(n); QC=Theta(sqrt(RC)) gives QC=Theta(sqrt(n)).", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-FC-C0-AND", "kind": "no_polynomial_bound", "A": "FC", "B": "C0", "exponent": null, "statement": "On AND_n, C0=1 and FC tends to infinity; no finite power of max(1,C0) bounds FC.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On AND_n, C0=1, while FC diverges. At the unique extreme input, the n singleton sensitive blocks give fractional block sensitivity n. LP duality and the indexed RC equivalence give RC=Theta(n); QC=Theta(sqrt(RC)) gives QC=Theta(sqrt(n)).", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-QC-C0-AND", "kind": "no_polynomial_bound", "A": "QC", "B": "C0", "exponent": null, "statement": "On AND_n, C0=1 and QC tends to infinity; no finite power of max(1,C0) bounds QC.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On AND_n, C0=1, while QC diverges. At the unique extreme input, the n singleton sensitive blocks give fractional block sensitivity n. LP duality and the indexed RC equivalence give RC=Theta(n); QC=Theta(sqrt(RC)) gives QC=Theta(sqrt(n)).", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-Q0-C0-AND", "kind": "no_polynomial_bound", "A": "Q0", "B": "C0", "exponent": null, "statement": "On AND_n, C0=1 and Q0 tends to infinity; no finite power of max(1,C0) bounds Q0.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On AND_n, C0=1, while Q0 diverges. Approximate degree is Omega(sqrt(n)) by the indexed block-sensitivity lower bound, since bs=n. Polynomial and query simulations, and s=bs=n with lambda>=sqrt(s), give divergence of all the remaining listed core/query measures; R1 dominates R and Q0 dominates Q up to constant amplification.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "total-unbounded-R1-C0-AND", "kind": "no_polynomial_bound", "A": "R1", "B": "C0", "exponent": null, "statement": "On AND_n, C0=1 and R1 tends to infinity; no finite power of max(1,C0) bounds R1.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On AND_n, C0=1, while R1 diverges. Approximate degree is Omega(sqrt(n)) by the indexed block-sensitivity lower bound, since bs=n. Polynomial and query simulations, and s=bs=n with lambda>=sqrt(s), give divergence of all the remaining listed core/query measures; R1 dominates R and Q0 dominates Q up to constant amplification.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02", "Aar02" ], "scalar_inference_allowed": true }, { "id": "forward-multislice-D-unbounded-domainDegree", "kind": "no_polynomial_bound", "A": "domainD", "B": "domainDegree", "exponent": null, "statement": "Forward-only all-distinct multislices have domainD=n/2 and domainDegree=1; no uniform finite polynomial comparison.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "single_multislice_forward_queries" ], "input_alphabet": "finite_symbols", "output_alphabet": "bits", "cost": "one_unique_element_answer_per_query", "error": "as_in_measure_definitions", "assumptions": [ "Domain is one complete coordinate-permutation orbit in [M]^n; queries reveal forward coordinate values only." ] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On permutations of [n], n even, use f(pi)=[the position of 1 is in the first half], with forward coordinate queries. A certificate is the single observed coordinate containing 1. These certificates partition both output classes, so C=UC=1. The sum of entry indicators for positions in the first half is an exact degree-one polynomial. Every sensitive change must include the original position of 1, so bs=fbs=s=1. An adversary avoids returning 1 for fewer than n/2 queries, leaving positions in both halves; querying a half suffices, hence D=n/2.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "DFLLV26", "BD16" ], "scalar_inference_allowed": false }, { "id": "forward-multislice-D-unbounded-domainAdeg", "kind": "no_polynomial_bound", "A": "domainD", "B": "domainAdeg", "exponent": null, "statement": "Forward-only all-distinct multislices have domainD=n/2 and domainAdeg=1; no uniform finite polynomial comparison.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "single_multislice_forward_queries" ], "input_alphabet": "finite_symbols", "output_alphabet": "bits", "cost": "one_unique_element_answer_per_query", "error": "as_in_measure_definitions", "assumptions": [ "Domain is one complete coordinate-permutation orbit in [M]^n; queries reveal forward coordinate values only." ] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On permutations of [n], n even, use f(pi)=[the position of 1 is in the first half], with forward coordinate queries. A certificate is the single observed coordinate containing 1. These certificates partition both output classes, so C=UC=1. The sum of entry indicators for positions in the first half is an exact degree-one polynomial. Every sensitive change must include the original position of 1, so bs=fbs=s=1. An adversary avoids returning 1 for fewer than n/2 queries, leaving positions in both halves; querying a half suffices, hence D=n/2.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "DFLLV26", "BD16" ], "scalar_inference_allowed": false }, { "id": "forward-multislice-D-unbounded-domainC", "kind": "no_polynomial_bound", "A": "domainD", "B": "domainC", "exponent": null, "statement": "Forward-only all-distinct multislices have domainD=n/2 and domainC=1; no uniform finite polynomial comparison.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "single_multislice_forward_queries" ], "input_alphabet": "finite_symbols", "output_alphabet": "bits", "cost": "one_unique_element_answer_per_query", "error": "as_in_measure_definitions", "assumptions": [ "Domain is one complete coordinate-permutation orbit in [M]^n; queries reveal forward coordinate values only." ] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On permutations of [n], n even, use f(pi)=[the position of 1 is in the first half], with forward coordinate queries. A certificate is the single observed coordinate containing 1. These certificates partition both output classes, so C=UC=1. The sum of entry indicators for positions in the first half is an exact degree-one polynomial. Every sensitive change must include the original position of 1, so bs=fbs=s=1. An adversary avoids returning 1 for fewer than n/2 queries, leaving positions in both halves; querying a half suffices, hence D=n/2.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "DFLLV26", "BD16" ], "scalar_inference_allowed": false }, { "id": "forward-multislice-D-unbounded-domainUC", "kind": "no_polynomial_bound", "A": "domainD", "B": "domainUC", "exponent": null, "statement": "Forward-only all-distinct multislices have domainD=n/2 and domainUC=1; no uniform finite polynomial comparison.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "single_multislice_forward_queries" ], "input_alphabet": "finite_symbols", "output_alphabet": "bits", "cost": "one_unique_element_answer_per_query", "error": "as_in_measure_definitions", "assumptions": [ "Domain is one complete coordinate-permutation orbit in [M]^n; queries reveal forward coordinate values only." ] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On permutations of [n], n even, use f(pi)=[the position of 1 is in the first half], with forward coordinate queries. A certificate is the single observed coordinate containing 1. These certificates partition both output classes, so C=UC=1. The sum of entry indicators for positions in the first half is an exact degree-one polynomial. Every sensitive change must include the original position of 1, so bs=fbs=s=1. An adversary avoids returning 1 for fewer than n/2 queries, leaving positions in both halves; querying a half suffices, hence D=n/2.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "DFLLV26", "BD16" ], "scalar_inference_allowed": false }, { "id": "forward-multislice-D-unbounded-domainBS", "kind": "no_polynomial_bound", "A": "domainD", "B": "domainBS", "exponent": null, "statement": "Forward-only all-distinct multislices have domainD=n/2 and domainBS=1; no uniform finite polynomial comparison.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "single_multislice_forward_queries" ], "input_alphabet": "finite_symbols", "output_alphabet": "bits", "cost": "one_unique_element_answer_per_query", "error": "as_in_measure_definitions", "assumptions": [ "Domain is one complete coordinate-permutation orbit in [M]^n; queries reveal forward coordinate values only." ] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On permutations of [n], n even, use f(pi)=[the position of 1 is in the first half], with forward coordinate queries. A certificate is the single observed coordinate containing 1. These certificates partition both output classes, so C=UC=1. The sum of entry indicators for positions in the first half is an exact degree-one polynomial. Every sensitive change must include the original position of 1, so bs=fbs=s=1. An adversary avoids returning 1 for fewer than n/2 queries, leaving positions in both halves; querying a half suffices, hence D=n/2.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "DFLLV26", "BD16" ], "scalar_inference_allowed": false }, { "id": "forward-multislice-D-unbounded-domainFBS", "kind": "no_polynomial_bound", "A": "domainD", "B": "domainFBS", "exponent": null, "statement": "Forward-only all-distinct multislices have domainD=n/2 and domainFBS=1; no uniform finite polynomial comparison.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "single_multislice_forward_queries" ], "input_alphabet": "finite_symbols", "output_alphabet": "bits", "cost": "one_unique_element_answer_per_query", "error": "as_in_measure_definitions", "assumptions": [ "Domain is one complete coordinate-permutation orbit in [M]^n; queries reveal forward coordinate values only." ] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On permutations of [n], n even, use f(pi)=[the position of 1 is in the first half], with forward coordinate queries. A certificate is the single observed coordinate containing 1. These certificates partition both output classes, so C=UC=1. The sum of entry indicators for positions in the first half is an exact degree-one polynomial. Every sensitive change must include the original position of 1, so bs=fbs=s=1. An adversary avoids returning 1 for fewer than n/2 queries, leaving positions in both halves; querying a half suffices, hence D=n/2.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "DFLLV26", "BD16" ], "scalar_inference_allowed": false }, { "id": "forward-multislice-D-unbounded-domainS", "kind": "no_polynomial_bound", "A": "domainD", "B": "domainS", "exponent": null, "statement": "Forward-only all-distinct multislices have domainD=n/2 and domainS=1; no uniform finite polynomial comparison.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "single_multislice_forward_queries" ], "input_alphabet": "finite_symbols", "output_alphabet": "bits", "cost": "one_unique_element_answer_per_query", "error": "as_in_measure_definitions", "assumptions": [ "Domain is one complete coordinate-permutation orbit in [M]^n; queries reveal forward coordinate values only." ] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "On permutations of [n], n even, use f(pi)=[the position of 1 is in the first half], with forward coordinate queries. A certificate is the single observed coordinate containing 1. These certificates partition both output classes, so C=UC=1. The sum of entry indicators for positions in the first half is an exact degree-one polynomial. Every sensitive change must include the original position of 1, so bs=fbs=s=1. An adversary avoids returning 1 for fewer than n/2 queries, leaving positions in both halves; querying a half suffices, hence D=n/2.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "DFLLV26", "BD16" ], "scalar_inference_allowed": false }, { "id": "completion-support-FC-fbs", "kind": "upper_bound", "A": "FC", "B": "fbs", "exponent": 1, "statement": "FC(f)=O(fbs(f)^1).", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "At each input the certificate covering LP and sensitive-block packing LP are dual: constraints are that each opposite-output disagreement block has covering weight at least one. Finite LP strong duality gives equality; maximize over inputs.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02" ], "scalar_inference_allowed": true }, { "id": "completion-support-fbs-FC", "kind": "upper_bound", "A": "fbs", "B": "FC", "exponent": 1, "statement": "fbs(f)=O(FC(f)^1).", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "The same primal/dual LP identity gives the reverse direction.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02" ], "scalar_inference_allowed": true }, { "id": "completion-support-QC-FC", "kind": "upper_bound", "A": "QC", "B": "FC", "exponent": 0.5, "statement": "QC(f)=O(FC(f)^0.5).", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Aar02 Theorem 7 and Lemma 5; repository LP identification", "proof_summary": "Use FC=Theta(RC) and Aaronson’s QC=Theta(sqrt(RC)) certificate characterization. Both are maximum-over-input verification costs with fixed constant rejection probability.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "Aar02" ], "scalar_inference_allowed": true }, { "id": "completion-support-FC-QC", "kind": "upper_bound", "A": "FC", "B": "QC", "exponent": 2, "statement": "FC(f)=O(QC(f)^2).", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Aar02 Theorem 7 and Lemma 5; repository LP identification", "proof_summary": "Square Aaronson’s RC=O(QC^2), then use FC=Theta(RC). No promise-completion minima are interchanged.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "Aar02" ], "scalar_inference_allowed": true }, { "id": "completion-support-RC-FC", "kind": "upper_bound", "A": "RC", "B": "FC", "exponent": 1, "statement": "RC(f)<=(ln 2)*FC(f) for expected classical verification accepting the claimed input with certainty and rejecting opposite outputs with probability at least 1/2.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "For a fixed claimed input x, choose a fractional certificate w. Query coordinate i independently with probability 1-2^(-w_i), rejecting on disagreement. For any opposite-output y, the probability of missing every disagreement is 2^(-sum_{i:x_i!=y_i} w_i)<=1/2. The expected query count is sum_i(1-2^(-w_i))<=(ln 2)*sum_i w_i. The verifier accepts x certainly; maximize over x.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "Aar02" ], "scalar_inference_allowed": true, "notes": "Repository constant refinement of fractional sampling, not a constant quoted from Aaronson. The ln 2 coefficient is asymptotically optimal for native measures over partial functions, as separately proved; sharpness for total functions or minimum-completion measures is not asserted." }, { "id": "completion-support-Q-Q0", "kind": "upper_bound", "A": "Q", "B": "Q0", "exponent": 1, "statement": "Q(f)=O(Q0(f)^1).", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "Repeat the fixed-cap never-wrong algorithm a constant number of times to reduce inconclusive probability from at most 1/2 to at most 1/3; output an arbitrary bit if still inconclusive.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "Aar02" ], "scalar_inference_allowed": true }, { "id": "address-unbounded-nRelevant-UC", "kind": "no_polynomial_bound", "A": "nRelevant", "B": "UC", "exponent": null, "statement": "Total address functions have nRelevant=2^k+k and UC=O(k+1), excluding every finite power.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "Use k address bits to select one of 2^k data bits. Every input bit is relevant, and reading the address followed by its selected data bit costs k+1. A deterministic tree supplies unambiguous certificate families from its leaves and simulates all the query measures here; fractional certificates and quantum verifiers cost at most the corresponding ordinary certificates/tree depth. Thus the right measure is at most O(k+1), while relevance is exponential in k. Total functions have a unique completion, so these witnesses lift unchanged.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02" ], "scalar_inference_allowed": true }, { "id": "address-unbounded-nRelevant-UC0", "kind": "no_polynomial_bound", "A": "nRelevant", "B": "UC0", "exponent": null, "statement": "Total address functions have nRelevant=2^k+k and UC0=O(k+1), excluding every finite power.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "Use k address bits to select one of 2^k data bits. Every input bit is relevant, and reading the address followed by its selected data bit costs k+1. A deterministic tree supplies unambiguous certificate families from its leaves and simulates all the query measures here; fractional certificates and quantum verifiers cost at most the corresponding ordinary certificates/tree depth. Thus the right measure is at most O(k+1), while relevance is exponential in k. Total functions have a unique completion, so these witnesses lift unchanged.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02" ], "scalar_inference_allowed": true }, { "id": "address-unbounded-nRelevant-UC1", "kind": "no_polynomial_bound", "A": "nRelevant", "B": "UC1", "exponent": null, "statement": "Total address functions have nRelevant=2^k+k and UC1=O(k+1), excluding every finite power.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "Use k address bits to select one of 2^k data bits. Every input bit is relevant, and reading the address followed by its selected data bit costs k+1. A deterministic tree supplies unambiguous certificate families from its leaves and simulates all the query measures here; fractional certificates and quantum verifiers cost at most the corresponding ordinary certificates/tree depth. Thus the right measure is at most O(k+1), while relevance is exponential in k. Total functions have a unique completion, so these witnesses lift unchanged.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02" ], "scalar_inference_allowed": true }, { "id": "address-unbounded-nRelevant-fbs", "kind": "no_polynomial_bound", "A": "nRelevant", "B": "fbs", "exponent": null, "statement": "Total address functions have nRelevant=2^k+k and fbs=O(k+1), excluding every finite power.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "Use k address bits to select one of 2^k data bits. Every input bit is relevant, and reading the address followed by its selected data bit costs k+1. A deterministic tree supplies unambiguous certificate families from its leaves and simulates all the query measures here; fractional certificates and quantum verifiers cost at most the corresponding ordinary certificates/tree depth. Thus the right measure is at most O(k+1), while relevance is exponential in k. Total functions have a unique completion, so these witnesses lift unchanged.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02" ], "scalar_inference_allowed": true }, { "id": "address-unbounded-nRelevant-FC", "kind": "no_polynomial_bound", "A": "nRelevant", "B": "FC", "exponent": null, "statement": "Total address functions have nRelevant=2^k+k and FC=O(k+1), excluding every finite power.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "Use k address bits to select one of 2^k data bits. Every input bit is relevant, and reading the address followed by its selected data bit costs k+1. A deterministic tree supplies unambiguous certificate families from its leaves and simulates all the query measures here; fractional certificates and quantum verifiers cost at most the corresponding ordinary certificates/tree depth. Thus the right measure is at most O(k+1), while relevance is exponential in k. Total functions have a unique completion, so these witnesses lift unchanged.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02" ], "scalar_inference_allowed": true }, { "id": "address-unbounded-nRelevant-QC", "kind": "no_polynomial_bound", "A": "nRelevant", "B": "QC", "exponent": null, "statement": "Total address functions have nRelevant=2^k+k and QC=O(k+1), excluding every finite power.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "Use k address bits to select one of 2^k data bits. Every input bit is relevant, and reading the address followed by its selected data bit costs k+1. A deterministic tree supplies unambiguous certificate families from its leaves and simulates all the query measures here; fractional certificates and quantum verifiers cost at most the corresponding ordinary certificates/tree depth. Thus the right measure is at most O(k+1), while relevance is exponential in k. Total functions have a unique completion, so these witnesses lift unchanged.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02" ], "scalar_inference_allowed": true }, { "id": "address-unbounded-nRelevant-Q0", "kind": "no_polynomial_bound", "A": "nRelevant", "B": "Q0", "exponent": null, "statement": "Total address functions have nRelevant=2^k+k and Q0=O(k+1), excluding every finite power.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "Use k address bits to select one of 2^k data bits. Every input bit is relevant, and reading the address followed by its selected data bit costs k+1. A deterministic tree supplies unambiguous certificate families from its leaves and simulates all the query measures here; fractional certificates and quantum verifiers cost at most the corresponding ordinary certificates/tree depth. Thus the right measure is at most O(k+1), while relevance is exponential in k. Total functions have a unique completion, so these witnesses lift unchanged.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02" ], "scalar_inference_allowed": true }, { "id": "address-unbounded-nRelevant-R1", "kind": "no_polynomial_bound", "A": "nRelevant", "B": "R1", "exponent": null, "statement": "Total address functions have nRelevant=2^k+k and R1=O(k+1), excluding every finite power.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository elementary proof; source measure conventions", "proof_summary": "Use k address bits to select one of 2^k data bits. Every input bit is relevant, and reading the address followed by its selected data bit costs k+1. A deterministic tree supplies unambiguous certificate families from its leaves and simulates all the query measures here; fractional certificates and quantum verifiers cost at most the corresponding ordinary certificates/tree depth. Thus the right measure is at most O(k+1), while relevance is exponential in k. Total functions have a unique completion, so these witnesses lift unchanged.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "BdW02" ], "scalar_inference_allowed": true }, { "id": "domain-added-domainDegree-domainQE", "A": "domainDegree", "B": "domainQE", "exponent": 1, "kind": "upper_bound", "statement": "domainDegree(f)<=2 domainQE(f).", "source": "DFLLV26", "locator": "Lemma 2.2(l), article p.13", "proof_summary": "Each query changes an amplitude polynomial by at most one degree in the input-entry indicators. A T-query algorithm has acceptance probability of degree at most 2T; exact acceptance equals f on the domain.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "composable_query_domain" ], "input_alphabet": "query_domain", "output_alphabet": "bits", "cost": "one_unique_element_answer_per_query", "error": "as_in_measure_definitions", "assumptions": [ "n-element set inputs; query family covers the universe; queries intersect every input in exactly one element." ] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "proof_sources": [ "DFLLV26" ], "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "domain-added-domainQ-domainR", "A": "domainQ", "B": "domainR", "exponent": 1, "kind": "upper_bound", "statement": "domainQ(f)<=O(domainR(f)).", "source": "DFLLV26", "locator": "Section 2.2 query model; repository simulation", "proof_summary": "Simulate each classical query with the quantum oracle, maintain the classical transcript and random seed, and use the same output distribution.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "composable_query_domain" ], "input_alphabet": "query_domain", "output_alphabet": "bits", "cost": "one_unique_element_answer_per_query", "error": "as_in_measure_definitions", "assumptions": [ "n-element set inputs; query family covers the universe; queries intersect every input in exactly one element." ] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": true, "proof_sources": [ "DFLLV26" ], "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "Goos15-D-UC0-square", "kind": "upper_bound", "A": "D", "B": "UC0", "exponent": 2, "statement": "D(f)<=UC0(f)*(UC0(f)+1)/2 for total Boolean functions; in particular D=O(UC0^2).", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "Goos15", "locator": "Proposition 5, p.4; complement symmetry; repository triangular-count refinement", "proof_summary": "Distinct unambiguous b-certificates conflict. Query all literals of one surviving certificate. If it matches, return b; otherwise each surviving certificate loses at least one queried variable. The maximum residual width decreases each round, so widths k,k-1,...,1 sum to k(k+1)/2. A zero-width surviving certificate fixes the output. Apply to the cheaper output side for UCmin.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "Goos15" ], "scalar_inference_allowed": true, "notes": "The original proposition states D<=UP^dt(f)^2, with UP^dt=UC1. The triangular bound is an explicit repository refinement of its proof, not quoted as a printed source constant. No sharp leading-constant optimality claimed." }, { "id": "Goos15-D-UC1-square", "kind": "upper_bound", "A": "D", "B": "UC1", "exponent": 2, "statement": "D(f)<=UC1(f)*(UC1(f)+1)/2 for total Boolean functions; in particular D=O(UC1^2).", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "Goos15", "locator": "Proposition 5, p.4; complement symmetry; repository triangular-count refinement", "proof_summary": "Distinct unambiguous b-certificates conflict. Query all literals of one surviving certificate. If it matches, return b; otherwise each surviving certificate loses at least one queried variable. The maximum residual width decreases each round, so widths k,k-1,...,1 sum to k(k+1)/2. A zero-width surviving certificate fixes the output. Apply to the cheaper output side for UCmin.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "Goos15" ], "scalar_inference_allowed": true, "notes": "The original proposition states D<=UP^dt(f)^2, with UP^dt=UC1. The triangular bound is an explicit repository refinement of its proof, not quoted as a printed source constant. No sharp leading-constant optimality claimed." }, { "id": "Goos15-D-UCmin-square", "kind": "upper_bound", "A": "D", "B": "UCmin", "exponent": 2, "statement": "D(f)<=UCmin(f)*(UCmin(f)+1)/2 for total Boolean functions; in particular D=O(UCmin^2).", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "Goos15", "locator": "Proposition 5, p.4; complement symmetry; repository triangular-count refinement", "proof_summary": "Distinct unambiguous b-certificates conflict. Query all literals of one surviving certificate. If it matches, return b; otherwise each surviving certificate loses at least one queried variable. The maximum residual width decreases each round, so widths k,k-1,...,1 sum to k(k+1)/2. A zero-width surviving certificate fixes the output. Apply to the cheaper output side for UCmin.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "Goos15" ], "scalar_inference_allowed": true, "notes": "The original proposition states D<=UP^dt(f)^2, with UP^dt=UC1. The triangular bound is an explicit repository refinement of its proof, not quoted as a printed source constant. No sharp leading-constant optimality claimed." }, { "id": "product-domainD-domainUC-square", "A": "domainD", "B": "domainUC", "exponent": 2, "kind": "upper_bound", "statement": "On full Cartesian products with coordinate symbol queries, domainD<=O(domainUC^2).", "source": "definitions", "locator": "Repository extension of certificate querying to full products", "proof_summary": "An unambiguous certificate family on a full Cartesian product has pairwise conflicting partial assignments: compatible assignments would have a common valid extension. Querying one surviving certificate reduces all others by at least one coordinate after failure, yielding k(k+1)/2 queries.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "full_product_coordinate_queries" ], "input_alphabet": "query_domain", "output_alphabet": "bits", "cost": "one_unique_element_answer_per_query", "error": "as_in_measure_definitions", "assumptions": [ "n-element set inputs; query family covers the universe; queries intersect every input in exactly one element." ] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "proof_sources": [ "Goos15", "BdW02" ], "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained.", "notes": "This is a profile-specific theorem. It is not inserted as a uniform bound over all composable query domains. Compatible partial assignments need not extend to a permutation or multislice input." }, { "id": "product-domainD-domainC-square", "A": "domainD", "B": "domainC", "exponent": 2, "kind": "upper_bound", "statement": "On full Cartesian products with coordinate symbol queries, domainD<=O(domainC^2).", "source": "definitions", "locator": "Repository extension of certificate querying to full products", "proof_summary": "Use D<=C0*C1: query a consistent positive certificate. At a fixed surviving negative input each failed round reveals a new coordinate of any fixed negative certificate, because opposite-output certificates conflict on a full product. At most C0 rounds of at most C1 queries suffice. Thus D<=C^2. Alphabets need not be binary; compatibility still implies a valid extension.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "full_product_coordinate_queries" ], "input_alphabet": "query_domain", "output_alphabet": "bits", "cost": "one_unique_element_answer_per_query", "error": "as_in_measure_definitions", "assumptions": [ "n-element set inputs; query family covers the universe; queries intersect every input in exactly one element." ] }, "loss": "none", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "proof_sources": [ "Goos15", "BdW02" ], "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained.", "notes": "This is a profile-specific theorem. It is not inserted as a uniform bound over all composable query domains. Compatible partial assignments need not extend to a permutation or multislice input." }, { "id": "domain-parameterized-domainD-domainC", "A": "domainD", "B": "domainC", "exponent": 2, "kind": "upper_bound", "statement": "D(f)<=max(Delta,n/M)*C(f)^2 for M>0. If C(f)<=M, D(f)<=Delta*C0(f)*C1(f)<=Delta*C(f)^2.", "source": "DFLLV26", "locator": "Lemma 3.3, article pp.19–20", "proof_summary": "Query a consistent positive certificate and all allowed queries mentioning its elements. Under the conflict threshold, each round reveals a new element of any surviving negative certificate. At most C0 rounds cost Delta*C1 each. If C>M, use D<=n and C0*C1>M; the global n/M prefactor covers this case.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "composable_query_domain" ], "input_alphabet": "query_domain", "output_alphabet": "bits", "cost": "one_unique_element_answer_per_query", "error": "as_in_measure_definitions", "assumptions": [ "M>0 for the global max(Delta,n/M) formula" ] }, "loss": "domain_parameters", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "notes": "M>0 required for n/M; no pure-power bound is asserted uniformly over domains.", "proof_sources": [ "DFLLV26" ], "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "profile-witnessing-degree-sensitivity", "A": "domainDegree", "B": "domainS", "exponent": 2, "kind": "upper_bound", "statement": "degree(f)<=alpha^(-1)*s(f)^2 on alpha-witnessing domains, alpha>0.", "source": "DFLLV26", "locator": "Theorem 5.3, article p.31", "proof_summary": "Choose a degree-witnessing pseudo-character correlated with f, restrict to its induced Boolean cube of dimension at least alpha*degree, and apply Huang’s sensitivity theorem.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "alpha_witnessing_composable_query_domain" ], "input_alphabet": "query_domain", "output_alphabet": "bits", "cost": "one_unique_element_answer_per_query", "error": "as_in_measure_definitions", "assumptions": [ "alpha>0 and alpha-witnessing domain" ] }, "loss": "domain_parameters", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "notes": "M>0 required for n/M; no pure-power bound is asserted uniformly over domains.", "proof_sources": [ "DFLLV26" ], "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "profile-certificate-sensitivity-five", "A": "domainC", "B": "domainS", "exponent": 5, "kind": "upper_bound", "statement": "C(f)<=6*beta_c^(-1)*alpha^(-2)*s(f)^5 when beta_c>0 and the domain is alpha-witnessing.", "source": "DFLLV26", "locator": "Lemmas 3.5–3.6 and Theorem 5.3; repository composition", "proof_summary": "Lemma 3.5 gives C<=beta_c^(-1)*s*bs. Lemma 3.6 gives bs<=6*adeg^2<=6*degree^2. Substitute Theorem 5.3, degree<=alpha^(-1)*s^2, retaining all constants.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "alpha_witnessing_composable_query_domain" ], "input_alphabet": "query_domain", "output_alphabet": "bits", "cost": "one_unique_element_answer_per_query", "error": "as_in_measure_definitions", "assumptions": [ "alpha>0 and alpha-witnessing domain", "beta_c>0" ] }, "loss": "domain_parameters", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "notes": "M>0 required for n/M; no pure-power bound is asserted uniformly over domains.", "proof_sources": [ "DFLLV26" ], "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "capped-D-R1-square", "kind": "upper_bound", "A": "D", "B": "R1", "exponent": 2, "statement": "For capped one-sided R1 with positive success at least p, D<=floor(R1/p)*C1<=R1^2/p. At p=2/3, D<=3 R1^2/2.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "definitions", "locator": "Repository certificate-query coupling proof; compare Nis89 one-sided argument and AIK26 capped convention", "proof_summary": "Query a consistent positive certificate each round. At any surviving zero input x, replace only its disagreeing literals in each queried certificate to obtain positive inputs y_j; their disagreement blocks are nonempty and disjoint. Couple an R1 computation on x and each y_j with the same randomness. Acceptance probability at least p on y_j and no false positives on x force querying block j with probability at least p on x. Hence kp<=T. After floor(T/p) rounds a nonconstant restriction would allow one more disjoint block, impossible. Each round costs at most C1. A positive accepting transcript is a certificate of width at most T, so C1<=T.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "independent_agent_mathematical_audit_passed_under_stored_conventions", "proof_sources": [ "Nis89", "AIK26" ], "scalar_inference_allowed": true, "notes": "R1 here is a worst-execution query cap, with no false positives. This constant is not transferred to the expected-cost Nisan convention. Constant functions use zero queries. No optimality claim for the quadratic leading constant.", "independent_audit_report": "docs/new-derivations-independent-audit.md" }, { "id": "uniform-hypermatching-D-sensitivity-lift", "A": "domainD", "B": "domainS", "exponent": 10, "kind": "upper_bound", "statement": "On uniform k-hypermatchings with vertex-to-hyperedge queries, D(f)<=10368*(2k-1)^10*s(f)^10, k>=2.", "source": "definitions", "locator": "Repository permutation-lift derivation from final-journal Lemmas 3.3,3.5,3.6 and Theorem 5.3", "proof_summary": "Lift f to permutations of kn vertices by grouping consecutive positions into k-sets. A permutation query or inverse query can be simulated with at most one hyperedge query by lazily assigning each newly discovered hyperedge to an unused position block. For t disjoint sensitive permutation transpositions, form a multigraph on original hyperedges. Sensitive swaps join distinct hyperedges, and each hyperedge has degree at most k because the transpositions use disjoint positions. Greedy matching gives at least t/(2k-1) swaps on disjoint hyperedge pairs, so s_perm<=(2k-1)*s_hyper. The permutation inequalities degree<=2s^2, bs<=6degree^2, C<=3s*bs and D<=2C^2 give D_perm<=10368*s_perm^10. Combine the query simulation and sensitivity comparison.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "uniform_hypermatching_vertex_queries" ], "input_alphabet": "query_domain", "output_alphabet": "bits", "cost": "one_unique_element_answer_per_query", "error": "as_in_measure_definitions", "assumptions": [ "All partitions of kn vertices into n unordered blocks of fixed size k>=2; oracle reveals the block containing a vertex.", "The coefficient depends on k and must be retained if k grows." ] }, "loss": "domain_parameters", "status": "repository_derivation_checked", "proof_audit": "proof_route_checked_not_independent_full_proof", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "notes": "A proved repository reduction with original-source inputs, not a priority or best-known claim. This does not establish an alpha-witnessing theorem or cover all lambda-partite hypermatching profiles. Empty/trivial domains are handled separately.", "proof_sources": [ "DFLLV26" ], "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "completion-support-FC-RC", "kind": "upper_bound", "A": "FC", "B": "RC", "exponent": 1, "statement": "FC(f)<=2*RC(f) under the accepting-x-certainly, rejecting-opposites-with-probability-at-least-1/2, expected-query convention.", "scope": { "object_kind": "decision_function", "totality": "partial_or_total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "repository_derivation_checked", "source": "Aar02", "locator": "Lemma 5 and the fractional certificate LP; repository directed extraction", "proof_summary": "Fix x and a verifier. Let q_i be its probability of querying i on x. Coupling to any opposite-output y forces a query to their disagreement set with probability at least 1/2. Union bound gives sum_{i:x_i!=y_i} 2q_i>=1, so 2q_i form a fractional certificate, of cost at most twice the expected number of queries. Clipping weights to one preserves feasibility. Maximize over x.", "last_checked": "2026-10-17", "best_known_certified": false, "proof_audit": "not_independently_audited", "proof_sources": [ "Aar02" ], "scalar_inference_allowed": true }, { "id": "C0-R1-2-sep-AIK", "kind": "separation", "A": "C0", "B": "R1", "exponent": 2, "statement": "The AIK26 underlying DNF has C0=Theta(n^2) and capped R1=O(n).", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "source": "AIK26", "locator": "Theorem 2.2; Lemma 3.4; Section 3.1", "proof_summary": "Lemma 3.4 ensures that any assignment of fewer than tau_n=n*(ceil(n/2)+1) variables consistent with all zeros still has a completion in the domain of H, hence a satisfying input of the underlying DNF. Therefore C0 at zero is Omega(n^2); input length gives the upper bound. The capped one-sided algorithm is Theorem 2.2.", "last_checked": "2026-10-17", "witness": "underlying DNF", "best_known_certified": false, "proof_audit": "not_independently_audited" }, { "id": "lambda-partite-D-sensitivity-product-lift", "A": "domainD", "B": "domainS", "exponent": 10, "kind": "upper_bound", "statement": "For lambda-partite perfect hypermatchings with k=sum_i lambda_i>=2, D<=10368*(2k-1)^10*s^10 under vertex-to-entire-hyperedge queries.", "source": "definitions", "locator": "Repository product-permutation lift using final-journal Theorems 5.3/5.5 and Lemmas 3.3,3.5,3.6", "proof_summary": "Represent a hypermatching by independent permutations of the lambda_i*n vertices in each color class, grouping lambda_i consecutive positions per common block label. Lazy simultaneous assignment of a discovered hyperedge to a free common block label simulates each forward/inverse permutation query using at most one hyperedge query. Products of the source permutation pseudo-characters span the product degree filtration and induce product cubes of at least half their degree, giving degree<=2s^2 on the permutation product. There Delta<=2, beta_c>=1/3, M=sum_i(lambda_i*n), and total set-input size is k*n. Thus C<=72s^5 and D<=2*C^2. Disjoint sensitive component transpositions form a multigraph on original hyperedges of maximum degree at most k; matching extraction gives s_product<=(2k-1)*s_hyper. Combine these inequalities. The simulation invariant is existential: every transcript admits a representative of the fixed input, not that all arbitrary completions project to it. Independent audit checked the tensor grading, full conflict threshold, simulation and matching argument.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "lambda_partite_hypermatching_vertex_queries" ], "input_alphabet": "query_domain", "output_alphabet": "bits", "cost": "one_unique_element_answer_per_query", "error": "as_in_measure_definitions", "assumptions": [ "All lambda_i positive; n>=2; all lambda-partite perfect hypermatchings are allowed. Vertex queries reveal the entire hyperedge.", "The coefficient depends only on total hyperedge size k; retain it when k grows." ] }, "loss": "domain_parameters", "status": "repository_derivation_checked", "proof_audit": "independent_agent_source_level_audit_passed_after_simulation_invariant_repair", "best_known_certified": false, "last_checked": "2026-10-17", "scalar_inference_allowed": false, "notes": "Repository derivation; separately audited by another assistant agent against original journal inputs. Not formal verification, human peer review, a printed source theorem, or a priority/best-known claim. The alpha=1/2 witnessing argument applies to the permutation product only, not directly to hypermatchings. Full-domain and oracle hypotheses remain essential.", "proof_sources": [ "DFLLV26" ], "journal_version_check": "Final 2026 journal definitions and main theorem/parameter statements checked; original priority source retained." }, { "id": "AIK26-Q0-Q-quadratic", "kind": "separation", "A": "Q0", "B": "Q", "exponent": 2, "statement": "The AIK26 total family gives Q0>=Omega_tilde(Q^2).", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "polylog", "status": "primary_statement_and_proof_route_checked", "source": "AIK26", "locator": "Theorems 2.1 and 2.3; Lemmas 7.3–7.4, article pp.4,19", "proof_summary": "Theorems 2.1 and 2.3 give Q<=O(sqrt(n)*log^3 n) and Q0=Theta(n*log^2 n). The zero-error lower bound restricts the count fields to AND on Theta(n*log^2 n) bits; a nonzero accepting amplitude for AND needs full degree. Exact term evaluation gives the matching Q0 upper bound. Together with C=Omega(n^2) and C=O(Q^4), logarithms of n and Q are comparable, yielding the quadratic lower exponent with polylog loss.", "last_checked": "2026-10-17", "witness": "F_n", "parameters": { "input_bits": "Theta(n^3 log(n)^2)", "n": "construction parameter, not input length" }, "best_known_certified": false, "proof_audit": "not_independently_audited", "notes": "New catalogue consequence of the original paper’s jointly specified family, not a new independent construction. Query caps, exactness and detectable-failure conventions match the source preliminary definitions." }, { "id": "AIK26-C-Q0-quadratic", "kind": "separation", "A": "C", "B": "Q0", "exponent": 2, "statement": "The same AIK26 total family gives C>=Omega_tilde(Q0^2).", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "polylog", "status": "primary_statement_and_proof_route_checked", "source": "AIK26", "locator": "Theorems 2.1 and 2.3; Lemmas 7.3–7.4, article pp.4,19", "proof_summary": "Theorem 2.1 gives C>=Omega(n^2); Theorem 2.3 and Lemmas 7.3–7.4 give Q0=Theta(n*log^2 n). Thus C>=Omega(Q0^2/log^4 Q0). The count-field AND restriction forces the zero-error cost, and exact evaluation supplies its upper bound.", "last_checked": "2026-10-17", "witness": "F_n", "parameters": { "input_bits": "Theta(n^3 log(n)^2)", "n": "construction parameter, not input length" }, "best_known_certified": false, "proof_audit": "not_independently_audited", "notes": "New catalogue consequence of the original paper’s jointly specified family, not a new independent construction. Query caps, exactness and detectable-failure conventions match the source preliminary definitions." }, { "id": "AIK26-C-QE-quadratic", "kind": "separation", "A": "C", "B": "QE", "exponent": 2, "statement": "The same AIK26 total family gives C>=Omega_tilde(QE^2).", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "polylog", "status": "primary_statement_and_proof_route_checked", "source": "AIK26", "locator": "Theorems 2.1 and 2.3; Lemmas 7.3–7.4, article pp.4,19", "proof_summary": "Theorem 2.1 gives C>=Omega(n^2), while Theorem 2.3 gives QE=Theta(n*log^2 n). Hence C>=Omega(QE^2/log^4 QE). The exact algorithm finds the unambiguous terms, reads the selected bits and checks the addressed cheat-sheet cell; the AND restriction gives its lower bound.", "last_checked": "2026-10-17", "witness": "F_n", "parameters": { "input_bits": "Theta(n^3 log(n)^2)", "n": "construction parameter, not input length" }, "best_known_certified": false, "proof_audit": "not_independently_audited", "notes": "New catalogue consequence of the original paper’s jointly specified family, not a new independent construction. Query caps, exactness and detectable-failure conventions match the source preliminary definitions." }, { "id": "bs-s-superquadratic-baseline", "A": "bs", "B": "s", "exponent": "2.000252407", "kind": "separation", "statement": "There are total Boolean functions with unbounded sensitivity and bs(f)>=s(f)^(2.000252407).", "statement_latex": "\\mathrm{bs}(F_k)\\ge s(F_k)^{2.000252407},\\qquad s(F_k)\\longrightarrow\\infty", "source": "OAIsens26", "proof_sources": [ "P2Msens26" ], "locator": "Original paper Theorem 1.1, Section 4 and Corollary 4.3; campaign OpenAI Baseline decimal certificate", "proof_summary": "Random edge labels exclude excessive simultaneous near-satisfied clauses. Four coupled profiles control ordinary sensitivity and simultaneous flips of distinct nested predicates. Block witnesses at zero multiply through recursion; OR balancing gives s<=2(d+2)M^(d+1) and bs at zero >=M^2(2M^2+1)^d for M>=9^(d+1). Composition multiplies the zero-input block lower bound and at most multiplies sensitivity.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "none", "status": "primary_statement_and_proof_route_checked", "proof_audit": "original_paper_selected_proof_route_checked_not_Lean_rebuilt", "best_known_certified": false, "last_checked": "2026-10-17", "witness_id": "sensitivity-baseline", "notes": "The decimal is the exact recorded target, not a rounded claim of a stronger power. Platform-attested does not mean independently kernel-checked by this catalogue. For campaign records the loss is conservatively retained at exponent level." }, { "id": "bs-s-superquadratic-depth20", "A": "bs", "B": "s", "exponent": "2.006470828", "kind": "separation", "statement": "There are total Boolean functions with unbounded sensitivity and bs(f)>=s(f)^(2.006470828 - o(1)).", "statement_latex": "\\forall\\varepsilon>0\\;\\forall S\\ge2\\;\\exists f:\\quad s(f)\\ge S,\\quad\\mathrm{bs}(f)\\ge s(f)^{2.006470828-\\varepsilon}", "source": "P2Mdepth2026", "proof_sources": [ "OAIsens26", "P2Msens26" ], "locator": "Mission description and completed goal; original general seed bounds in OAIsens26 Section 4", "proof_summary": "Specialize the same general seed bounds to depth twenty, certify the stated decimal below log(B)/log(A), and invoke finite-seed amplification. This is a stronger specialization, not a different base architecture.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "subpolynomial", "status": "primary_platform_theorem_and_attestation_checked", "proof_audit": "platform_attestation_and_goal_checked_accepted_proof_not_rebuilt", "best_known_certified": false, "last_checked": "2026-10-17", "witness_id": "sensitivity-depth20", "notes": "The decimal is the exact recorded target, not a rounded claim of a stronger power. Platform-attested does not mean independently kernel-checked by this catalogue. For campaign records the loss is conservatively retained at exponent level." }, { "id": "bs-s-superquadratic-2007", "A": "bs", "B": "s", "exponent": "2.007", "kind": "separation", "statement": "There are total Boolean functions with unbounded sensitivity and bs(f)>=s(f)^(2.007 - o(1)).", "statement_latex": "\\forall\\varepsilon>0\\;\\forall S\\ge2\\;\\exists f:\\quad s(f)\\ge S,\\quad\\mathrm{bs}(f)\\ge s(f)^{2.007-\\varepsilon}", "source": "P2M200726", "proof_sources": [ "OAIsens26", "P2Msens26" ], "locator": "Campaign completed entry; mission goal and certificate framework", "proof_summary": "The campaign records the asymptotic goal as proved. Its stated route is to certify a finite seed with s<=A, bs at a chosen input >=B, and A^2.007<=B, then use the shared normalization/composition theorem. The accepted proof body was not recovered here; this is a summary of the published certificate framework, not an assertion of its exact implementation.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "subpolynomial", "status": "primary_platform_theorem_and_attestation_checked", "proof_audit": "platform_attestation_and_goal_checked_accepted_proof_not_rebuilt", "best_known_certified": false, "last_checked": "2026-10-17", "witness_id": "sensitivity-2007", "notes": "The decimal is the exact recorded target, not a rounded claim of a stronger power. Platform-attested does not mean independently kernel-checked by this catalogue. For campaign records the loss is conservatively retained at exponent level." }, { "id": "bs-s-superquadratic-20205", "A": "bs", "B": "s", "exponent": "2.0205", "kind": "separation", "statement": "There are total Boolean functions with unbounded sensitivity and bs(f)>=s(f)^(2.0205 - o(1)).", "statement_latex": "\\forall\\varepsilon>0\\;\\forall S\\ge2\\;\\exists f:\\quad s(f)\\ge S,\\quad\\mathrm{bs}(f)\\ge s(f)^{2.0205-\\varepsilon}", "source": "P2M2020526", "proof_sources": [ "OAIsens26", "P2Msens26" ], "locator": "Mission construction description, specified parameters, exact certificate and PROVED goal", "proof_summary": "Retain four profiles, reduce the target-row count to fifteen, and split the gate estimate by whether the candidate tournament has a sink. The resulting finite certificate satisfies A^4041=s(f)^(2.0391 - o(1)).", "statement_latex": "\\forall\\varepsilon>0\\;\\forall S\\ge2\\;\\exists f:\\quad s(f)\\ge S,\\quad\\mathrm{bs}(f)\\ge s(f)^{2.0391-\\varepsilon}", "source": "P2M2039126", "proof_sources": [ "OAIsens26", "P2Msens26" ], "locator": "Mission goal, explicit PROVED badge, campaign completed entry", "proof_summary": "The source marks the goal proved and records accepted activity. It formulates the finite-certificate route s<=A, bs at a chosen input >=B, A^2.0391<=B. The accepted proof body and actual seed parameters were not inspected; no more specific construction mechanism is asserted.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "subpolynomial", "status": "primary_platform_theorem_and_attestation_checked", "proof_audit": "platform_attestation_and_goal_checked_accepted_proof_not_rebuilt", "best_known_certified": false, "last_checked": "2026-10-17", "witness_id": "sensitivity-20391", "notes": "The decimal is the exact recorded target, not a rounded claim of a stronger power. Platform-attested does not mean independently kernel-checked by this catalogue. For campaign records the loss is conservatively retained at exponent level." }, { "id": "bs-s-superquadratic-205", "A": "bs", "B": "s", "exponent": "2.05", "kind": "separation", "statement": "There are total Boolean functions with unbounded sensitivity and bs(f)>=s(f)^(2.05 - o(1)).", "statement_latex": "\\forall\\varepsilon>0\\;\\forall S\\ge2\\;\\exists f:\\quad s(f)\\ge S,\\quad\\mathrm{bs}(f)\\ge s(f)^{2.05-\\varepsilon}", "source": "P2M20526", "proof_sources": [ "OAIsens26", "P2Msens26" ], "locator": "Mission SensitivityCampaign.reach_205; explicit PROVED goal and accepted activity; campaign completed entry", "proof_summary": "The inspected mission marks reach_205 proved and displays accepted activity attributed to Robin Kothari. The stated certificate criterion is s<=A, bs at a chosen input >=B, and A^(41/20)<=B, followed by the shared amplification theorem. The accepted proof body was not retrieved, so its exact seed and proof route remain source-extraction gaps.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "subpolynomial", "status": "primary_platform_theorem_and_attestation_checked", "proof_audit": "platform_attestation_and_goal_checked_accepted_proof_not_rebuilt", "best_known_certified": false, "last_checked": "2026-10-17", "witness_id": "sensitivity-205", "notes": "The decimal is the exact recorded target, not a rounded claim of a stronger power. Platform-attested does not mean independently kernel-checked by this catalogue. For campaign records the loss is conservatively retained at exponent level." }, { "id": "bs-s-superquadratic-20635", "A": "bs", "B": "s", "exponent": "2.0635", "kind": "separation", "statement": "For every epsilon>0 and every integer S>=2 there is a total Boolean function f with s(f)>=S and bs(f)>=s(f)^(alpha-epsilon), where alpha=2.0635.", "statement_latex": "\\forall\\varepsilon>0\\;\\forall S\\ge2\\;\\exists f:\\quad s(f)\\ge S,\\quad\\mathrm{bs}(f)\\ge s(f)^{2.0635-\\varepsilon}", "source": "P2M2063526", "proof_sources": [ "OAIsens26", "P2Msens26" ], "locator": "Public quantified theorem, construction/provenance notes and dated campaign completed entry; theorem f4a6abf1-69b0-4785-a7d9-cd45649231b2", "proof_summary": "The tracker now records the exact target 4127/2000 proved. Later Kothari/Codex sources identify accepted submission f80a3f37-72fa-4d56-a094-3e4d351764f3 as their foundation. The theorem provenance still contains stale prose saying full verification is outstanding; the dated proved badge and tracker supersede that status wording. Actual seed and proof details remain unextracted. Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "subpolynomial", "status": "primary_platform_theorem_and_attestation_checked", "proof_audit": "public_theorem_and_construction_summary_checked_authenticated_proof_not_replayed", "best_known_certified": false, "last_checked": "2026-10-22", "witness_id": "sensitivity-20635", "notes": "Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas." }, { "id": "bs-s-superquadratic-2075", "A": "bs", "B": "s", "exponent": "2.075", "kind": "separation", "statement": "For every epsilon>0 and every integer S>=2 there is a total Boolean function f with s(f)>=S and bs(f)>=s(f)^(alpha-epsilon), where alpha=2.075.", "statement_latex": "\\forall\\varepsilon>0\\;\\forall S\\ge2\\;\\exists f:\\quad s(f)\\ge S,\\quad\\mathrm{bs}(f)\\ge s(f)^{2.075-\\varepsilon}", "source": "P2M207526", "proof_sources": [ "OAIsens26", "P2Msens26" ], "locator": "Public quantified theorem, construction/provenance notes and dated campaign completed entry; theorem 49fdd176-c319-4f1e-b15f-b1fd5130f0bc", "proof_summary": "Joint-change bounds and finite labeling existence control the nine sensitivity profiles; construct disjoint block witnesses and certify the finite exponent 83/40 by exact integer arithmetic. The public record identifies Full2075.lean, SHA-256 b8b3d440c3c148ee8401967fc8f2270df9c8c142dfb63f8fde28dfb88bf8c152, and accepted submission 819ad4c6-aa25-48ca-82e9-c8df39c182c0. Apply the campaign finite-certificate theorem. The actual seed integers are not exposed in the public summary. Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "subpolynomial", "status": "primary_platform_theorem_and_attestation_checked", "proof_audit": "public_theorem_and_construction_summary_checked_authenticated_proof_not_replayed", "best_known_certified": false, "last_checked": "2026-10-22", "witness_id": "sensitivity-2075", "notes": "Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas." }, { "id": "bs-s-superquadratic-2085", "A": "bs", "B": "s", "exponent": "2.085", "kind": "separation", "statement": "For every epsilon>0 and every integer S>=2 there is a total Boolean function f with s(f)>=S and bs(f)>=s(f)^(alpha-epsilon), where alpha=2.085.", "statement_latex": "\\forall\\varepsilon>0\\;\\forall S\\ge2\\;\\exists f:\\quad s(f)\\ge S,\\quad\\mathrm{bs}(f)\\ge s(f)^{2.085-\\varepsilon}", "source": "P2M208526", "proof_sources": [ "OAIsens26", "P2Msens26" ], "locator": "Public quantified theorem, construction/provenance notes and dated campaign completed entry; theorem 7e54a28e-196b-40a3-a2c0-957e75db9e4f", "proof_summary": "Refine the finite sensitivity-profile certificate and assemble actual Boolean functions, then invoke finite-seed amplification. The source names Sensitivity20855.finite_seed and achieves_20855 and accepted submission c6096517-61e7-4f51-bc68-c1521d61eb65. The public target is exactly 2.085; the internal declaration name alone is not evidence for a 2.0855 bound. Seed integers remain unavailable. Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "subpolynomial", "status": "primary_platform_theorem_and_attestation_checked", "proof_audit": "public_theorem_and_construction_summary_checked_authenticated_proof_not_replayed", "best_known_certified": false, "last_checked": "2026-10-22", "witness_id": "sensitivity-2085", "notes": "Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas." }, { "id": "bs-s-superquadratic-2119", "A": "bs", "B": "s", "exponent": "2.119", "kind": "separation", "statement": "For every epsilon>0 and every integer S>=2 there is a total Boolean function f with s(f)>=S and bs(f)>=s(f)^(alpha-epsilon), where alpha=2.119.", "statement_latex": "\\forall\\varepsilon>0\\;\\forall S\\ge2\\;\\exists f:\\quad s(f)\\ge S,\\quad\\mathrm{bs}(f)\\ge s(f)^{2.119-\\varepsilon}", "source": "P2M211926", "proof_sources": [ "OAIsens26", "P2Msens26" ], "locator": "Public quantified theorem, construction/provenance notes and dated campaign completed entry; theorem cf005af0-93b2-44c1-991f-ddab41d85874", "proof_summary": "An exact two-stage invariant improves the profile recurrence in the inherited nine-predicate framework. The source reports standalone verification and accepted submission bb3251a4-f00f-4b06-962a-97be2de5db1b. The public summary does not specify each parameter’s recurrence role, so no unprinted formula is inferred from their names. Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "subpolynomial", "status": "primary_platform_theorem_and_attestation_checked", "proof_audit": "public_theorem_and_construction_summary_checked_authenticated_proof_not_replayed", "best_known_certified": false, "last_checked": "2026-10-22", "witness_id": "sensitivity-2119", "notes": "Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas." }, { "id": "bs-s-superquadratic-21", "A": "bs", "B": "s", "exponent": "2.1", "kind": "separation", "statement": "For every epsilon>0 and every integer S>=2 there is a total Boolean function f with s(f)>=S and bs(f)>=s(f)^(alpha-epsilon), where alpha=2.1.", "statement_latex": "\\forall\\varepsilon>0\\;\\forall S\\ge2\\;\\exists f:\\quad s(f)\\ge S,\\quad\\mathrm{bs}(f)\\ge s(f)^{2.1-\\varepsilon}", "source": "P2M2126", "proof_sources": [ "OAIsens26", "P2Msens26" ], "locator": "Public quantified theorem, construction/provenance notes and dated campaign completed entry; theorem 34794a98-e5bd-4623-ae65-526fc87e9ad1", "proof_summary": "Use the current campaign completed status for the exact formal 21/10 goal. The stronger informal 2.105 label is not imported as a verified exponent: the public formal target at this theorem remains 2.10. Accepted proof details are not public without authentication. Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "subpolynomial", "status": "primary_platform_theorem_and_attestation_checked", "proof_audit": "public_theorem_and_construction_summary_checked_authenticated_proof_not_replayed", "best_known_certified": false, "last_checked": "2026-10-22", "witness_id": "sensitivity-21", "notes": "Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas." }, { "id": "bs-s-superquadratic-2127294", "A": "bs", "B": "s", "exponent": "2.127294", "kind": "separation", "statement": "For every epsilon>0 and every integer S>=2 there is a total Boolean function f with s(f)>=S and bs(f)>=s(f)^(alpha-epsilon), where alpha=2.127294.", "statement_latex": "\\forall\\varepsilon>0\\;\\forall S\\ge2\\;\\exists f:\\quad s(f)\\ge S,\\quad\\mathrm{bs}(f)\\ge s(f)^{2.127294-\\varepsilon}", "source": "P2M212729426", "proof_sources": [ "OAIsens26", "P2Msens26" ], "locator": "Public quantified theorem, construction/provenance notes and dated campaign completed entry", "proof_summary": "Sharpen joint boundary estimates in the twelve-predicate recursion, then prove the campaign asymptotic goal. The tracker records completion on 20 October 2026; exact seed parameters and the accepted proof body remain unavailable in the inspected public record. Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "subpolynomial", "status": "primary_platform_theorem_and_attestation_checked", "proof_audit": "public_theorem_and_construction_summary_checked_authenticated_proof_not_replayed", "best_known_certified": false, "last_checked": "2026-10-22", "witness_id": "sensitivity-2127294", "notes": "Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas." }, { "id": "bs-s-superquadratic-21301", "A": "bs", "B": "s", "exponent": "2.1301", "kind": "separation", "statement": "For every epsilon>0 and every integer S>=2 there is a total Boolean function f with s(f)>=S and bs(f)>=s(f)^(alpha-epsilon), where alpha=2.1301.", "statement_latex": "\\forall\\varepsilon>0\\;\\forall S\\ge2\\;\\exists f:\\quad s(f)\\ge S,\\quad\\mathrm{bs}(f)\\ge s(f)^{2.1301-\\varepsilon}", "source": "P2M2130126", "proof_sources": [ "OAIsens26", "P2Msens26" ], "locator": "Public quantified theorem, construction/provenance notes and dated campaign completed entry; theorem ff7ef69d-ca91-413b-872c-cf760dc244dc", "proof_summary": "Multiple defect tolerances and coupled target/guard profiles improve a periodic recursion. Verify the shared labeling conditions and actual disjoint block assembly, then obtain the finite-family bounds and the logarithmic exponent log(G)/log(R)>2.1301. The construction must bound sensitivity at every input and exhibit nonempty pairwise disjoint sensitive blocks at zero. Fixed prefactors are absorbed into an arbitrary positive exponent loss; the known polynomial relation between block sensitivity and sensitivity ensures unbounded sensitivity. Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "subpolynomial", "status": "primary_platform_theorem_and_attestation_checked", "proof_audit": "public_theorem_and_construction_summary_checked_authenticated_proof_not_replayed", "best_known_certified": false, "last_checked": "2026-10-22", "witness_id": "sensitivity-21301", "notes": "Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas." }, { "id": "bs-s-superquadratic-21303", "A": "bs", "B": "s", "exponent": "2.1303", "kind": "separation", "statement": "For every epsilon>0 and every integer S>=2 there is a total Boolean function f with s(f)>=S and bs(f)>=s(f)^(alpha-epsilon), where alpha=2.1303.", "statement_latex": "\\forall\\varepsilon>0\\;\\forall S\\ge2\\;\\exists f:\\quad s(f)\\ge S,\\quad\\mathrm{bs}(f)\\ge s(f)^{2.1303-\\varepsilon}", "source": "P2M2130326", "proof_sources": [ "OAIsens26", "P2Msens26", "P2Msharedfamily26", "MT10" ], "locator": "Public quantified theorem, construction/provenance notes and dated campaign completed entry; theorem 637bbe88-761a-4bb9-bad9-78a8c1bb90c7", "proof_summary": "Apply the finite asymmetric Lovasz local lemma to simultaneous bad-label events, proving joint independence, dependency counts and an exact integer activity certificate. A coupled four-stage profile recurrence and actual Boolean/block assembly give the finite family. Its exponent log(G)/log(R)>2.1303. The construction must bound sensitivity at every input and exhibit nonempty pairwise disjoint sensitive blocks at zero. Fixed prefactors are absorbed into an arbitrary positive exponent loss; the known polynomial relation between block sensitivity and sensitivity ensures unbounded sensitivity. The mission description preserves a historical server-timeout note; the supporting finite-family theorem is now publicly marked Proved. Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "subpolynomial", "status": "primary_platform_theorem_and_attestation_checked", "proof_audit": "public_theorem_and_construction_summary_checked_authenticated_proof_not_replayed", "best_known_certified": false, "last_checked": "2026-10-22", "witness_id": "sensitivity-21303", "notes": "Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas." }, { "id": "bs-s-superquadratic-213", "A": "bs", "B": "s", "exponent": "2.13", "kind": "separation", "statement": "For every epsilon>0 and every integer S>=2 there is a total Boolean function f with s(f)>=S and bs(f)>=s(f)^(alpha-epsilon), where alpha=2.13.", "statement_latex": "\\forall\\varepsilon>0\\;\\forall S\\ge2\\;\\exists f:\\quad s(f)\\ge S,\\quad\\mathrm{bs}(f)\\ge s(f)^{2.13-\\varepsilon}", "source": "P2M21326", "proof_sources": [ "OAIsens26", "P2Msens26" ], "locator": "Public quantified theorem, construction/provenance notes and dated campaign completed entry", "proof_summary": "The tracker records the 2.13 goal completed. Independently, AchievesExponent(2.1301) implies AchievesExponent(2.13) by replacing epsilon with epsilon+0.0001. This is a valid implication from the public stronger theorem, not a claim about the unretrieved accepted 2.13 proof implementation. Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "subpolynomial", "status": "primary_platform_theorem_and_attestation_checked", "proof_audit": "public_theorem_and_construction_summary_checked_authenticated_proof_not_replayed", "best_known_certified": false, "last_checked": "2026-10-22", "witness_id": "sensitivity-213", "notes": "Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas." }, { "id": "bs-s-superquadratic-21308255793978239", "A": "bs", "B": "s", "exponent": "2.1308255793978239", "kind": "separation", "statement": "For every epsilon>0 and every integer S>=2 there is a total Boolean function f with s(f)>=S and bs(f)>=s(f)^(alpha-epsilon), where alpha=log(14055^8)/log(3738277295413674).", "statement_latex": "\\forall\\varepsilon>0\\;\\forall S\\ge2\\;\\exists f:\\quad s(f)\\ge S,\\quad\\mathrm{bs}(f)\\ge s(f)^{\\frac{\\log(14055^8)}{\\log(3738277295413674)}-\\varepsilon}", "source": "P2M2130825579397823926", "proof_sources": [ "OAIsens26", "P2Msens26" ], "locator": "Public quantified theorem, construction/provenance notes and dated campaign completed entry; theorem 01bb63ef-ced2-4ae9-8b81-b5ed295526c6", "proof_summary": "Share one labeling across six ordinary defect tolerances and a source tolerance; feed the simultaneous bounds into a recursive Boolean sensitivity profile and block amplification. The public primitive theorem explicitly expands sensitivity and block sensitivity as finite maxima, proves the full asymptotic quantifiers, and strictly improves log(14055^8)/log(3738289747988852). The campaign wrapper transfers the primitive definitions to the shared ones. Accepted construction submission: a9d25c46-43bd-4fb1-826b-0ed075b8cf0a. The exact exponent is logarithmic, not the truncated tracker decimal. Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "subpolynomial", "status": "primary_platform_theorem_and_attestation_checked", "proof_audit": "public_theorem_and_construction_summary_checked_authenticated_proof_not_replayed", "best_known_certified": false, "last_checked": "2026-10-22", "witness_id": "sensitivity-21308255793978239", "notes": "Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas.", "exponent_expression": { "op": "log_ratio", "numerator": "1522813785050025472152230187890625", "denominator": "3738277295413674" }, "exponent_decimal_role": "Display/ranking approximation only; exponent_expression is authoritative." }, { "id": "bs-s-superquadratic-coupled-exact", "A": "bs", "B": "s", "exponent": "2.130124795190623811814339398329880183773916962501898656874710981831468974402490401064", "kind": "separation", "statement": "The campaign quantified separation holds at exact exponent alpha=log(201838849)/log(7922).", "statement_latex": "\\alpha=\\frac{\\log(201838849)}{\\log(7922)},\\qquad\\forall\\varepsilon>0\\;\\forall S\\ge2\\;\\exists f:\\;s(f)\\ge S,\\;\\mathrm{bs}(f)\\ge s(f)^{\\alpha-\\varepsilon}", "source": "P2Mcoupledexact26", "proof_sources": [ "OAIsens26", "P2Msens26" ], "locator": "Public exact endpoint/finite-family theorem ddfe4bf3-88be-4876-a196-33c20c0319ff", "proof_summary": "Multiple defect tolerances and coupled target/guard profiles improve a periodic recursion. Verify the shared labeling conditions and actual disjoint block assembly, then obtain the finite-family bounds and the logarithmic exponent log(G)/log(R)>2.1301. The construction must bound sensitivity at every input and exhibit nonempty pairwise disjoint sensitive blocks at zero. Fixed prefactors are absorbed into an arbitrary positive exponent loss; the known polynomial relation between block sensitivity and sensitivity ensures unbounded sensitivity. Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "subpolynomial", "status": "primary_platform_theorem_and_attestation_checked", "proof_audit": "public_theorem_and_construction_summary_checked_authenticated_proof_not_replayed", "best_known_certified": false, "last_checked": "2026-10-22", "witness_id": "sensitivity-21301", "notes": "Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas.", "exponent_expression": { "op": "log_ratio", "numerator": "201838849", "denominator": "7922" }, "exponent_decimal_role": "Display/ranking approximation only; exponent_expression is authoritative." }, { "id": "bs-s-superquadratic-shared-exact", "A": "bs", "B": "s", "exponent": "2.130319939591316261213482755199411041844981809669880905177220379594601397047542035920", "kind": "separation", "statement": "The campaign quantified separation holds at exact exponent alpha=log(25260794820729601)/log(50063687).", "statement_latex": "\\alpha=\\frac{\\log(25260794820729601)}{\\log(50063687)},\\qquad\\forall\\varepsilon>0\\;\\forall S\\ge2\\;\\exists f:\\;s(f)\\ge S,\\;\\mathrm{bs}(f)\\ge s(f)^{\\alpha-\\varepsilon}", "source": "P2Msharedexact26", "proof_sources": [ "OAIsens26", "P2Msens26", "P2Msharedfamily26", "MT10" ], "locator": "Public exact endpoint/finite-family theorem 2bc2b8ee-797d-4500-9dc8-9b23c457e131", "proof_summary": "Apply the finite asymmetric Lovasz local lemma to simultaneous bad-label events, proving joint independence, dependency counts and an exact integer activity certificate. A coupled four-stage profile recurrence and actual Boolean/block assembly give the finite family. Its exponent log(G)/log(R)>2.1303. The construction must bound sensitivity at every input and exhibit nonempty pairwise disjoint sensitive blocks at zero. Fixed prefactors are absorbed into an arbitrary positive exponent loss; the known polynomial relation between block sensitivity and sensitivity ensures unbounded sensitivity. The mission description preserves a historical server-timeout note; the supporting finite-family theorem is now publicly marked Proved. Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "subpolynomial", "status": "primary_platform_theorem_and_attestation_checked", "proof_audit": "public_theorem_and_construction_summary_checked_authenticated_proof_not_replayed", "best_known_certified": false, "last_checked": "2026-10-22", "witness_id": "sensitivity-21303", "notes": "Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas.", "exponent_expression": { "op": "log_ratio", "numerator": "25260794820729601", "denominator": "50063687" }, "exponent_decimal_role": "Display/ranking approximation only; exponent_expression is authoritative." }, { "id": "bs-s-superquadratic-refined-exact", "A": "bs", "B": "s", "exponent": "2.130320333220365838965754517205705352241211113936833114071341079654743300887191423144", "kind": "separation", "statement": "The campaign quantified separation holds at exact exponent alpha=log(25260794820729601)/log(50063523).", "statement_latex": "\\alpha=\\frac{\\log(25260794820729601)}{\\log(50063523)},\\qquad\\forall\\varepsilon>0\\;\\forall S\\ge2\\;\\exists f:\\;s(f)\\ge S,\\;\\mathrm{bs}(f)\\ge s(f)^{\\alpha-\\varepsilon}", "source": "P2Mrefinedexact26", "proof_sources": [ "OAIsens26", "P2Msens26", "P2Msharedfamily26", "MT10" ], "locator": "Public exact endpoint/finite-family theorem 62961329-4b50-48c7-a1fd-11159da9de26", "proof_summary": "Keep the shared labeling and block amplification of the four-stage construction, refine the finite sensitivity recipe from growth factor 50063687 to 50063523, and retain block growth 12607^4. Absorb the fixed prefactor into positive exponent loss. Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "loss": "subpolynomial", "status": "primary_platform_theorem_and_attestation_checked", "proof_audit": "public_theorem_and_construction_summary_checked_authenticated_proof_not_replayed", "best_known_certified": false, "last_checked": "2026-10-22", "witness_id": "sensitivity-21303", "notes": "Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas.", "exponent_expression": { "op": "log_ratio", "numerator": "25260794820729601", "denominator": "50063523" }, "exponent_decimal_role": "Display/ranking approximation only; exponent_expression is authoritative." } ], "conjectures": [ { "id": "sensitivity-polynomial", "statement": "bs(f) <= poly(s(f)) for all total Boolean functions", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions" }, "source": "HKP11", "locator": "Central sensitivity conjecture", "status": "resolved", "resolution": [ "bs-s4" ], "implications": [ "The recorded Huang and Nisan–Szegedy bounds give bs(f) <= 2 s(f)^4.", "Later constant improvements are recorded separately (bs-deg2-Tal, bs-deg2-Pro, and the Wellens bound); the displayed historical consequence is not the sharpest coefficient." ], "status_source": "Hua19" }, { "id": "D-bs-quadratic", "statement": "D(f) = O(bs(f)^2)", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions" }, "source": "ABKBRT21", "locator": "Conjecture 1", "status": "searched_no_resolution_found_current_openness_not_certified", "implications": [ "Would imply D=O(R^2) using bs=O(R).", "Would imply D=O(adeg^4) using bs=O(adeg^2)." ], "implication_audit": "Relevant bs lower-bound edges still require extraction.", "status_history": [ { "superseded_by_audit": "2026-10-23", "status": "historical_conjecture_current_status_unverified" } ], "status_date": "2026-10-23", "status_audit": { "checked": "2026-10-23", "verdict": "no_resolution_found", "summary": "The cubic-to-quadratic unrestricted problem remains unresolved in the checked CPS proceedings; quadratic results there require zebra or bounded-alternation hypotheses. A search through 23 October 2026 found no primary resolution.", "evidence": [ { "source": "ABKBRT21", "locator": "Conjecture 1", "role": "original target" }, { "source": "CPS23", "locator": "Question 1 and Theorems 4–5", "role": "restricted progress; not a general resolution" }, { "source": "CPS23", "locator": "Theorem 5 and Proposition 29(1)", "role": "Parameter-dependent progress: bounded alternation only" } ], "queries": [ "\"D(f)\" \"bs(f)\" quadratic conjecture 2026 site:arxiv.org", "\"Decision Tree Complexity\" \"Block Sensitivity\" 2025 2026 conjecture", "\"block sensitivity\" \"quadratic\" \"decision tree\" 2025 2026 conjecture" ], "limitations": "No exhaustive literature or independent full-proof correctness certification is claimed.", "followup_checked": "2026-10-23", "clarifications": [ { "title": "What the restricted theorem proves", "text": "CPS23 gives a bound depending on alternation. Replacing alternation by a universal constant changes the class of functions, so it cannot close the unrestricted quadratic question.", "latex": "D(f)=O(\\operatorname{alt}(f)\\,s(f)\\,\\operatorname{bs}(f))." } ] }, "status_sources": [ "ABKBRT21", "CPS23" ] }, { "id": "R-Q-cubic", "statement": "R(f) = O(Q(f)^3)", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions" }, "source": "ABKBRT21", "locator": "Conjecture 2", "status": "reported_refuted_in_primary_preprint", "implications": [ "A near-quartic separation would refute this." ], "leads": [ "quartic-RQ-2026" ], "resolution": [ "R-Q4-sep-2026" ], "status_source": "OAI26", "status_date": "2026-10-17", "notes": "Original Theorem 1.1 and construction inspected; independent full-proof audit and publication review remain distinct." }, { "id": "rational-polynomial", "statement": "deg(f) <= poly(rdeg(f)) for total Boolean functions", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions" }, "source": "IJKKSWW23", "locator": "Introduction / abstract", "status": "resolved_in_primary_paper", "implications": [ "Would relate exact rational representations to the usual total-function polynomial cluster." ], "resolution": [ "deg-rdeg3" ], "status_source": "KKWY26", "status_date": "2026-10-17", "notes": "Latest version v3 has four authors and cubic exponent up to logarithms; the January v1 quartic statement is historical." }, { "id": "bs-lambda-quadratic", "statement": "bs(f)=O(lambda(f)^2) for all total Boolean f", "source": "ABKBRT21", "locator": "Open problems / spectral sensitivity questions", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "status": "reported_refuted_in_primary_preprint", "status_source": "Mei26", "resolution": [ "bs-lambda-sep-2026" ], "implications": [ "The spectral counterexample alone does not refute bs=O(s^2). The stronger ordinary-sensitivity quadratic question is separately refuted by OAIsens26 and improved campaign entries." ] }, { "id": "Saks-Wigderson-maximal-speedup", "source": "SW86", "locator": "Conjecture 1.6, p.31", "statement": "In the source’s zero-error convention, R(f)>=Omega(D(f)^log_2((1+sqrt(33))/4)) for all total Boolean f.", "status": "refuted", "resolution": [ "D-R0-sep-ABB" ], "implications": [ "Would make balanced NAND trees extremal for randomized speedup. Pointer functions in ABBLS17 refute this historical conjecture." ] }, { "id": "extremal-forrelation-sqrt", "source": "GS26", "locator": "Conjecture 6", "statement": "The randomized complexity of Extremal Forrelation is tildeOmega(2^(n/2)).", "status": "open_in_checked_recent_primary_source", "implications": [ "Would improve the proven tildeOmega(2^(n/4)) lower bound. The paper reports a 2^(n/4+o(n)) distinguisher for its own two hard distributions, so new distributions are needed.", "CCM26 v2 proves a near-half-exponent NONADAPTIVE lower bound. It does not resolve this adaptive conjecture. Appendix D explains a gap in the stronger v1 claim." ], "status_checked": "2026-10-17", "status_sources": [ "GS26", "CCM26" ], "status_history": [ { "superseded_by_audit": "2026-10-23", "status": "conjectured_in_inspected_2026_source_current_status_unverified", "status_sources": [ "CCM26" ] } ], "status_date": "2026-10-23", "status_audit": { "checked": "2026-10-23", "verdict": "primary_reaffirmed_open", "summary": "The latest CCM v2 is nonadaptive. Its Appendix B.4 explicitly distinguishes the remaining adaptive Girish–Servedio conjecture; the stronger adaptive claim in v1 has a gap.", "evidence": [ { "source": "GS26", "locator": "Conjecture 6, Section 1.3", "role": "original adaptive question" }, { "source": "CCM26", "locator": "v2 Theorems 1/3, Appendix B.4, Appendix D", "role": "current restricted result and explicit remaining adaptive conjecture" } ], "queries": [ "\"extremal forrelation\" \"2026\" conjecture randomized", "arXiv:2602.07503 current version history" ], "limitations": "No exhaustive literature or independent full-proof correctness certification is claimed." } }, { "id": "extremal-k-forrelation", "source": "GS26", "locator": "Conjecture 7", "statement": "For k>2, the randomized complexity of extremal k-Forrelation is tildeOmega(2^(n*(1-1/k))).", "status": "searched_no_resolution_found_current_openness_not_certified", "implications": [ "Would approach the best bounded-error quantum/classical separations using exact quantum algorithms." ], "status_history": [ { "superseded_by_audit": "2026-10-23", "status": "conjectured_in_inspected_2026_source_current_status_unverified" } ], "status_date": "2026-10-23", "status_audit": { "checked": "2026-10-23", "verdict": "no_resolution_found", "summary": "No primary resolution of the k>2 extremal/exact-quantum conjecture was found. The checked CCM theorem is about the two-function extremal problem with a nonadaptive classical lower bound; ordinary bounded-error k-Forrelation separations are different.", "evidence": [ { "source": "GS26", "locator": "Conjecture 7, Section 1.3", "role": "original k>2 extremal question" }, { "source": "CCM26", "locator": "v2 Introduction and Theorem 3", "role": "nearby k=2 restricted result, not a resolution" }, { "source": "BS21", "locator": "Abstract and main theorem", "role": "Ordinary bounded-error k-Forrelation; different extremal promise" } ], "queries": [ "\"extremal forrelation\" \"2026\" conjecture randomized", "\"extremal k-forrelation\" lower bound", "\"extremal\" \"k-Forrelation\"", "\"k-Forrelation\" \"exact\" conjecture" ], "limitations": "No exhaustive literature or independent full-proof correctness certification is claimed.", "followup_checked": "2026-10-23", "clarifications": [ { "title": "Two different promises", "text": "BS21 proves the optimal ordinary bounded-error k-Forrelation separation. GS26 Conjecture 7 asks for extremal instances supporting exact quantum computation. The adjective extremal in a discussion of optimal separations alone does not identify the extremal promise.", "latex": "R(\\mathrm{Extremal}\\ k\\text{-Forrelation})\\stackrel{?}{=}\\widetilde\\Omega(2^{n(1-1/k)}),\\quad k>2." } ] }, "status_sources": [ "GS26", "CCM26", "BS21" ] }, { "id": "BK24-search-QE-R", "source": "BK24", "locator": "Section 1.3, Open Problem 1", "statement": "Can a total search problem have an exponential QE/R separation in either direction?", "status": "searched_no_resolution_found_current_openness_not_certified", "implications": [ "The 2024 paper does not establish exact-query separations; its Q and bounded-polynomial constructions alone do not answer this." ], "status_history": [ { "superseded_by_audit": "2026-10-23", "status": "open_question_as_of_inspected_2024_source_current_status_unverified" } ], "status_date": "2026-10-23", "status_audit": { "checked": "2026-10-23", "verdict": "no_resolution_found", "summary": "No resolution found. The October Pelagic Fourier Fishing paper gives an optimal bounded-error query separation, while its exact result concerns a different communication problem. Neither answers BK24’s exact query question; pseudo-deterministic quantum results also use a different output requirement.", "evidence": [ { "source": "BK24", "locator": "Section 1.3, Open Problem 1, PDF pp.8–9", "role": "original precise relation question; latest listed v1" }, { "source": "AGL26", "locator": "Final Theorem 4 and Section 3.1", "role": "Pseudo-deterministic quantum, not exact arbitrary-output search" }, { "source": "ADGG26", "locator": "Theorem 1.1, Section 1.1.1; Theorem 1.3", "role": "Recent bounded-error query separation; separate exact communication result" } ], "queries": [ "\"total search\" \"exact quantum\" \"exponential\" 2026 query complexity", "\"total search\" \"conical\" \"2026\"", "\"total search\" \"adversary\" \"approximate degree\" \"open\"", "\"Separations in query complexity for total search problems\"", "\"total search\" \"exact quantum\" separation", "\"Separations in query complexity for total search problems\" 2025 2026" ], "limitations": "No exhaustive literature or independent full-proof correctness certification is claimed.", "followup_checked": "2026-10-23", "clarifications": [ { "title": "October progress does not give exact query search", "text": "ADGG26 Pelagic Fourier Fishing has one quantum query with success at least 3/4 and randomized cost linear in truth-table length. Its Theorem 1.3 is exact quantum communication for a different problem, with a fixed-degree polynomial classical lower bound. Neither establishes the exponential exact-query separation asked by BK24. AGL26 restricts outputs to a canonical answer with high probability; exact relation algorithms may output many different valid answers.", "latex": "Q(\\mathrm{Pelagic})=1,\\quad R(\\mathrm{Pelagic})=\\Omega(N),\\quad N=2^n.", "proof_summary": "For the upper bound, Fourier sampling chooses y with probability |hhat(y)|^2. Each excluded tail carries less than 1/8 mass, so total success is at least 3/4. A T-query classical algorithm yields a two-way protocol exchanging at most two bits per query to h=fg, transferring the communication lower bound. This verifies the scope/reduction, not every step of the 65-page lower-bound proof." } ] }, "status_sources": [ "BK24", "AGL26", "ADGG26" ] }, { "id": "BK24-search-poly-conical", "source": "BK24", "locator": "Section 1.3, Open Problem 2", "statement": "Can total search exponentially separate bounded approximate polynomial degree from bounded approximate nonnegative degree? What about exact degree versus approximate degree?", "status": "searched_no_resolution_found_current_openness_not_certified", "implications": [ "Uses relation polynomial systems, not the ordinary unrestricted polynomial of a Boolean function." ], "status_history": [ { "superseded_by_audit": "2026-10-23", "status": "open_question_as_of_inspected_2024_source_current_status_unverified" } ], "status_date": "2026-10-23", "status_audit": { "checked": "2026-10-23", "verdict": "no_resolution_found", "summary": "Both approximate-degree/conical-degree and exact-degree/approximate-degree subquestions remain listed in the unchanged BK24 source. Searches found no primary answer. Relation polynomial systems and their boundedness conventions remain essential.", "evidence": [ { "source": "BK24", "locator": "Section 1.3, Open Problem 2, PDF pp.8–9", "role": "original precise relation question; latest listed v1" }, { "source": "BK24", "locator": "Definitions 10–12, PDF pp.11–12", "role": "Precise global nonnegativity and normalization constraints" } ], "queries": [ "\"total search\" \"exact quantum\" \"exponential\" 2026 query complexity", "\"total search\" \"conical\" \"2026\"", "\"total search\" \"adversary\" \"approximate degree\" \"open\"", "\"Separations in query complexity for total search problems\"", "\"total search\" \"conical\" degree 2025 2026", "\"total search\" \"approximate degree\" 2026 2025 separation \"Ben-David\"" ], "limitations": "No exhaustive literature or independent full-proof correctness certification is claimed.", "followup_checked": "2026-10-23", "clarifications": [ { "title": "Two subquestions with a shared convention", "text": "The first asks whether approximate conical degree can be exponentially larger than approximate relation degree. The second asks whether exact relation degree can be exponentially larger than approximate relation degree. A relation is represented by one nonnegative polynomial per output; their sum is at most one on the whole cube and valid-output mass is at least 1-epsilon on the domain. Exact degree means epsilon=0. Scalar-function or unbounded-off-promise polynomial separations do not suffice.", "latex": "\\widetilde{\\deg}_{+2}(R)\\gg\\widetilde{\\deg}(R)\\qquad\\text{or}\\qquad\\deg(R)\\gg\\widetilde{\\deg}(R)." } ] }, "status_sources": [ "BK24" ] }, { "id": "BK24-search-Adv-degree", "source": "BK24", "locator": "Section 1.3, Open Problem 3", "statement": "Does Adv(S)=O(rel_adeg(S)^2) hold for total search? Can Adv be exponentially larger than rel_adeg?", "status": "searched_no_resolution_found_current_openness_not_certified", "implications": [ "Do not extend the partial decision-function theorem ABK21 to relations automatically. BK24 supplies an exponential separation in the opposite direction." ], "status_history": [ { "superseded_by_audit": "2026-10-23", "status": "open_question_as_of_inspected_2024_source_current_status_unverified" } ], "status_date": "2026-10-23", "status_audit": { "checked": "2026-10-23", "verdict": "no_resolution_found", "summary": "No primary proof or counterexample to the relation Adv=O(approximate-degree squared) direction was found. BK24’s exponential separation is in the opposite direction; partial-function theorems do not automatically transfer to total relations.", "evidence": [ { "source": "BK24", "locator": "Section 1.3, Open Problem 3, PDF pp.8–9", "role": "original precise relation question; latest listed v1" }, { "source": "BK24", "locator": "Definition 13 and Theorem 8", "role": "Positive-weight adversary and direction of existing separation" } ], "queries": [ "\"total search\" \"exact quantum\" \"exponential\" 2026 query complexity", "\"total search\" \"conical\" \"2026\"", "\"total search\" \"adversary\" \"approximate degree\" \"open\"", "\"Separations in query complexity for total search problems\"", "\"total search\" \"adversary\" \"approximate degree\"" ], "limitations": "No exhaustive literature or independent full-proof correctness certification is claimed.", "followup_checked": "2026-10-23", "clarifications": [ { "title": "Which direction remains unproved", "text": "BK24 Theorem 8 separates approximate degree above the positive adversary bound. Open Problem 3 asks about the reverse ordering. Adv here is the positive-weight relation bound in Definition 13, not the negative-weight adversary bound.", "latex": "\\operatorname{Adv}(R)\\stackrel{?}{=}O(\\widetilde{\\deg}(R)^2)." } ] }, "status_sources": [ "BK24" ] }, { "id": "KLMY-randomized-AND-derandomization", "source": "KLMY21survey", "locator": "Conjecture 3.16", "statement": "DandPos(f)<=poly(RandPos(f),log n) for total Boolean functions.", "status": "resolved_as_stated_by_2023", "resolution": [ "and-D-R3" ], "implications": [ "CDMRS23 Theorem 4.5 gives DandPos=O(RandPos^3 log^4 n). Input-length logarithms remain essential: and-threshold-no-pure-power shows that a pure RandPos power cannot hold." ] }, { "id": "KLMY-monomial-approximation", "source": "KLMY21survey", "locator": "Conjecture 3.19 and Claim 3.20", "statement": "Exact 0/1 monomial sparsity is polynomially bounded by approximate 0/1 monomial sparsity for total Boolean functions.", "status": "refuted_as_literally_stated", "implications": [ "The literal absolute-sparsity statement in Conjecture 3.19 is false. Sparse25 Remark 1.2 gives OR and near-top threshold counterexamples. The later positive result concerns logarithms and includes log n; it must not be recorded as a proof of the literal conjecture." ], "resolution": [ "monomial-sparsity-no-pure-approx-power" ] }, { "id": "KLMY-pure-monomial-AND", "source": "KLMY21", "locator": "Conjecture 6.1", "statement": "DandPos(f)<=poly(log(2+monomialSparsity(f))) for total Boolean functions, eliminating input-length dependence.", "status": "searched_no_resolution_found_current_openness_not_certified", "implications": [ "Theorem 1.2 establishes O(log^5(2+spar)*log(2n)); the conjecture would eliminate its input-length logarithm. Do not confuse this with polynomial derandomization allowing log n factors." ], "status_history": [ { "superseded_by_audit": "2026-10-23", "status": "conjectured_in_inspected_2020_preprint_current_status_unverified" } ], "status_date": "2026-10-23", "status_audit": { "checked": "2026-10-23", "verdict": "no_resolution_found", "summary": "The dimension-free AND-query target remains conjectural in the checked sources. The 2026 quantum/classical result is for communication and retains polylog(n) factors; neither it nor hardness incondensability removes the log n in the query bound.", "evidence": [ { "source": "KLMY21", "locator": "Conjecture 6.1", "role": "original dimension-free query target" }, { "source": "KLMY21survey", "locator": "Conjecture 3.13 and Theorem 3.11", "role": "same target in survey" }, { "source": "BBCLD26", "locator": "Section 1.1, PDF pp.3–5", "role": "related communication result with input-length losses" }, { "source": "NPS26", "locator": "Sections 3–4", "role": "related condensation question, not this conjecture" }, { "source": "Sparse25", "locator": "Theorems 4.6–4.7 and discussion following them", "role": "De Morgan representations with input-length dependence, different from the pure query target" } ], "queries": [ "\"Log-Rank and Lifting for AND-Functions\" conjecture 2025 2026", "\"monomial sparsity\" \"AND\" \"2026\" conjecture", "\"AND\" \"monomial sparsity\" \"log\" 2026", "\"Restriction Trees for Sparsity and Applications\" conjecture AND log n" ], "limitations": "No exhaustive literature or independent full-proof correctness certification is claimed.", "followup_checked": "2026-10-23", "clarifications": [ { "title": "Which sparsity and which model", "text": "The target uses ordinary 0/1 monomial sparsity and AND-query depth. Sparse25 compares exact and approximate De Morgan/generalized-monomial sparsity and retains input-length factors; BBCLD26 is a communication theorem. Those are substantive adjacent results but do not establish this dimension-free query bound.", "latex": "D_{\\land}(f)\\stackrel{?}{\\leq}\\operatorname{poly}(\\log(2+\\operatorname{spar}(f)))." } ] }, "status_sources": [ "KLMY21", "KLMY21survey", "BBCLD26", "NPS26", "Sparse25" ] }, { "id": "Wellens-relevant-sensitivity", "source": "Wel22", "locator": "Conjecture 3.17", "statement": "For every total Boolean function, nRelevant(f)=O(4^s(f)). The source also conjectures a stronger bounded coordinate-sensitivity potential.", "status": "searched_no_resolution_found_current_openness_not_certified", "implications": [ "Would remove the factor s from Simon’s bound. Monotone address functions show the exponential base 4 cannot be improved uniformly. The stronger potential requires its own definition; it is not entered as an additional scalar theorem." ], "status_history": [ { "superseded_by_audit": "2026-10-23", "status": "conjectured_in_2022_source_current_status_unverified" } ], "status_date": "2026-10-23", "status_audit": { "checked": "2026-10-23", "verdict": "no_resolution_found", "summary": "No primary resolution of nRelevant=O(4^s), or the stronger coordinate-sensitivity potential, was found. The latest listed Wellens version remains v2 (December 2022). Bounds involving polynomial degree and hardness condensation address different questions.", "evidence": [ { "source": "Wel22", "locator": "Conjecture 3.17; discussion near end of paper", "role": "exact sensitivity-variable conjecture" }, { "source": "GNSTW16", "locator": "Theorem 8, proceedings p.61", "role": "Historical Simon bound; not the sharper conjecture" }, { "source": "Wel22", "locator": "Section 3.5, Conjecture 3.17 and Theorem 3.18", "role": "Strong potential target versus mixed-measure progress" } ], "queries": [ "\"Wellens\" \"relevant variables\" \"conjecture\" sensitivity 2026", "\"n(f)\" \"4\" \"Wellens\" \"conjecture\"", "\"relevant variables\" \"sensitivity\" \"2026\" bound", "\"Wellens\" \"Conjecture 3.17\"", "\"relevant variables\" \"sensitivity\" Wellens conjecture" ], "limitations": "No exhaustive literature or independent full-proof correctness certification is claimed.", "followup_checked": "2026-10-23", "clarifications": [ { "title": "Removing a factor versus changing the exponent", "text": "The known Simon bound has a prefactor s; Wellens asks to remove it uniformly. The stronger potential assertion would imply that removal because each coordinate sensitivity is at most twice maximum sensitivity. The mixed sensitivity/degree theorem is not a sensitivity-only solution.", "latex": "n_{\\mathrm{rel}}(f)\\leq s(f)4^{s(f)},\\qquad n_{\\mathrm{rel}}(f)\\stackrel{?}{=}O(4^{s(f)})." } ] }, "status_sources": [ "Wel22", "GNSTW16" ] }, { "id": "HKMP-natural-perturbation", "source": "HKMP26", "locator": "Conjecture 72", "statement": "Characterize polynomial approximate-degree natural completions by perturbing Fourier coefficients, allowing basis terms up to polynomial degree.", "status": "formulation_still_needs_clarification", "implications": [ "The exact support, coefficient norm and sign-preservation constraints need clarification. Without constraints, every target polynomial in the enlarged basis is a coefficient perturbation, so that literal formulation is a representation identity. Lemma 71 does not currently provide a verified normalization route." ], "formulation_audit": { "fixed_completion": "For a fixed F extending f and a full degree-t basis, q approximates F iff Delta=coeff(q)-coeff(p) gives p_Delta=q. This is an exact representation equivalence, not a new characterization theorem.", "fixed_sign_completion": "If F is required to equal sign(p) off the promise, existence of some other low-degree completion does not imply this fixed F is easy. Kernel preservation on the promise does not establish off-promise sign agreement.", "restricted_perturbation": "Support, norm, degree, margin and the quantifier fixing F must be specified to obtain a nontrivial restricted claim. No resolution of that intended restricted formulation is asserted." }, "uniformity_and_norm_audit": { "uniform_polynomial": "A family interpretation must fix polynomial overhead constants independently of n and the particular function; allowing a different constant for every single finite function makes poly(d) vacuous.", "bounded_target_coefficients": "For globally bounded p and q, each Fourier coefficient has magnitude at most one, so unrestricted Delta=qhat-phat has coefficient infinity norm at most two. Thus that norm restriction alone does not distinguish bounded target approximants in the enlarged full basis.", "exact_domain_kernel": "Delta in ker(A_D) forces q=p pointwise on the promise, stronger than q merely approximating the same Boolean labels. It is not automatic for an arbitrary target approximant.", "fixed_target": "For fixed F and a full permitted basis, existence of q approximating F and existence of an unrestricted coefficient perturbation yielding such q are equivalent by coefficient subtraction. An intended stronger statement must add restrictions beyond that representation." }, "bounded_radius_followup": { "status": "unrestricted_dependent_support_still_unverified", "sufficient_result": "quantitative-kernel-line-slab-avoidance", "missing_control": "Uniform polynomial bounds on distinct constraints, nonzero directional slopes, resulting coefficient norm and function amplification are not obtained from support minimality.", "scope_warning": "The separate lambda-partite sensitivity lift has no implication for the natural-completion characterization." }, "bounded_polytope_followup": "Support minimality gives a full-rank stack of domain rows and active full-cube bounds at an interpolation-polytope vertex. This is not a proof of full rank for the domain matrix or polynomial kernel amplification.", "status_history": [ { "superseded_by_audit": "2026-10-23", "status": "formulation_or_current_resolution_unverified" } ], "status_date": "2026-10-23", "status_audit": { "checked": "2026-10-23", "verdict": "formulation_unresolved", "summary": "Literal enlarged-basis perturbation is coefficient subtraction after fixing the same completion and degree scale. Definition 59 uses native approximate degree, whereas Conjecture 72 writes deg(p); an explicit parity family diagnoses this difference for arbitrary p. The intended restricted normalization statement remains unresolved.", "evidence": [ { "source": "HKMP26", "locator": "v3 Conjecture 72 and Lemma 71", "role": "source formulation and flagged normalization step" }, { "source": "HKMP26", "locator": "Definition 59, manuscript p.27; Definition 68; Lemma 71 and Conjecture 72, pp.33–34", "role": "Fixes the target completion and exposes distinct degree parameters" } ], "queries": [ "\"natural completions\" \"perturbation\" 2026", "\"completion\" \"perturbation\" \"Huffstutler\"" ], "limitations": "The intended restricted formulation and the dependent-support normalization step remain unresolved in the atlas. Version-history confirmation is not a mathematical repair.", "followup_checked": "2026-10-23", "clarifications": [ { "title": "Fix both the target and the degree scale", "text": "Definition 59 fixes F_p(x)=+1 when p(x)>=0 and -1 otherwise. Its no-blowup condition is measured against native approximate degree a. Conjecture 72 instead writes polynomial in d=deg(p). For a family, require d=poly(a) with common constants before treating these scales as interchangeable. At zeros, F_p=+1; a replacement approximant need not share those zeros.", "latex": "a=\\widetilde{\\deg}(f),\\quad d=\\deg(p),\\quad F_p(x)=\\begin{cases}+1&p(x)\\geq0,\\\\-1&p(x)<0.\\end{cases}" }, { "title": "A diagnostic family for the degree mismatch", "text": "On the promise product_{i=2}^n x_i=1, let f_n(x)=x_1 and choose p_n(x)=product_{i=1}^n x_i. Native approximate degree is one, but the chosen polynomial and its fixed natural completion have approximate degree n. Delta=0 satisfies the polynomial-in-d approximant condition. Thus arbitrary p and a native-degree left side cannot be combined literally; this diagnoses a missing degree hypothesis rather than refuting an intended optimal-degree version.", "latex": "\\widetilde{\\deg}(f_n)=1,\\quad d_n=n,\\quad F_{p_n}=\\chi_{[n]},\\quad\\widetilde{\\deg}(F_{p_n})=n.", "proof_summary": "Both labels occur on the promise, so no constant approximates f_n; x_1 does. For any polynomial r of degree less than n, orthogonality gives E[r chi_[n]]=0. Uniform approximation to chi_[n] with error below one would instead force this expectation positive. Hence the natural parity completion has approximate degree n." }, { "title": "Literal unrestricted perturbation is an equivalence of coordinates", "text": "With a full degree-t basis, t>=d, every approximant q to this same fixed F_p has Delta=qhat-phat and p_Delta=q. Conversely such a perturbation already supplies an approximant. This implication needs neither a kernel condition nor Lemma 71. It is an algebraic observation about the literal enlarged-basis statement, not evidence for a stronger restricted characterization.", "latex": "q=p+\\sum_{|S|\\leq t}(\\widehat q(S)-\\widehat p(S))\\chi_S." }, { "title": "A precise sufficient normalization condition", "text": "An editorial sufficient statement, not a replacement author conjecture: q has degree t, full-cube norm at most K, full-cube margin gamma>0 and sign(q)=F_p everywhere. Then fixed-error approximate degree of F_p is O(tK/gamma). To obtain polynomial native-degree overhead, t and K/gamma must be polynomial in a with uniform constants. Keeping only promise signs gives some completion, not necessarily F_p. A coefficient infinity-norm bound alone does not control K.", "latex": "\\|q\\|_{\\infty}\\leq K,\\quad\\min_x|q(x)|\\geq\\gamma\\quad\\Longrightarrow\\quad\\widetilde{\\deg}(\\operatorname{sign}q)=O(tK/\\gamma).", "proof_summary": "Normalize by K, then compose with a polynomial sign approximant on the two intervals separated from zero by gamma/K. The unrestricted original-support dependent-kernel case remains unverified." } ] }, "status_sources": [ "HKMP26" ], "formulation_clarifications": [ { "title": "Fix both the target and the degree scale", "text": "Definition 59 fixes F_p(x)=+1 when p(x)>=0 and -1 otherwise. Its no-blowup condition is measured against native approximate degree a. Conjecture 72 instead writes polynomial in d=deg(p). For a family, require d=poly(a) with common constants before treating these scales as interchangeable. At zeros, F_p=+1; a replacement approximant need not share those zeros.", "latex": "a=\\widetilde{\\deg}(f),\\quad d=\\deg(p),\\quad F_p(x)=\\begin{cases}+1&p(x)\\geq0,\\\\-1&p(x)<0.\\end{cases}" }, { "title": "A diagnostic family for the degree mismatch", "text": "On the promise product_{i=2}^n x_i=1, let f_n(x)=x_1 and choose p_n(x)=product_{i=1}^n x_i. Native approximate degree is one, but the chosen polynomial and its fixed natural completion have approximate degree n. Delta=0 satisfies the polynomial-in-d approximant condition. Thus arbitrary p and a native-degree left side cannot be combined literally; this diagnoses a missing degree hypothesis rather than refuting an intended optimal-degree version.", "latex": "\\widetilde{\\deg}(f_n)=1,\\quad d_n=n,\\quad F_{p_n}=\\chi_{[n]},\\quad\\widetilde{\\deg}(F_{p_n})=n.", "proof_summary": "Both labels occur on the promise, so no constant approximates f_n; x_1 does. For any polynomial r of degree less than n, orthogonality gives E[r chi_[n]]=0. Uniform approximation to chi_[n] with error below one would instead force this expectation positive. Hence the natural parity completion has approximate degree n." }, { "title": "Literal unrestricted perturbation is an equivalence of coordinates", "text": "With a full degree-t basis, t>=d, every approximant q to this same fixed F_p has Delta=qhat-phat and p_Delta=q. Conversely such a perturbation already supplies an approximant. This implication needs neither a kernel condition nor Lemma 71. It is an algebraic observation about the literal enlarged-basis statement, not evidence for a stronger restricted characterization.", "latex": "q=p+\\sum_{|S|\\leq t}(\\widehat q(S)-\\widehat p(S))\\chi_S." }, { "title": "A precise sufficient normalization condition", "text": "An editorial sufficient statement, not a replacement author conjecture: q has degree t, full-cube norm at most K, full-cube margin gamma>0 and sign(q)=F_p everywhere. Then fixed-error approximate degree of F_p is O(tK/gamma). To obtain polynomial native-degree overhead, t and K/gamma must be polynomial in a with uniform constants. Keeping only promise signs gives some completion, not necessarily F_p. A coefficient infinity-norm bound alone does not control K.", "latex": "\\|q\\|_{\\infty}\\leq K,\\quad\\min_x|q(x)|\\geq\\gamma\\quad\\Longrightarrow\\quad\\widetilde{\\deg}(\\operatorname{sign}q)=O(tK/\\gamma).", "proof_summary": "Normalize by K, then compose with a polynomial sign approximant on the two intervals separated from zero by gamma/K. The unrestricted original-support dependent-kernel case remains unverified." } ] }, { "id": "HKMP-dense-almost-completion", "source": "HKMP26", "locator": "Conjecture 78", "statement": "If Dom(f) has density at least epsilon and bounded approximate degree d, can one discard a delta fraction of the promise and extend the remainder to a total function with approximate degree poly(d,1/epsilon,1/delta)?", "status": "open_in_checked_recent_primary_source", "implications": [ "The extension is of a subfunction, not necessarily of f on its whole domain. It is a polynomial-degree analogue of the average-case simulation question recalled as Conjecture 77." ], "status_history": [ { "superseded_by_audit": "2026-10-23", "status": "formulation_or_current_resolution_unverified" } ], "status_date": "2026-10-23", "status_audit": { "checked": "2026-10-23", "verdict": "primary_reaffirmed_open", "summary": "Conjecture 78 remains explicitly posed in the latest v3. It permits discarding a delta fraction of the promise and extends a subfunction; this is not completion of every original promise input. No resolution was found.", "evidence": [ { "source": "HKMP26", "locator": "v3 Conjectures 77–78, manuscript p.36", "role": "explicit current conjecture and subfunction caveat" } ], "queries": [ "\"dense\" \"completion\" \"approximate degree\" \"2026\" conjecture", "arXiv:2603.29256 current version history" ], "limitations": "No exhaustive literature or independent full-proof correctness certification is claimed." }, "status_sources": [ "HKMP26" ] }, { "id": "domain-robust-low-degree", "source": "DFLLV22author", "locator": "Section 9, Robust versions", "statement": "A Boolean function on S_n close in uniform squared L2 to degree d should be close to a Boolean function computable by a decision tree of polynomial depth in d; perhaps to a Boolean degree-d function. A corresponding question is proposed for perfect matchings.", "status": "open_in_checked_recent_primary_source", "implications": [ "A robustness conjecture, not a consequence of exact low-degree polynomial equivalence. The source gives no explicit uniform error modulus for the full conjecture." ], "status_sources": [ "DFLLV26", "FLM26" ], "journal_locator": "Section 8, article pp.48–49", "query_lift_warning": "The repository hypermatching lift is an exact query/sensitivity comparison. It does not prove the robust low-degree approximation question or construct hypermatching degree witnesses.", "status_history": [ { "superseded_by_audit": "2026-10-23", "status": "still_proposed_in_2026_journal_current_resolution_unverified", "status_sources": [ "DFLLV26" ] } ], "status_date": "2026-10-23", "status_audit": { "checked": "2026-10-23", "verdict": "primary_reaffirmed_open", "summary": "The final 2026 journal explicitly retains the arbitrary-degree robust decision-tree question on the symmetric group and perfect matchings. Degree-one Grassmann FKN structure is related progress on a different domain, not a solution.", "evidence": [ { "source": "DFLLV26", "locator": "Section 8, article pp.48–49", "role": "explicit open robustness direction" }, { "source": "FLM26", "locator": "Theorems 1.2–1.3", "role": "related degree-one Grassmann structural result only" } ], "queries": [ "\"robust\" \"symmetric group\" \"degree\" \"2026\" Boolean", "\"An FKN Theorem for the Binary Grassmann Scheme\"" ], "limitations": "No exhaustive literature or independent full-proof correctness certification is claimed." } }, { "id": "domain-q-analog-framework", "source": "DFLLV22author", "locator": "Section 9, q-analogs", "statement": "Can the polynomial-equivalence framework be extended to Grassmann, bilinear and general linear group domains, or can nontrivial counterexamples be found?", "status": "open_in_checked_recent_primary_source", "implications": [ "A research direction rather than a proved scalar exponent comparison; query models and domain constants must be specified." ], "status_sources": [ "DFLLV26", "FLM26" ], "journal_locator": "Section 8, article pp.48–49", "status_history": [ { "superseded_by_audit": "2026-10-23", "status": "still_proposed_in_2026_journal_current_resolution_unverified", "status_sources": [ "DFLLV26" ] } ], "status_date": "2026-10-23", "status_audit": { "checked": "2026-10-23", "verdict": "primary_reaffirmed_open", "summary": "The final 2026 journal says it can neither extend its general comparison theorem to q-analogs nor give nontrivial counterexamples. The 2026 binary Grassmann FKN theorem concerns degree-one structure and does not establish a full query-measure equivalence framework.", "evidence": [ { "source": "DFLLV26", "locator": "Section 8, q-analogs, article p.49", "role": "explicit current open direction" }, { "source": "FLM26", "locator": "Introduction and Theorem 1.3", "role": "adjacent structural progress, different target" } ], "queries": [ "\"Grassmann\" \"sensitivity\" \"Boolean\" conjecture", "\"An FKN Theorem for the Binary Grassmann Scheme\"" ], "limitations": "No exhaustive literature or independent full-proof correctness certification is claimed." } }, { "id": "AA-folklore-average-simulation", "source": "AA14", "locator": "Conjecture 1.5 and Appendix Conjecture 7.1", "statement": "A T-query quantum acceptance probability can be approximated within epsilon on at least 1-delta of uniform inputs by a deterministic classical tree of depth poly(T,1/epsilon,1/delta).", "status": "open_in_checked_recent_primary_source", "implications": [ "The paper explicitly calls this folklore. Appendix 7 gives the equivalent dense-promise formulation. HKMP26 Conjecture 77 recalls that formulation; do not label the influence conjecture equivalent to it without a converse proof." ], "status_sources": [ "LMparallel26", "GMNR26", "BSW22", "HKMP26" ], "status_history": [ { "superseded_by_audit": "2026-10-23", "status": "conjectured_in_original_and_recalled_in_2026_primary_source_no_resolution_verified", "status_sources": [ "GMNR26", "HKMP26" ] } ], "status_date": "2026-10-23", "status_audit": { "checked": "2026-10-23", "verdict": "primary_reaffirmed_open", "summary": "The unrestricted adaptive simulation question remains explicitly posed in LM v2 (20 October 2026). Its Theorems 1.6–1.8 prove parallel, limited-preprocessing and constant-layer cases; layer-dependent exponents do not give a uniform polynomial for arbitrary adaptivity. GMNR establishes equivalences, not truth of the conjecture.", "evidence": [ { "source": "LMparallel26", "locator": "Conjecture 1.1, Theorems 1.6–1.8, PDF pp.3,7–8", "role": "recent explicit conjecture and restricted progress" }, { "source": "GMNR26", "locator": "Introduction and Theorem 1.1", "role": "distributional equivalences only" }, { "source": "BSW22", "locator": "Conjecture 1.1 and Section 4", "role": "earlier formulation and restricted case" }, { "source": "HKMP26", "locator": "Conjecture 77", "role": "equivalent dense-promise formulation" } ], "queries": [ "\"Aaronson Ambainis conjecture\" 2026 resolution", "\"Parallel Quantum Advantage with Limited Adaptivity Requires Structure\"" ], "limitations": "No exhaustive literature or independent full-proof correctness certification is claimed." } }, { "id": "AA-bounded-polynomial-influence", "source": "AA14", "locator": "Conjecture 1.7; Theorem 1.8", "statement": "Every full-cube [0,1]-bounded degree-d polynomial with uniform variance at least v has a coordinate influence at least (v/d)^O(1).", "status": "open_in_checked_recent_primary_source", "implications": [ "This implies the folklore simulation conjecture. GMNR26 restates it as Conjecture 4.2 and proves equivalence of its uniform, fixed-bias product and fixed-density slice analogues, not its truth." ], "status_sources": [ "LMparallel26", "GMNR26", "BSW22", "Bha26" ], "status_history": [ { "superseded_by_audit": "2026-10-23", "status": "conjectured_in_original_and_recalled_in_2026_primary_source_no_resolution_verified", "status_sources": [ "GMNR26", "HKMP26" ] } ], "status_date": "2026-10-23", "status_audit": { "checked": "2026-10-23", "verdict": "primary_reaffirmed_open", "summary": "The unrestricted real bounded-polynomial influence conjecture remains explicitly open in the checked October LM version and September GMNR paper. Completely bounded block-multilinear forms and random restrictions under variance hypotheses are positive special cases, not a general proof. This stronger polynomial question is kept distinct from algorithmic simulation.", "evidence": [ { "source": "LMparallel26", "locator": "Introduction, PDF p.4", "role": "explicit general-open statement" }, { "source": "GMNR26", "locator": "Conjecture 4.2 and Theorem 4.3", "role": "same conjecture and equivalences" }, { "source": "BSW22", "locator": "Conjecture 1.2 and Theorem 1.3", "role": "completely bounded positive case" }, { "source": "Bha26", "locator": "Conjecture 1.3 and main result", "role": "random-restriction positive case" } ], "queries": [ "\"Aaronson Ambainis conjecture\" 2026 resolution", "\"Aaronson-Ambainis Conjecture Is True For Random Restrictions\"" ], "limitations": "No exhaustive literature or independent full-proof correctness certification is claimed." } }, { "id": "FLLV20-hypergraphical-sensitivity-question", "source": "FLLV20notes", "locator": "Section 7, p.7", "statement": "Historical question: extend sensitivity theorems to hypergraphical analogues of the symmetric group and perfect matching scheme.", "status": "formulation_still_needs_clarification", "implications": [ "Separate coarse polynomial equivalence for fixed hyperedge size from a direct quadratic intrinsic degree/sensitivity theorem or a witnessing framework.", "The separately audited repository query lift gives coarse parameterized polynomial bounds on full lambda-partite hypermatching domains; no published priority or resolution of every intended formulation is claimed." ], "status_sources": [ "FLLV20notes", "DFLLV26" ], "notes": "The original notes closing sentence does not quantify the desired exponent or parameter uniformity. Do not replace this ambiguity by a stronger or weaker conjecture without qualification. The nearby notes context emphasizes Huang-type estimates; a direct quadratic reading is plausible, but no retroactive resolution of this ambiguous historical formulation is asserted.", "status_history": [ { "superseded_by_audit": "2026-10-23", "status": "historical_question_precise_exponent_and_current_literature_status_unverified", "status_sources": [ "DFLLV26" ] } ], "status_date": "2026-10-23", "status_audit": { "checked": "2026-10-23", "verdict": "formulation_unresolved", "summary": "The historical question is now separated into explicit editorial variants: a fixed-hyperedge-size coarse query comparison, a quadratic intrinsic-degree bound, and stronger uniformity or witness requirements. The atlas lift establishes the stated coarse variant in its oracle model; it does not settle the others or establish published priority.", "evidence": [ { "source": "FLLV20notes", "locator": "Section 7, p.7", "role": "historical ambiguous formulation" }, { "source": "DFLLV26", "locator": "Section 8 and query-domain framework", "role": "related later journal context" }, { "source": "FLLV20notes", "locator": "Sections 1, 3 and 7", "role": "Huang-type quantitative context versus unquantified closing question" }, { "source": "DFLLV26", "locator": "Section 5 opening paragraph", "role": "Limitation of the representation-theoretic method, not a counterexample" } ], "queries": [ "\"hypergraphical\" \"sensitivity\" Boolean functions", "\"Grassmann\" \"sensitivity\" \"Boolean\" conjecture", "\"hypergraphical\" \"sensitivity\"" ], "limitations": "No exhaustive literature or independent full-proof correctness certification is claimed.", "followup_checked": "2026-10-23", "clarifications": [ { "title": "Specify the domain and oracle first", "text": "Use all lambda-partite perfect hypermatchings, with fixed positive lambda_i and k=sum_i lambda_i. A query at a vertex reveals its entire hyperedge. Degree is minimum degree in hyperedge-indicator coordinates, and sensitivity uses the native disjoint-support convention. Restricting the allowed hypermatchings or changing to membership queries is a different problem." }, { "title": "Coarse polynomial equivalence", "text": "One possible question asks for a uniform-in-n polynomial when k is fixed. The separately audited atlas lift supplies the displayed bound in this oracle model. This is a repository derivation, with published priority unverified. It establishes this particular coarse statement, rather than every reading of the historical question.", "latex": "D(f)\\leq 10368(2k-1)^{10}s(f)^{10}." }, { "title": "Direct quadratic intrinsic-degree theorem", "text": "A stronger possible target is a Huang-type intrinsic degree bound with exponent two. Neither the coarse query lift nor the final journal method establishes it for these hypermatching domains. This is an editorial quantitative reading suggested by Sections 1 and 3 of the 2020 notes, not an exact conjecture attributed to the authors.", "latex": "\\deg_{\\mathcal H_{\\lambda,n}}(f)\\stackrel{?}{\\leq}C_{\\lambda}s(f)^2." }, { "title": "Uniformity and witness framework are further questions", "text": "A constant independent of lambda, or a direct witnessing/representation-theoretic proof, is stronger or method-specific. The final journal says its technique does not capture hypergraphical perfect matchings; this is a limitation of that technique, not a theorem that no polynomial comparison exists. The historical closing sentence does not choose among these variants." } ] }, "formulation_clarifications": [ { "title": "Specify the domain and oracle first", "text": "Use all lambda-partite perfect hypermatchings, with fixed positive lambda_i and k=sum_i lambda_i. A query at a vertex reveals its entire hyperedge. Degree is minimum degree in hyperedge-indicator coordinates, and sensitivity uses the native disjoint-support convention. Restricting the allowed hypermatchings or changing to membership queries is a different problem." }, { "title": "Coarse polynomial equivalence", "text": "One possible question asks for a uniform-in-n polynomial when k is fixed. The separately audited atlas lift supplies the displayed bound in this oracle model. This is a repository derivation, with published priority unverified. It establishes this particular coarse statement, rather than every reading of the historical question.", "latex": "D(f)\\leq 10368(2k-1)^{10}s(f)^{10}." }, { "title": "Direct quadratic intrinsic-degree theorem", "text": "A stronger possible target is a Huang-type intrinsic degree bound with exponent two. Neither the coarse query lift nor the final journal method establishes it for these hypermatching domains. This is an editorial quantitative reading suggested by Sections 1 and 3 of the 2020 notes, not an exact conjecture attributed to the authors.", "latex": "\\deg_{\\mathcal H_{\\lambda,n}}(f)\\stackrel{?}{\\leq}C_{\\lambda}s(f)^2." }, { "title": "Uniformity and witness framework are further questions", "text": "A constant independent of lambda, or a direct witnessing/representation-theoretic proof, is stronger or method-specific. The final journal says its technique does not capture hypergraphical perfect matchings; this is a limitation of that technique, not a theorem that no polynomial comparison exists. The historical closing sentence does not choose among these variants." } ] }, { "id": "bs-s-quadratic", "statement": "bs(f)=O(s(f)^2) for all total Boolean functions", "statement_latex": "\\mathrm{bs}(f)=O(s(f)^2)", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "source": "NS94", "locator": "Section 4 quadratic suggestion; original attribution checked against OAIsens26 introduction", "status": "refuted_in_primary_preprint_and_platform_improvements_recorded", "status_source": "OAIsens26", "resolution": [ "bs-s-superquadratic-baseline", "bs-s-superquadratic-21308255793978239" ], "implications": [ "The qualitative polynomial sensitivity conjecture remains resolved positively; its quadratic strengthening fails. The recorded universal upper exponent remains 4.", "On 22 October 2026 the campaign records exact achieved exponent log(14055^8)/log(3738277295413674), with arbitrary positive exponent loss. The 2.15 target remains open." ] } ], "leads": [ { "id": "quartic-RQ-2026", "title": "A Nearly Quartic Separation Between Randomized and Quantum Query Complexity", "url": "https://github.com/openai/math/blob/main/preprints/A-Nearly-Quartic-Separation-Between-Randomized-and-Quantum-Query-Complexity-October-5-2026/quartic-query-separation.pdf", "status": "source_extracted_followup_audit_needed", "action": "Primary PDF, pinned commit, theorem and proof route inspected and recorded. Remaining: independent full-proof audit, subsequent corrections and publication tracking.", "audit_update": "Definitions and selected primary theorem/proof routes entered 2026-10-17; broad literature completeness remains separate." }, { "id": "search-definitions", "source": "BK24", "status": "source_extracted_followup_audit_needed", "action": "Extract every relation polynomial definition, all theorem parameters, and exact proof summaries. Relation degree cannot inherit the decision-function definition.", "audit_update": "Definitions and selected primary theorem/proof routes entered 2026-10-17; broad literature completeness remains separate." }, { "id": "certification-2026", "source": "KLLPV26", "status": "source_extracted_followup_audit_needed", "action": "Certification transform, eighteen instantiated measures and main comparisons extracted. Remaining: every parameterized variant, complete proof audit and original attribution chain.", "audit_update": "Definitions and selected primary theorem/proof routes entered 2026-10-17; broad literature completeness remains separate." }, { "id": "games", "source": "CG22", "status": "source_extracted_followup_audit_needed", "action": "Private/public/entangled/nonsignaling and single-bit variants extracted. Remaining: source-wide alias audit, partial-certificate conventions and every separating example.", "audit_update": "Definitions and selected primary theorem/proof routes entered 2026-10-17; broad literature completeness remains separate." }, { "id": "new-block-measures", "source": "CG26", "status": "source_extracted_followup_audit_needed", "action": "Block/brick definitions and sensitivity counting bounds extracted. Remaining: all DNF/tree-size applications and cross-version notation.", "audit_update": "Definitions and selected primary theorem/proof routes entered 2026-10-17; broad literature completeness remains separate." }, { "id": "expectation", "source": "KLW14", "status": "source_extracted_followup_audit_needed", "action": "Expectation-model definitions and literal-degree/SOS exact characterizations extracted. Remaining: Boolean-output sharper comparisons and all auxiliary polynomial variants.", "audit_update": "Definitions and selected primary theorem/proof routes entered 2026-10-17; broad literature completeness remains separate." }, { "id": "noisy", "source": "ARS26", "status": "source_extracted_followup_audit_needed", "action": "Separate negligent, dephasing, hybrid, and time-dependent models and extract multivariate tradeoffs.", "audit_update": "Definitions and selected primary theorem/proof routes entered 2026-10-17; broad literature completeness remains separate." }, { "id": "core-table-completion", "source": "ABKBRT21", "status": "total_and_promise_core_grids_filled_other_models_incomplete", "action": "Total core has 132 evidence chains. Promise core has all 132 cells classified: 49 matched finite power exponents and 83 non-polynomial pairs, using weak certificates and unrestricted promise degrees. Complete search, globally bounded polynomial, other-measure and priority coverage remains pending." }, { "id": "notation-completion", "status": "core_historical_occurrences_extracted_inventory_incomplete", "action": "Historical core aliases and source clashes extracted; unsupported seed attributions repaired. Finish candidate occurrences and expand beyond the checked-source inventory. CDM23 input-directed s_b/bs_b, GIPS search notation, and nonadaptive/restricted-query conventions are now extracted; output-conditioned s_b/bs_b remain separate." }, { "id": "search-model-survey", "url": "https://drops.dagstuhl.de/storage/00lipics/lipics-vol272-mfcs2023/LIPIcs.MFCS.2023.34/LIPIcs.MFCS.2023.34.pdf", "status": "source_extracted_followup_audit_needed", "action": "Read Query Complexity of Search Problems; reconcile cubic/quartic pseudo-deterministic statements with later formulations and their output conventions.", "audit_update": "Definitions and selected primary theorem/proof routes entered 2026-10-17; broad literature completeness remains separate." }, { "id": "additional-polynomial-measures", "status": "pending", "action": "Add finite-field/modular degree, threshold/sign degree, approximate rational degree, postselection complexity, one-sided/nonnegative/SOS/block-multilinear/completely-bounded degrees, query lower-bound LPs and partition measures, with explicit definitions before relations." }, { "id": "restricted-classes", "status": "pending", "action": "Add monotone, transitive, symmetric, graph properties, read-once formulas, composition families, and distributional/average-case restrictions; restrictions are intersected, not unioned." }, { "id": "bibliography-completion", "status": "87_versioned_references_original_text_gaps_remain", "action": "Publication metadata and original sources added for size/AND queries and quantum tree simulation (CDMRS23, KLMY21, LM19, LL16, BT20, CMP22), plus KLMY21survey and ZPP26. EH89 bibliography and attribution are complete but its original full proof is uninspected; Nis91 original-source access gap also remains.", "audit_update": "User-supplied EH89 and Nis89 originals now inspected for definitions and selected proofs. Nis91 journal text remains unchecked; exact one-sided expected-cost constant discrepancy remains pending." }, { "id": "rational-degree-version-discrepancy", "source": "IJKKSWW23", "status": "reconciled", "action": "Version-specific authors and bounds recorded under KKWY26. v1 quartic, v2/v3 cubic with logarithmic loss; retain both historical conjecture and resolution provenance." }, { "id": "monotone-variants", "source": "BJKM24", "status": "source_extracted_followup_audit_needed", "action": "Extract mbs, fmbs, MCC, monotone sensitivity, and their names/notations. These variants are not the ordinary measures restricted to monotone functions.", "audit_update": "Definitions and selected primary theorem/proof routes entered 2026-10-17; broad literature completeness remains separate." }, { "id": "rank-size-and-instance", "source": "DM23", "status": "source_extracted_followup_audit_needed", "action": "Extract decision-tree rank, size, conjunctive depth, and sparsity bounds; then process LM26 for instance complexity.", "audit_update": "Definitions and selected primary theorem/proof routes entered 2026-10-17; broad literature completeness remains separate." }, { "id": "hazard-free", "source": "BKMNSS25", "status": "source_extracted_followup_audit_needed", "action": "Extend scope schema to ternary inputs/outputs and add hazard-free measures. Do not put hazard-free upper bounds in Boolean-function rows.", "audit_update": "Definitions and selected primary theorem/proof routes entered 2026-10-17; broad literature completeness remains separate." }, { "id": "search-frontier-completion", "source": "BK24", "status": "partially_extracted_not_exhaustive", "action": "Six scope-separated search evidence matrices generated; finish unindexed comparisons, exact/zero-error search, full relation degree/conical gaps and forward-citation checks on BK24 questions. Do not declare its 2024 questions open today." }, { "id": "search-size-and-distribution-completion", "source": "CDM23full", "status": "final_MFCS_size_ANDOR_and_GIPS_average_results_extracted", "action": "Closed the final-version ApproxMaj/canonical-size, unlifted CNF, AND/OR, strong average-cost and quantum-size error-convention gaps. Remaining: complete exact/zero-error search frontiers, optimal size endpoints and error-dependent comparisons; original EH89 proof and complete attribution chains still pending.", "audit_update": "User-supplied EH89 and Nis89 originals now inspected for definitions and selected proofs. Nis91 journal text remains unchecked; exact one-sided expected-cost constant discrepancy remains pending." }, { "id": "partial-noncore-completion", "source": "ABK21", "status": "core_finished_extended_inventory_incomplete", "action": "Extend the complete core promise grid to globally bounded polynomial degrees, strong certificates, adversary and error/local variants; complete original-priority and historical-name inventories." }, { "id": "general-search-error-reduction", "source": "CMP22", "status": "identified_and_guarded_further_audit_pending", "action": "Q_{2/5} relation simulation kept separate from default Q_{1/3}. Inventory error-dependent search measures and reductions with verification or canonical output; do not assume Boolean majority amplification for relations." }, { "id": "strong-average-cost-observations", "source": "GIPS21", "status": "source_extracted_repository_sharpenings_explicit", "action": "GIPS21 Theorems 26–29 and Proposition 25 extracted. Strict finite nonuniform Las Vegas averages admit fixed-coins equality, and dense Find1 admits a distribution-dependent deterministic scan of mean cost below two. These are explicitly repository derivations, not first-priority claims; retain distinctions from failure-flagged caps and uniform algorithms." }, { "id": "new-domain-and-completion-extraction", "title": "Full completion and non-cube domain inventories", "source": "HKMP26", "status": "source_extracted_followup_audit_needed", "action": "Extract promised sensitivity/bs, critical certificates and minimum-over-completions measures with all bounds; audit HKMP26 Conjectures 72,77,78. Complete DFLLV21 general-domain parameter matrix, perfect matching/multislice definitions and journal-version check; follow 2025/2026 Hamming-graph relevant-variable papers.", "audit_update": "23 minimum-completion measures have all 506 directions indexed, including 442 finite intervals and 64 no-finite-power directions. Seven parameterized domain-profile upper grids cover 924 cells. Final DFLLV26 journal selected results are checked; the lambda lift passed separate-agent audit after correction. Remaining: unmatched exponent intervals, unrestricted dependent-support normalization, precise conjecture formulations, current-best/priority certification, wider measure/domain extraction and pending journal texts." } ], "witnesses": [ { "id": "OR_n", "definition": "OR of n input bits", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric", "monotone" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "parameter": "n = input length", "source": "BBCMW01", "locator": "Introduction and AND/OR characterization", "values": { "D": "n", "R": "Theta(n)", "Q": "Theta(sqrt(n))", "deg": "n", "adeg": "Theta(sqrt(n))" }, "notes": "Use to witness quadratic lower exponents for D/Q, R/Q, deg/Q, and deg/adeg. Not a quartic witness." }, { "id": "Imbalance_n", "source": "IJKKSWW23", "locator": "v2 Section 6.2", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "values": { "rdeg": "O(1)", "adeg": "Omega(n)", "Q": "Omega(n)" }, "definition_status": "formal_promise_extraction_pending" }, { "id": "MajOrNone_n", "source": "IJKKSWW23", "locator": "v2 Section 6.1", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "values": { "Q": "O(1)", "rdeg": "Omega(n)" }, "definition_status": "formal_promise_extraction_pending" }, { "id": "promise-unit-OR", "definition": "Domain {0^m,e_1,...,e_m}; output 0 at zero and 1 on unit vectors.", "source": null, "locator": "Repository derivation", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "definition_status": "explicit_promise_and_derivation_checked", "proof_summary": "Unrestricted polynomial sum_i x_i is exact and has degree one. Zero requires inspecting all coordinates classically; each unit position is a disjoint sensitive singleton at zero. The sensitivity graph is a star with norm sqrt(m). Grover search gives Q=O(sqrt(m)); the star adversary gives the matching lower bound. For exact search, apply BHMT02 Theorem 4 with hypothesized marked fraction 1/m and verify the returned index. When there is one mark it is found with certainty; when there are none verification rejects. This also proves QE=O(sqrt(m)).", "parameter": "m (growing positive integer)", "growth_lower": { "D": 1, "R0": 1, "R": 1, "C": 1, "RC": 1, "bs": 1, "s": 1, "lambda": 0.5, "Q": 0.5, "QE": 0.5, "deg": 0, "adeg": 0 }, "growth_upper": { "D": 1, "R0": 1, "R": 1, "C": 1, "RC": 1, "bs": 1, "s": 1, "lambda": 0.5, "Q": 0.5, "deg": 0, "adeg": 0, "QE": 0.5 }, "values": { "D": "Theta(m^1)", "R0": "Theta(m^1)", "R": "Theta(m^1)", "C": "Theta(m^1)", "RC": "Theta(m^1)", "bs": "Theta(m^1)", "s": "Theta(m^1)", "lambda": "Theta(m^0.5)", "Q": "Theta(m^0.5)", "deg": "Theta(m^0)", "adeg": "Theta(m^0)", "QE": "Theta(sqrt(m))" }, "notes": "Growth exponents are bounds in the construction parameter, not claims of optimal separation exponents in all models. A zero upper exponent denotes O(1), including zero. No original-priority claim is made for repository derivations.", "proof_sources": [ "BHMT02", "BBCMW01" ] }, { "id": "promise-doubled-parity", "definition": "Domain x_(2i-1)=x_(2i), i=1,...,m. Output parity of the m decoded bits.", "source": null, "locator": "Repository derivation", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "definition_status": "explicit_promise_and_derivation_checked", "proof_summary": "There are no promised pairs at Hamming distance one, hence s=lambda=0. Substitution x_(2i-1)=x_(2i)=z_i converts any interpolating/approximating polynomial to parity on m bits, giving degree m. Flip whole pairs for bs=m. One representative bit per pair supplies the classical upper bound; standard parity lower bounds and verification give linear remaining measures.", "parameter": "m (growing positive integer)", "growth_lower": { "D": 1, "R0": 1, "R": 1, "C": 1, "RC": 1, "bs": 1, "QE": 1, "deg": 1, "Q": 1, "adeg": 1 }, "growth_upper": { "D": 1, "R0": 1, "R": 1, "C": 1, "RC": 1, "bs": 1, "QE": 1, "deg": 1, "Q": 1, "adeg": 1, "s": 0, "lambda": 0 }, "values": { "D": "Theta(m^1)", "R0": "Theta(m^1)", "R": "Theta(m^1)", "C": "Theta(m^1)", "RC": "Theta(m^1)", "bs": "Theta(m^1)", "QE": "Theta(m^1)", "deg": "Theta(m^1)", "Q": "Theta(m^1)", "adeg": "Theta(m^1)" }, "notes": "Growth exponents are bounds in the construction parameter, not claims of optimal separation exponents in all models. A zero upper exponent denotes O(1), including zero. No original-priority claim is made for repository derivations." }, { "id": "promise-dense-two-axes", "definition": "Input (a,b) in {0,1}^{2m}. Domain: (|a|>=ceil(m/2), b=0^m) OR (a=0^m, |b|>=ceil(m/2)); output identifies the nonzero half.", "source": null, "locator": "Repository derivation", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "definition_status": "explicit_promise_and_derivation_checked", "proof_summary": "Each promised input has a one-bit weak certificate: any of its 1 positions identifies the half. Sampling uniformly until a 1 is found is zero-error and uses at most four queries in expectation, so R0,R,Q,RC,bs are O(1). There are no opposite-output adjacent inputs. To lower-bound exact degree, symmetrize p(a,0) and p(0,b) separately. If degree is less than floor(m/2)+1, the first univariate polynomial is identically its required constant and the second is identically the other constant, because both agree at at least floor(m/2)+1 consecutive weights. Their values at zero must agree, a contradiction. Thus deg=Omega(m), QE>=deg/2, and D>=deg. Deterministic evaluation gives O(m) upper bounds.", "parameter": "m (growing positive integer)", "growth_lower": { "D": 1, "QE": 1, "deg": 1 }, "growth_upper": { "D": 1, "QE": 1, "deg": 1, "R0": 0, "R": 0, "C": 0, "RC": 0, "bs": 0, "s": 0, "lambda": 0, "Q": 0, "adeg": 0 }, "values": { "D": "Theta(m^1)", "QE": "Theta(m^1)", "deg": "Theta(m^1)" }, "notes": "Growth exponents are bounds in the construction parameter, not claims of optimal separation exponents in all models. A zero upper exponent denotes O(1), including zero. No original-priority claim is made for repository derivations." }, { "id": "promise-nonempty-two-axes", "definition": "Input (a,b) in {0,1}^{2m}. Domain: a nonzero and b=0^m OR a=0^m and b nonzero. Output identifies the nonzero half.", "source": null, "locator": "Repository derivation", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "definition_status": "explicit_promise_and_derivation_checked", "proof_summary": "Again C=RC=fbs=bs=1 and s=lambda=0. If p approximates the side label on the promise, amplify it on its two promised ranges to error 1/12 using a constant-degree polynomial. Then q(z)=p(0,z)-p(z,0), with sign chosen for the label convention, is zero at z=0 and within 1/6 of 1 for every nonzero z. It approximates total OR, so degree(p)=Omega(sqrt(m)). A bounded-error algorithm searches for a marked bit; the lower bound follows from this degree argument. Linear exact degree follows because the restrictions p(a,0) and p(0,b) would otherwise force incompatible values at their common origin.", "parameter": "m (growing positive integer)", "growth_lower": { "D": 1, "QE": 1, "deg": 1, "Q": 0.5, "adeg": 0.5 }, "growth_upper": { "C": 0, "RC": 0, "bs": 0, "s": 0, "lambda": 0, "D": 1, "QE": 1, "deg": 1 }, "values": { "D": "Theta(m^1)", "QE": "Theta(m^1)", "deg": "Theta(m^1)" }, "notes": "Growth exponents are bounds in the construction parameter, not claims of optimal separation exponents in all models. A zero upper exponent denotes O(1), including zero. No original-priority claim is made for repository derivations." }, { "id": "promise-projective-plane", "definition": "Domain consists of zero and the incidence vectors of lines in a projective plane of order t. Output distinguishes zero from a line. Use m=t as the growth parameter.", "source": "AKPVZ18", "locator": "Theorem 17 with k=1; LP/verifier and fixed-line consequences derived here", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "definition_status": "explicit_promise_and_derivation_checked", "proof_summary": "Any two lines intersect, so bs=1. Put weight 1/(t+1) on every line to obtain fbs=Theta(t), hence RC=Theta(t). A fixed line meets all lines, so querying its t+1 points decides the problem, and yields C,D,R,R0=O(t); C,RC lower bounds give matching growth. Lines have weight t+1, so unrestricted degree is one. Promised opposite pairs are never adjacent.", "parameter": "m (growing positive integer)", "growth_lower": { "D": 1, "R0": 1, "R": 1, "C": 1, "RC": 1 }, "growth_upper": { "D": 1, "R0": 1, "R": 1, "C": 1, "RC": 1, "bs": 0, "s": 0, "lambda": 0, "deg": 0, "adeg": 0 }, "values": { "D": "Theta(m^1)", "R0": "Theta(m^1)", "R": "Theta(m^1)", "C": "Theta(m^1)", "RC": "Theta(m^1)" }, "notes": "Growth exponents are bounds in the construction parameter, not claims of optimal separation exponents in all models. A zero upper exponent denotes O(1), including zero. No original-priority claim is made for repository derivations." }, { "id": "promise-extremal-forrelation", "definition": "Joint truth tables f,g with forrelation exactly +1 or -1; output its sign. Parameter m=2^n; input length 2m.", "source": "GS26", "locator": "Theorem 4", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "definition_status": "explicit_promise_and_derivation_checked", "proof_summary": "Use GS26 Theorem 4. Exact quantum cost is constant in standard bit-oracle conventions, while R=tildeOmega(m^(1/4)). The logarithmic loss does not affect the unboundedness consequence used here.", "parameter": "m (growing positive integer)", "growth_lower": { "R": "1/4-o(1)", "R0": "1/4-o(1)", "D": "1/4-o(1)" }, "growth_upper": { "QE": 0, "Q": 0, "deg": 0, "adeg": 0 }, "values": {}, "notes": "Growth exponents are bounds in the construction parameter, not claims of optimal separation exponents in all models. A zero upper exponent denotes O(1), including zero. No original-priority claim is made for repository derivations." }, { "id": "promise-restricted-DJ", "definition": "For even m, domain {0^m} union all strings of weight m/2. Output 0 at zero and 1 on balanced strings.", "source": null, "locator": "Repository derivation", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "definition_status": "explicit_promise_and_derivation_checked", "proof_summary": "The phase-query algorithm prepares the uniform state, applies phases (-1)^x_i and tests overlap with the uniform state. Its amplitude is 1 at zero and 0 on balanced inputs, hence QE=1. A weak certificate for zero needs m/2+1 zero positions, so C,D,R0=Theta(m). Random sampling gives R,RC=O(1); at zero exactly two disjoint half-size sensitive blocks can be packed, so bs=2. The promise has no opposite-output adjacent pair for m>=4. The unrestricted polynomial 2 sum_i x_i/m is exact of degree one.", "parameter": "m (growing positive integer)", "growth_lower": { "D": 1, "R0": 1, "C": 1 }, "growth_upper": { "D": 1, "R0": 1, "C": 1, "R": 0, "RC": 0, "bs": 0, "s": 0, "lambda": 0, "Q": 0, "QE": 0, "deg": 0, "adeg": 0 }, "values": { "D": "Theta(m^1)", "R0": "Theta(m^1)", "C": "Theta(m^1)" }, "notes": "Growth exponents are bounds in the construction parameter, not claims of optimal separation exponents in all models. A zero upper exponent denotes O(1), including zero. No original-priority claim is made for repository derivations." }, { "id": "promise-unique-mark-color", "definition": "Domain consists of unit vectors in {0,1}^{2m}; output whether the unique 1 lies in the first or second half.", "source": "AKPVZ18", "locator": "Theorem 9 and its proof", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "definition_status": "explicit_promise_and_derivation_checked", "proof_summary": "A one-bit weak certificate reads the marked position, so C=RC=fbs=bs=1. AKPVZ18 Theorem 9 uses the complete bipartite relation between the two halves to prove classical adversary Omega(m), hence R=Omega(m), and R0,D are at least as large. Scanning all coordinates gives matching upper bounds. The sum of the second-half input bits is an exact unrestricted degree-one polynomial. No two promised inputs are adjacent.", "parameter": "m (growing positive integer)", "growth_lower": { "D": 1, "R0": 1, "R": 1 }, "growth_upper": { "D": 1, "R0": 1, "R": 1, "C": 0, "RC": 0, "bs": 0, "s": 0, "lambda": 0, "deg": 0, "adeg": 0 }, "values": { "D": "Theta(m^1)", "R0": "Theta(m^1)", "R": "Theta(m^1)" }, "notes": "Growth exponents are bounds in the construction parameter, not claims of optimal separation exponents in all models. A zero upper exponent denotes O(1), including zero. No original-priority claim is made for repository derivations." }, { "id": "permutation-single-entry", "name": "Single permutation entry indicator", "source": "DFLLV21", "definition": "f(pi)=1 iff pi(1)=1 on S_n.", "proof_summary": "One forward query computes f. All sensitive transpositions at any input involve position 1, so the disjoint-transposition sensitivity is 1. At pi(1)=1 exactly n-1 transpositions change the output, so edge sensitivity is n-1.", "status": "repository_example_checked" }, { "id": "relevant-complete-address-tree", "name": "Complete fresh-variable decision tree", "source": "CHS20", "definition": "f_1(x)=x; f_d=(1-z)*f_{d-1}(L)+z*f_{d-1}(R) on disjoint fresh variable sets and a new root z.", "proof_summary": "There are 2^d-1 relevant variables, degree d and decision-tree depth d. Every core measure is O(d), so no finite power of any core measure bounds relevant-variable count.", "status": "repository_witness_derivation_checked" }, { "id": "sensitivity-baseline", "name": "OpenAI baseline", "source": "OAIsens26", "proof_sources": [ "P2Msens26" ], "definition": "The base predicates are indicators of unions of separated Hamming balls. Recursive rows are indexed by a regular labelled tournament: each accepting clause requires strong predicates throughout its target row and weaker blocking predicates at labelled positions in outgoing rows. An OR of disjoint copies balances the two output sensitivities; disjoint self-composition amplifies the seed.", "proof_summary": "Random edge labels exclude excessive simultaneous near-satisfied clauses. Four coupled profiles control ordinary sensitivity and simultaneous flips of distinct nested predicates. Block witnesses at zero multiply through recursion; OR balancing gives s<=2(d+2)M^(d+1) and bs at zero >=M^2(2M^2+1)^d for M>=9^(d+1). Composition multiplies the zero-input block lower bound and at most multiplies sensitivity.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "parameter": "Self-composition depth k, after a fixed finite seed; campaign targets quantify arbitrarily large sensitivity.", "values": { "bs": ">= s^(2.000252407)" }, "status": "original_construction_and_amplification_checked", "notes": "The 22 October snapshot records 2.0635 and 2.10 completed; 2.15 is still an open target. Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas.", "claim_ids": [ "bs-s-superquadratic-baseline" ], "construction_latex": "\\begin{gathered}d=9,\\ M=9^{10},\\ A=22M^{10},\\ B=M^2(2M^2+1)^9,\\\\ s(f)\\le A,\\quad\\mathrm{bs}(f,0)\\ge B>A^2,\\\\ F_{k+1}=f\\circ F_k,\\quad s(F_k)\\le A^k,\\quad\\mathrm{bs}(F_k,0)\\ge B^k.\\end{gathered}" }, { "id": "sensitivity-depth20", "name": "Depth-20 parameter improvement", "source": "P2Mdepth2026", "proof_sources": [ "OAIsens26", "P2Msens26" ], "definition": "The base predicates are indicators of unions of separated Hamming balls. Recursive rows are indexed by a regular labelled tournament: each accepting clause requires strong predicates throughout its target row and weaker blocking predicates at labelled positions in outgoing rows. An OR of disjoint copies balances the two output sensitivities; disjoint self-composition amplifies the seed. Set d=20 and M=9^21; use A=44M^21 and B=M^2(2M^2+1)^20.", "proof_summary": "Specialize the same general seed bounds to depth twenty, certify the stated decimal below log(B)/log(A), and invoke finite-seed amplification. This is a stronger specialization, not a different base architecture.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "parameter": "Self-composition depth k, after a fixed finite seed; campaign targets quantify arbitrarily large sensitivity.", "values": { "bs": ">= s^(2.006470828 - o(1))" }, "status": "original_construction_and_amplification_checked", "notes": "The 22 October snapshot records 2.0635 and 2.10 completed; 2.15 is still an open target. Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas.", "claim_ids": [ "bs-s-superquadratic-depth20" ], "construction_latex": "d=20,\\quad M=9^{21},\\quad A=44M^{21},\\quad B=M^2(2M^2+1)^{20}" }, { "id": "sensitivity-2007", "name": "Campaign exponent 2.007", "source": "P2M200726", "proof_sources": [ "OAIsens26", "P2Msens26" ], "definition": "A completed campaign improvement to exponent 2.007, starting from the OpenAI construction as background. The public goal specifies a finite-certificate route but does not expose the actual accepted seed construction or its parameters; the baseline architecture must not be assumed to be that proof’s implementation.", "proof_summary": "The campaign records the asymptotic goal as proved. Its stated route is to certify a finite seed with s<=A, bs at a chosen input >=B, and A^2.007<=B, then use the shared normalization/composition theorem. The accepted proof body was not recovered here; this is a summary of the published certificate framework, not an assertion of its exact implementation.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "parameter": "Self-composition depth k, after a fixed finite seed; campaign targets quantify arbitrarily large sensitivity.", "values": { "bs": ">= s^(2.007 - o(1))" }, "status": "platform_attested_construction_details_partially_extracted", "notes": "The 22 October snapshot records 2.0635 and 2.10 completed; 2.15 is still an open target. Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas.", "claim_ids": [ "bs-s-superquadratic-2007" ] }, { "id": "sensitivity-20205", "name": "Sharper tournament recurrence: 2.0205", "source": "P2M2020526", "proof_sources": [ "OAIsens26", "P2Msens26" ], "definition": "The base predicates are indicators of unions of separated Hamming balls. Recursive rows are indexed by a regular labelled tournament: each accepting clause requires strong predicates throughout its target row and weaker blocking predicates at labelled positions in outgoing rows. An OR of disjoint copies balances the two output sensitivities; disjoint self-composition amplifies the seed. The mission specifies d=17, M=1475863, label alphabet r=88, and K=1439236 disjoint OR copies.", "proof_summary": "Retain four profiles, reduce the target-row count to fifteen, and split the gate estimate by whether the candidate tournament has a sink. The resulting finite certificate satisfies A^4041= s^(2.0205 - o(1))" }, "status": "platform_attested_construction_details_partially_extracted", "notes": "The 22 October snapshot records 2.0635 and 2.10 completed; 2.15 is still an open target. Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas.", "claim_ids": [ "bs-s-superquadratic-20205" ] }, { "id": "sensitivity-20391", "name": "Campaign exponent 2.0391", "source": "P2M2039126", "proof_sources": [ "OAIsens26", "P2Msens26" ], "definition": "A finite total-Boolean seed or family satisfying the completed campaign goal. The inspected public goal record does not expose the accepted seed construction or its parameters.", "proof_summary": "The source marks the goal proved and records accepted activity. It formulates the finite-certificate route s<=A, bs at a chosen input >=B, A^2.0391<=B. The accepted proof body and actual seed parameters were not inspected; no more specific construction mechanism is asserted.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "parameter": "Self-composition depth k, after a fixed finite seed; campaign targets quantify arbitrarily large sensitivity.", "values": { "bs": ">= s^(2.0391 - o(1))" }, "status": "platform_attested_construction_details_partially_extracted", "notes": "The 22 October snapshot records 2.0635 and 2.10 completed; 2.15 is still an open target. Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas.", "claim_ids": [ "bs-s-superquadratic-20391" ] }, { "id": "sensitivity-205", "name": "Campaign exponent 2.05", "source": "P2M20526", "proof_sources": [ "OAIsens26", "P2Msens26" ], "definition": "A total-Boolean construction certified by the completed reach_205 mission. The public goal text specifies the asymptotic theorem and a sufficient finite-certificate criterion, but not the accepted seed parameters.", "proof_summary": "The inspected mission marks reach_205 proved and displays accepted activity attributed to Robin Kothari. The stated certificate criterion is s<=A, bs at a chosen input >=B, and A^(41/20)<=B, followed by the shared amplification theorem. The accepted proof body was not retrieved, so its exact seed and proof route remain source-extraction gaps.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "parameter": "Self-composition depth k, after a fixed finite seed; campaign targets quantify arbitrarily large sensitivity.", "values": { "bs": ">= s^(2.05 - o(1))" }, "status": "platform_attested_construction_details_partially_extracted", "notes": "The 22 October snapshot records 2.0635 and 2.10 completed; 2.15 is still an open target. Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas.", "claim_ids": [ "bs-s-superquadratic-205" ] }, { "id": "sensitivity-20635", "name": "Campaign exponent 2.0635", "source": "P2M2063526", "proof_sources": [ "OAIsens26", "P2Msens26" ], "definition": "The public theorem identifies an analytic construction extending the attributed 2.0391 and 2.05 developments. Its exact seed and analytic argument are not exposed in the current public description.", "proof_summary": "The tracker now records the exact target 4127/2000 proved. Later Kothari/Codex sources identify accepted submission f80a3f37-72fa-4d56-a094-3e4d351764f3 as their foundation. The theorem provenance still contains stale prose saying full verification is outstanding; the dated proved badge and tracker supersede that status wording. Actual seed and proof details remain unextracted.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "parameter": "Finite seed amplification or the stated finite-family parameter m; the theorem quantifies every positive exponent loss and sensitivity threshold.", "values": { "bs": ">= s^(alpha-o(1)); alpha=2.0635" }, "status": "public_primary_construction_summary_checked_proof_download_requires_authentication", "definition_status": "public_primary_construction_overview_not_full_formal_definition", "notes": "Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas.", "claim_ids": [ "bs-s-superquadratic-20635" ] }, { "id": "sensitivity-2075", "name": "Seven-level nine-predicate certificate", "source": "P2M207526", "proof_sources": [ "OAIsens26", "P2Msens26" ], "definition": "Use nested predicates in a labelled regular-tournament recursion. Target clauses require strong predicates in their own row and weaker guard predicates on labelled outgoing coordinates; an OR of disjoint copies balances the output sensitivities. Use seven levels and nine nested predicates, retaining joint boundary information between related predicates.", "proof_summary": "Joint-change bounds and finite labeling existence control the nine sensitivity profiles; construct disjoint block witnesses and certify the finite exponent 83/40 by exact integer arithmetic. The public record identifies Full2075.lean, SHA-256 b8b3d440c3c148ee8401967fc8f2270df9c8c142dfb63f8fde28dfb88bf8c152, and accepted submission 819ad4c6-aa25-48ca-82e9-c8df39c182c0. Apply the campaign finite-certificate theorem. The actual seed integers are not exposed in the public summary.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "parameter": "Finite seed amplification or the stated finite-family parameter m; the theorem quantifies every positive exponent loss and sensitivity threshold.", "values": { "bs": ">= s^(alpha-o(1)); alpha=2.075" }, "status": "public_primary_construction_summary_checked_proof_download_requires_authentication", "definition_status": "public_primary_construction_overview_not_full_formal_definition", "notes": "Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas.", "claim_ids": [ "bs-s-superquadratic-2075" ] }, { "id": "sensitivity-2085", "name": "Eight-layer nine-predicate refinement", "source": "P2M208526", "proof_sources": [ "OAIsens26", "P2Msens26" ], "definition": "Use nested predicates in a labelled regular-tournament recursion. Target clauses require strong predicates in their own row and weaker guard predicates on labelled outgoing coordinates; an OR of disjoint copies balances the output sensitivities. Refine the accepted nine-predicate construction to eight layers.", "proof_summary": "Refine the finite sensitivity-profile certificate and assemble actual Boolean functions, then invoke finite-seed amplification. The source names Sensitivity20855.finite_seed and achieves_20855 and accepted submission c6096517-61e7-4f51-bc68-c1521d61eb65. The public target is exactly 2.085; the internal declaration name alone is not evidence for a 2.0855 bound. Seed integers remain unavailable.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "parameter": "Finite seed amplification or the stated finite-family parameter m; the theorem quantifies every positive exponent loss and sensitivity threshold.", "values": { "bs": ">= s^(alpha-o(1)); alpha=2.085" }, "status": "public_primary_construction_summary_checked_proof_download_requires_authentication", "definition_status": "public_primary_construction_overview_not_full_formal_definition", "notes": "Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas.", "claim_ids": [ "bs-s-superquadratic-2085" ] }, { "id": "sensitivity-2119", "name": "Periodic two-stage profile invariant", "source": "P2M211926", "proof_sources": [ "OAIsens26", "P2Msens26" ], "definition": "Use nested predicates in a labelled regular-tournament recursion. Target clauses require strong predicates in their own row and weaker guard predicates on labelled outgoing coordinates; an OR of disjoint copies balances the output sensitivities. Use a two-stage periodic profile invariant with the published parameters K=44759, R=24501 and C=83673433898775.", "proof_summary": "An exact two-stage invariant improves the profile recurrence in the inherited nine-predicate framework. The source reports standalone verification and accepted submission bb3251a4-f00f-4b06-962a-97be2de5db1b. The public summary does not specify each parameter’s recurrence role, so no unprinted formula is inferred from their names.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "parameter": "Finite seed amplification or the stated finite-family parameter m; the theorem quantifies every positive exponent loss and sensitivity threshold.", "values": { "bs": ">= s^(alpha-o(1)); alpha=2.119" }, "status": "public_primary_construction_summary_checked_proof_download_requires_authentication", "definition_status": "public_primary_construction_overview_not_full_formal_definition", "notes": "Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas.", "claim_ids": [ "bs-s-superquadratic-2119" ] }, { "id": "sensitivity-21", "name": "Campaign checkpoint 2.10", "source": "P2M2126", "proof_sources": [ "OAIsens26", "P2Msens26" ], "definition": "A total-Boolean construction satisfying the completed 2.10 campaign target. Its public goal still contains the original challenge wording; a later 2.127294 proposal refers to this proof informally as 2.105.", "proof_summary": "Use the current campaign completed status for the exact formal 21/10 goal. The stronger informal 2.105 label is not imported as a verified exponent: the public formal target at this theorem remains 2.10. Accepted proof details are not public without authentication.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "parameter": "Finite seed amplification or the stated finite-family parameter m; the theorem quantifies every positive exponent loss and sensitivity threshold.", "values": { "bs": ">= s^(alpha-o(1)); alpha=2.1" }, "status": "public_primary_construction_summary_checked_proof_download_requires_authentication", "definition_status": "public_primary_construction_overview_not_full_formal_definition", "notes": "Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas.", "claim_ids": [ "bs-s-superquadratic-21" ] }, { "id": "sensitivity-2127294", "name": "Twelve-predicate checkpoint 2.127294", "source": "P2M212729426", "proof_sources": [ "OAIsens26", "P2Msens26" ], "definition": "Use nested predicates in a labelled regular-tournament recursion. Target clauses require strong predicates in their own row and weaker guard predicates on labelled outgoing coordinates; an OR of disjoint copies balances the output sensitivities. The public proposal specifies twelve nested predicates and sharper joint boundary bounds, extending the Kothari 2.075 and subsequently accepted 2.10 developments.", "proof_summary": "Sharpen joint boundary estimates in the twelve-predicate recursion, then prove the campaign asymptotic goal. The tracker records completion on 20 October 2026; exact seed parameters and the accepted proof body remain unavailable in the inspected public record.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "parameter": "Finite seed amplification or the stated finite-family parameter m; the theorem quantifies every positive exponent loss and sensitivity threshold.", "values": { "bs": ">= s^(alpha-o(1)); alpha=2.127294" }, "status": "public_primary_construction_summary_checked_proof_download_requires_authentication", "definition_status": "public_primary_construction_overview_not_full_formal_definition", "notes": "Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas.", "claim_ids": [ "bs-s-superquadratic-2127294" ] }, { "id": "sensitivity-21301", "name": "Coupled target and guard profiles", "source": "P2M2130126", "proof_sources": [ "OAIsens26", "P2Msens26" ], "definition": "Use nested predicates in a labelled regular-tournament recursion. Target clauses require strong predicates in their own row and weaker guard predicates on labelled outgoing coordinates; an OR of disjoint copies balances the output sensitivities. Use fifteen nested predicates and simultaneous defect controls. The published coupled family has K=14207, R=7922, G=201838849=K^2 and C=32057676379612508934192581; for every m>=0 there is a total function with s<=C R^m and bs at zero >=G^m.", "proof_summary": "Multiple defect tolerances and coupled target/guard profiles improve a periodic recursion. Verify the shared labeling conditions and actual disjoint block assembly, then obtain the finite-family bounds and the logarithmic exponent log(G)/log(R)>2.1301. The construction must bound sensitivity at every input and exhibit nonempty pairwise disjoint sensitive blocks at zero. Fixed prefactors are absorbed into an arbitrary positive exponent loss; the known polynomial relation between block sensitivity and sensitivity ensures unbounded sensitivity.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "parameter": "Finite seed amplification or the stated finite-family parameter m; the theorem quantifies every positive exponent loss and sensitivity threshold.", "values": { "bs": ">= s^(alpha-o(1)); alpha=2.1301" }, "status": "public_primary_construction_summary_checked_proof_download_requires_authentication", "definition_status": "public_primary_construction_overview_not_full_formal_definition", "notes": "Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas.", "claim_ids": [ "bs-s-superquadratic-21301", "bs-s-superquadratic-coupled-exact" ], "construction_latex": "s(f_m)\\le 32057676379612508934192581\\cdot7922^m,\\quad\\mathrm{bs}(f_m,0)\\ge 201838849^m" }, { "id": "sensitivity-21303", "name": "Shared local-lemma labeling, four-stage cycle", "source": "P2M2130326", "proof_sources": [ "OAIsens26", "P2Msens26", "P2Mrefinedexact26", "P2Msharedfamily26", "MT10" ], "definition": "Use nested predicates in a labelled regular-tournament recursion. Target clauses require strong predicates in their own row and weaker guard predicates on labelled outgoing coordinates; an OR of disjoint copies balances the output sensitivities. Use fifteen nested predicates, K=12607, joint defect thresholds (23,33,45,57,71), gamma=254/253 and one edge labeling shared across four repeated stages. For every m>=0, s<=C R^m and bs at zero >=G^m, where R=50063687, G=12607^4=25260794820729601 and C=3418946681687790375032. The subsequent exact recipe refinement retains block growth and reduces sensitivity growth to 50063523.", "proof_summary": "Apply the finite asymmetric Lovasz local lemma to simultaneous bad-label events, proving joint independence, dependency counts and an exact integer activity certificate. A coupled four-stage profile recurrence and actual Boolean/block assembly give the finite family. Its exponent log(G)/log(R)>2.1303. The construction must bound sensitivity at every input and exhibit nonempty pairwise disjoint sensitive blocks at zero. Fixed prefactors are absorbed into an arbitrary positive exponent loss; the known polynomial relation between block sensitivity and sensitivity ensures unbounded sensitivity. The mission description preserves a historical server-timeout note; the supporting finite-family theorem is now publicly marked Proved.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "parameter": "Finite seed amplification or the stated finite-family parameter m; the theorem quantifies every positive exponent loss and sensitivity threshold.", "values": { "bs": ">= s^(alpha-o(1)); alpha=2.1303" }, "status": "public_primary_construction_summary_checked_proof_download_requires_authentication", "definition_status": "public_primary_construction_overview_not_full_formal_definition", "notes": "Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas.", "claim_ids": [ "bs-s-superquadratic-21303", "bs-s-superquadratic-shared-exact", "bs-s-superquadratic-refined-exact" ], "construction_latex": "s(f_m)\\le 3418946681687790375032\\cdot50063687^m,\\quad\\mathrm{bs}(f_m,0)\\ge 12607^{4m}" }, { "id": "sensitivity-213", "name": "Campaign checkpoint 2.13", "source": "P2M21326", "proof_sources": [ "OAIsens26", "P2Msens26" ], "definition": "A total-Boolean family satisfying the exact 2.13 campaign checkpoint. The stronger public 2.1301 construction also implies this target by choosing a smaller positive exponent loss.", "proof_summary": "The tracker records the 2.13 goal completed. Independently, AchievesExponent(2.1301) implies AchievesExponent(2.13) by replacing epsilon with epsilon+0.0001. This is a valid implication from the public stronger theorem, not a claim about the unretrieved accepted 2.13 proof implementation.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "parameter": "Finite seed amplification or the stated finite-family parameter m; the theorem quantifies every positive exponent loss and sensitivity threshold.", "values": { "bs": ">= s^(alpha-o(1)); alpha=2.13" }, "status": "public_primary_construction_summary_checked_proof_download_requires_authentication", "definition_status": "public_primary_construction_overview_not_full_formal_definition", "notes": "Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas.", "claim_ids": [ "bs-s-superquadratic-213" ] }, { "id": "sensitivity-21308255793978239", "name": "Six shared defect tolerances", "source": "P2M2130825579397823926", "proof_sources": [ "OAIsens26", "P2Msens26" ], "definition": "Use nested predicates in a labelled regular-tournament recursion. Target clauses require strong predicates in their own row and weaker guard predicates on labelled outgoing coordinates; an OR of disjoint copies balances the output sensitivities. Use one binary tournament labeling with ordinary defect cutoffs (24,31,38,44,50,55) and source cutoff 22, followed by the exact recursive Boolean profile. The exponent is log(14055^8)/log(3738277295413674), with numerator 1522813785050025472152230187890625.", "proof_summary": "Share one labeling across six ordinary defect tolerances and a source tolerance; feed the simultaneous bounds into a recursive Boolean sensitivity profile and block amplification. The public primitive theorem explicitly expands sensitivity and block sensitivity as finite maxima, proves the full asymptotic quantifiers, and strictly improves log(14055^8)/log(3738289747988852). The campaign wrapper transfers the primitive definitions to the shared ones. Accepted construction submission: a9d25c46-43bd-4fb1-826b-0ed075b8cf0a. The exact exponent is logarithmic, not the truncated tracker decimal.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "parameter": "Finite seed amplification or the stated finite-family parameter m; the theorem quantifies every positive exponent loss and sensitivity threshold.", "values": { "bs": ">= s^(alpha-o(1)); alpha=log(14055^8)/log(3738277295413674)" }, "status": "public_primary_construction_summary_checked_proof_download_requires_authentication", "definition_status": "public_primary_construction_overview_not_full_formal_definition", "notes": "Public construction summary and quantified theorem checked; accepted Lean proof body is behind authentication and has not been downloaded or replayed by this atlas.", "claim_ids": [ "bs-s-superquadratic-21308255793978239" ], "construction_latex": "\\alpha=\\frac{\\log(14055^8)}{\\log(3738277295413674)}=2.1308255793978239476\\ldots;\\quad(q_1,\\ldots,q_6)=(24,31,38,44,50,55),\\quad q_{\\mathrm{src}}=22" } ], "pair_summaries": [ { "A": "D", "B": "Q", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "lower": 4, "upper": 4, "upper_claim": "D-Q4", "lower_claim": "D-Q4-sep", "endpoint_upper": true, "lower_loss": "polylog", "status": "matched_exponents_in_recorded_sources" }, { "A": "deg", "B": "Q", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "lower": 2, "upper": 2, "upper_claim": "deg-Q2", "lower_claim": "OR-deg-Q", "endpoint_upper": true, "lower_loss": "none", "status": "matched_exponents_in_recorded_sources" }, { "A": "deg", "B": "adeg", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "lower": 2, "upper": 2, "upper_claim": "deg-adeg2", "lower_claim": "OR-deg-adeg", "endpoint_upper": true, "lower_loss": "none", "status": "matched_exponents_in_recorded_sources" }, { "A": "C", "B": "R", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "lower": 2, "upper": 2, "upper_claim": "C-R2", "lower_claim": "C-R2-sep-AIK", "endpoint_upper": true, "lower_loss": "none", "status": "matched_exponents_in_recorded_sources" }, { "A": "C", "B": "Q", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "lower": 4, "upper": 4, "upper_claim": "C-Q4", "lower_claim": "C-Q4-sep-AIK", "endpoint_upper": true, "lower_loss": "polylog", "status": "matched_exponents_in_recorded_sources" }, { "A": "Q", "B": "C", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "lower": 2, "upper": 2, "upper_claim": "Q-C2-upper", "lower_claim": "Q-C2-sep", "endpoint_upper": true, "lower_loss": "polylog", "status": "matched_exponents_in_recorded_sources" }, { "A": "Q", "B": "adeg", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "lower": 4, "upper": 4, "upper_claim": "Q-adeg4-upper", "lower_claim": "Q-adeg4-sep", "endpoint_upper": true, "lower_loss": "subpolynomial", "status": "matched_exponents_in_recorded_sources" }, { "A": "R", "B": "Q", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "lower": 4, "upper": 4, "upper_claim": "R-Q4", "lower_claim": "R-Q4-sep-2026", "endpoint_upper": true, "lower_loss": "arbitrarily_small_power_and_polylog", "status": "matched_exponents_including_repository_preprint", "note": "OAI26 primary theorem inspected; independent full-proof validation is not asserted." }, { "A": "R", "B": "Q", "scope": { "object_kind": "decision_function", "totality": "partial", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "lower": "infinity", "upper": null, "upper_claim": null, "lower_claim": "partial-R-Q", "endpoint_upper": false, "lower_loss": "polylog", "status": "no_finite_exponent_in_recorded_sources" }, { "A": "D", "B": "Q", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [ "symmetric" ], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "lower": 2, "upper": 2, "upper_claim": "sym-D-Q2", "lower_claim": "OR-D-Q", "endpoint_upper": true, "lower_loss": "none", "status": "matched_exponents_in_recorded_sources" }, { "A": "C", "B": "adeg", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "lower": 4, "upper": 4, "upper_claim": "C-adeg4-upper", "lower_claim": "C-adeg4-sep", "endpoint_upper": true, "lower_loss": "polylog_in_witness_parameter", "status": "matched_exponents_in_recorded_sources" }, { "A": "R0", "B": "Q", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "lower": 4, "upper": 4, "upper_claim": "R0-Q4-upper", "lower_claim": "R0-Q4-sep", "endpoint_upper": true, "lower_loss": "polylog_in_witness_parameter", "status": "matched_exponents_in_recorded_sources" } ], "measure_families": [ { "id": "certification", "name": "certification transform", "definition": "For a measure M defined on partial Boolean functions, M^cert(f)=max_z M(f_z^cert), where f_z^cert accepts z and rejects all inputs with opposite f-value.", "source": "KLLPV26", "locator": "Definitions 1–2" }, { "id": "CGsingle", "name": "single-bit certificate game variants", "definition": "Restrict the input pairs of a certificate game to Hamming distance one; retain the randomness/resource model.", "source": "CG22", "locator": "Section 8" }, { "id": "noisyQ", "name": "noisy-oracle quantum query models", "definition": "Separate measures for negligent, dephasing, and hybrid oracle access; these models are not interchangeable.", "source": "ARS26", "locator": "Section 1" } ], "audit_date": "2026-10-17", "matrix_baseline_witness": { "id": "parity-core", "source": "BBCMW01", "locator": "Section 6.2 (PARITY); elementary certificates and cube spectrum", "measures": [ "D", "R0", "R", "C", "RC", "bs", "s", "lambda", "QE", "deg", "Q", "adeg" ], "statement": "For n-bit parity, each of these twelve measures is Theta(n).", "proof_summary": "Every differing singleton is sensitive, so bs, certificate and randomized certificate costs are linear. The sensitivity graph is the n-cube, of norm n. Fourier orthogonality forces approximate degree n. The polynomial method and reading all bits sandwich the query costs.", "scope": { "object_kind": "decision_function", "totality": "total", "classes": [], "input_alphabet": "bits", "output_alphabet": "bits", "cost": "as_in_measure_definitions", "error": "as_in_measure_definitions", "assumptions": [] }, "status": "primary_result_plus_elementary_calculation", "note": "Establishes lower exponent 1 in every ordered core pair; it is not an upper-bound assertion." }, "priority_audits": [ { "id": "classical-pseudodeterminism-cubic", "sources": [ "CDM23", "CDM23full", "AGL26", "GGR13" ], "finding": "Use CDM23 Theorem 3.2: D=O(psR^3) for total relations. AGL26 compares its quantum result to an older quartic classical theorem; that does not supersede the cubic bound. Original GGR13 Theorem 4.1(3) was now read directly; its quartic bound has no input-length logarithm. Treat the extra logarithm in AGL26’s discussion as a weaker summary, not part of the original theorem.", "status": "original_statements_and_full_proof_route_checked" }, { "id": "partial-bounded-degree-quantum", "sources": [ "AB23", "BK24" ], "finding": "AB23 separates Q from exact degree bounded on the entire cube (Boolean degree O(log N)); BK24 separates Q even from bounded approximate degree and conical degree. Neither statement is an unrestricted promise-degree result in disguise.", "status": "original_definitions_and_theorems_checked" }, { "id": "classical-adversary-authors", "sources": [ "AKPV18", "AKPVZ18" ], "finding": "Four-author STACS publication and five-author arXiv v3 are separate records. Theorems 9, 15 and 17 inspected here are the v3 locators.", "status": "versions_separated" }, { "id": "ABK21-published-constant-repair", "sources": [ "ABK21" ], "finding": "Final CCC Lemma 27 repairs arXiv v1: q(x,i)=q_x(i). This closes the statement/proof factor discrepancy and supports CAdv<=cfbs with constant one. Lemma 26 supplies cfbs<=2 CAdv.", "status": "preprint_and_final_proof_constructions_compared" }, { "id": "CDM23-final-report-new-results", "sources": [ "CDM23", "CDM23full" ], "finding": "The final MFCS Theorems 4.5–4.6 and 5.5–5.7 contain the ApproxMaj completion lower bound, exponential canonical-size gap and AND/OR extensions absent from the inspected earlier ECCC report. Its Theorem 4.5 expressly credits the corruption argument to Swagato Sanyal.", "status": "versions_compared_and_final_proof_read" }, { "id": "CDMRS23-original-size-derandomization", "sources": [ "CDMRS23", "CDM23" ], "finding": "Replaced secondary-only use of logarithmic size derandomization by the original STOC Theorem 3.1, and original AND/OR and AND-query bounds (Theorems 4.1 and 4.5). Proposition 2.11 attributes deterministic rank-size to Ehrenfeucht–Haussler; randomized extension is syntactic.", "status": "original_theorems_and_proof_routes_checked" }, { "id": "zero-answer-versus-zero-error", "sources": [ "CDMRS23" ], "finding": "R^{0-dt} counts zero answers, while R0 is expected zero-error cost. HSC_0/FHSC_0 refer to input-zero sensitive blocks, not output-zero certificates. Registered as separate measures / qualified aliases.", "status": "original_definitions_checked" }, { "id": "CMP22-node-size-and-error", "sources": [ "CMP22", "CDM23", "CDMRS23" ], "finding": "CMP22 DTSize/RDTSize count total nodes, unlike CDM23/CDMRS23 leaf size. Its randomized-relation simulation is Q_{2/5}, not default Q_{1/3}; Boolean majority reduction is only used in a decision-function record. Coupled rank/depth tree bounds are stored without multiplying separately minimized measures.", "status": "original_definitions_and_error_composition_checked" }, { "id": "guessing-simulation-priority", "sources": [ "LL16", "BT20", "CMP22" ], "finding": "LL16 supplies the original guessing-algorithm O(sqrt(TG)) simulation and bomb-query characterization. BT20 generalizes input/output alphabets and expected random-algorithm parameters using span programs and adversary SDPs. CMP22 identifies guessing complexity with Strahler rank and optimizes edge weights to obtain the sqrt(node-size) bound.", "status": "original_results_and_proof_routes_read" }, { "id": "input-hitting-set-historical-clash", "sources": [ "KLMY21", "CDMRS23", "LM19" ], "finding": "KLMY21 bare HSC/FHSC are input-directed and match MCC/fmbs. CDMRS23 bare HSC is ordinary certificate complexity and HSC_0/FHSC_0 are the directed variants. KLMY21 also uses P^{wedge-dt}/P^{0-dt}, where later papers use D. Its Claim 4.4 reproduces the zero-depth/conjunction proof and attributes it to Loff–Mukhopadhyay 2019; the original STACS Definitions 9–10 and Lemma 11 are now inspected. Its omitted proof is supplied by KLMY21.", "status": "original_definitions_and_reproduced_proof_checked" }, { "id": "monomial-versus-Fourier-sparsity", "sources": [ "KLMY21", "KLMY21survey" ], "finding": "KLMY21 spar counts nonzero 0/1 monomials, whereas the parity-query survey uses Walsh Fourier sparsity. AND_n has monomial sparsity 1 and Fourier sparsity 2^n. A separate measure prevents transferring its hitting-set and AND-query bounds to the wrong basis.", "status": "original_basis_definitions_checked" }, { "id": "literal-sparsity-conjecture-correction", "sources": [ "KLMY21survey", "DMbasis25", "Sparse25" ], "finding": "Conjecture 3.19 says spar<=poly(approx_spar) literally. Sparse25 Remark 1.2 disproves that scale while proving a log-scale relationship with input length. Keep the negative and positive results separate.", "status": "literal_statement_and_counterexample_checked" }, { "id": "Forrelation-v1-v2-correction", "sources": [ "GS26", "CCM26" ], "finding": "The October 2026 revision restricts its near-half-exponent lower bound to nonadaptive algorithms and explicitly identifies the v1 gap in Appendix D. GS26 adaptive Conjecture 6 remains unresolved by this source.", "status": "revision_and_model_checked" }, { "id": "CDM25-HTML-PDF-disagreement", "sources": [ "CDM23", "CDM25" ], "finding": "Publisher HTML abstract names CONJ and ApproxHW, whereas the pinned 57-page journal PDF defines positive AND/OR queries and uses ApproxHamWt. Do not silently identify signed conjunction queries with positive AND/OR queries. Journal theorem numbers are separately recorded.", "status": "publisher_pdf_checked" }, { "id": "EH89-Nis91-original-access", "sources": [ "EH89", "Nis89", "Nis91" ], "finding": "Earlier publisher full-text requests failed. User subsequently supplied original EH89 journal and Nis89 conference PDFs; statements and proof routes now inspected. Nis91 journal full text remains uninspected. See user-originals-EH89-Nis89 for conventions and the one-sided constant discrepancy.", "status": "EH89_and_Nis89_access_resolved_Nis91_journal_pending" }, { "id": "user-originals-EH89-Nis89", "sources": [ "EH89", "Nis89", "Nis91" ], "finding": "User supplied original EH89 journal and Nis89 conference PDFs. Rank counts total nodes in EH89; Nis89 uses expected R_1/R_2 with success 1/2 and 3/4. Lemma 2.5 credits earlier work, perhaps Blum, via Blum–Impagliazzo. Journal Nis91 proof text remains uninspected; conference source closes the original cubic-proof access gap.", "status": "original_statements_and_proof_routes_checked" }, { "id": "Nisan-expectation-constant-repair", "sources": [ "Nis89", "Nis91" ], "finding": "Literal factor 2 disproved by one-bit expected-cost solver. The certificate-query proof gives sharp factor 4 at success 1/2; repository derivation, not a claimed published erratum. Nis91 journal text still uninspected.", "status": "literal_counterexample_and_repair_checked" }, { "id": "Wellens-inner-theorem-constant", "sources": [ "Wel22" ], "finding": "Theorem 3.13 gives 4.3935, sharper than 4.394 in introductory Theorem 1.1. Formula visually checked; source Maple LP computation not independently repeated.", "status": "source_constant_and_proof_route_checked" }, { "id": "DFLLV-2022-author-version", "sources": [ "DFLLV21", "DFLLV22author", "DFLLV26" ], "finding": "The 62-page May 2022 author manuscript confirms the main indexed general-domain/sensitivity statements and adds a Circuits section. Open questions move to Section 9. It is not identified as the 2026 journal proof; direct journal PDF attempts returned non-PDF responses.", "status": "later_full_manuscript_checked_journal_pending" }, { "id": "HKMP71-coefficient-versus-function-norm", "sources": [ "HKMP26" ], "finding": "The Lemma 71 proof normalizes by coefficient infinity norm but uses it as a function sup-norm bound. These differ by potentially the Fourier support size. The specific proof step is blocked; the final lemma is not claimed refuted.", "status": "normalization_step_unverified" }, { "id": "DFLLV26-journal-obtained", "sources": [ "DFLLV21", "DFLLV22author", "DFLLV26" ], "finding": "Normal browser access and download obtained the final 2026 journal PDF after shell/web downloads failed. It has 60 article pages plus repository cover. Main bound constants, R0 failure probability 2/3 and the oracle conventions are preserved. The journal omits the author manuscript Circuits section; open questions are in Section 8, not Section 9. Robust/q-analog questions are still proposed there.", "status": "selected_final_journal_statements_and_proof_routes_checked" }, { "id": "single-orbit-priority-BD16", "sources": [ "BD16", "DFLLV26", "HKMP26" ], "finding": "The final domain journal paper credits BD16 for uniform D/Q polynomial equivalence on single multislices with coordinate queries. The original TQC paper proves D=O(Q^18), D=O(R^9) and C<=3bs*s_2. These do not imply uniform D/degree equivalence on unbalanced domains; later full-slice refinements are separate.", "status": "original_source_and_model_checked" }, { "id": "lambda-lift-independent-and-priority-audit", "sources": [ "FLLV20notes", "DFLLV26" ], "finding": "Separate-agent source-level audit validates the product lift after correcting an overstrong simulation sentence. It strengthens the product conflict bound to M=N_total and improves the coefficient to 10368*(2k-1)^10. Original 24 June 2020 author notes Section 7 say hypergraphical sensitivity remains open, but do not specify the precise exponent/uniformity in that closing sentence. The final journal records limitations of its direct representation-theoretic method. These do not certify current openness or novelty of this coarse fixed-k query-lift bound. Limited targeted primary searches found no printed exact lift theorem; priority remains unverified.", "status": "independent_agent_mathematical_audit_passed_priority_unverified" }, { "id": "verification-constants-and-corrected-Aaronson-route", "sources": [ "Aar02", "Tal13" ], "status": "mathematical_audit_passed_exact_constant_priority_unverified", "finding": "Tal original Section 5.1.1 Claim 5.5 proves the constant-factor equivalence using independent coordinate sampling; it does not state the sharp ln(2) coefficient or the partial-promise sharpness example. Limited exact-phrase searches located no printed sharp-constant theorem. No novelty claim. Later Aaronson author manuscript explicitly repairs its old Theorem 10 proof; the four independently audited derivations do not use that proof." }, { "id": "Tal-fractional-packing-constant-priority", "sources": [ "Tal13", "CL12" ], "status": "Boolean_statement_original_checked_earlier_combinatorial_originals_pending", "finding": "Original Tal Section 5.1.2 prints fbs<=bs*(s-1+1/s). Chan–Lau later author-hosted journal-layout text credits the unweighted hypergraph theorem to Füredi (1981), and the weighted extension to Füredi–Kahn–Seymour (1993). Do not assign the packing theorem priority to Tal or Chan–Lau. Older texts remain to inspect." } ], "reductions": [ { "id": "BK24-search-to-partial-decision", "source": "BK24", "locator": "Definition 18 and Theorem 19; Lemmas 20–22", "input_scope": "total query-TFNP relation with polylogarithmic certificate verification", "output_scope": "partial Boolean decision function", "construction": "For relation S(x), auxiliary string y indexed by all certificate outputs. Promise all y_c for c in S(x) agree; the decision value is that common bit.", "bounds": [ "D(decision)=tildeTheta(D(search))", "R(decision)=tildeTheta(R(search))", "Q(decision)<=O(Q(search))", "Q(decision)>=tildeOmega(Q(search)^(1/3))", "cubeBoundedAdeg(decision)<=O(rel_adeg(search))", "conical(decision)<=O(rel_conical(search))", "CAdv(decision)>=tildeOmega(rel_CAdv(search))", "AdvPlus(decision)>=tildeOmega(rel_Adv(search))" ], "proof_summary": "Index an auxiliary bit y_c by every solution certificate c. Promise that all bits y_c for certificates valid on x have the same value, and ask for that value. Polynomial systems transfer via sum_c p_c(x)y_c. In the reverse direction, measure a decision algorithm’s query register on auxiliary input all zero and verify sampled certificates. The quantum hybrid argument gives Q(search)<=tildeO(Q(decision)^3), hence the cube-root lower-bound transfer.", "notes": "Input length may grow with the number of certificates. Theorem 19 explicitly gives the cube-root quantum lower transfer; the stronger-looking quantum statement printed in Lemma 21 is not used in this catalogue." } ], "problem_classes": [ { "id": "query_TFZPP", "name": "query analogue of total NP search in zero-error randomized polynomial time", "aliases": [ "TFZPP^{dt}" ], "source": "ZPP26", "locator": "Definition 14", "definition": "Total relations with polylogarithmic certificate verification and a polylog-depth randomized decision tree that returns only valid answers or bottom, succeeding with probability at least 2/3 on each input.", "scalar_characterization": "Rfail=polylog(n) and efficient certificate verification; equivalently expected zero-error cost polylog(n) after independent repetition.", "notes": "Class membership also includes the verification restriction; small R0 alone does not place an arbitrary relation in query-TFNP. Theorem 23 uses explicit 3-XOR instances to separate this class from PLSdt and PPPdt via SoS lower bounds. No scalar query exponent is inferred from these reduction-class separations." } ], "algorithmic_bounds": [ { "id": "CMP22-tree-rank-depth", "source": "CMP22", "locator": "Theorem 4 and Corollary 9(1)", "object_kind": "deterministic_decision_tree", "statement": "Given one correct tree T, Q(S)=O(sqrt(rank(T)*depth(T))).", "proof_summary": "Identify rank(T) with optimal guessing mistakes and apply the Lin–Lin simulation, or the optimized weighted span-program construction.", "notes": "Both factors belong to the SAME tree. This does not justify replacing them by separately minimized rank(f) and D(f). Original guessing-algorithm simulation is LL16 Theorems 5.4–5.5; BT20 Theorem 4 generalizes alphabets and uses span programs/adversary SDPs. CMP22 proves the guessing-rank identity.", "proof_sources": [ "LL16", "BT20", "CMP22" ] }, { "id": "CMP22-random-tree-rank-depth", "source": "CMP22", "locator": "Theorem 5 and Corollary 27", "object_kind": "randomized_decision_tree", "statement": "Given one randomized tree family T, Q_{2/5}(S)=O(sqrt(rrank(T)*depth(T))).", "proof_summary": "Sample a support tree, then run a high-success quantum reached-leaf algorithm. Multiply the two pointwise success guarantees.", "notes": "Depth and worst-support rank belong to the same algorithm. Preserve epsilon=2/5 for general relations." }, { "id": "BT20-expected-guessing", "source": "BT20", "locator": "Theorem 4(ii)", "object_kind": "randomized_generalized_decision_tree", "statement": "For a single randomized algorithm computing a function f:D subset [ell]^n -> [m], let T=max_x E[depth on x], G=max_x E[red mistakes on x]. Then Q(f)=O(sqrt(TG)).", "proof_summary": "Construct a feasible dual adversary solution from the colored query algorithm; average its contributions over the classical random choices. Nonbinary span programs provide the deterministic counterpart.", "notes": "The bounds are expectations for the SAME algorithm and a single-valued function. Do not replace them by independently optimal measures or import them as a general multivalued-relation amplification theorem." }, { "id": "completion-optimal-cross-evaluation", "source": "HKMP26", "locator": "Lemma 48", "statement": "If a fixed set of measures is uniformly polynomially equivalent on total functions, then their evaluations on any completions optimizing any measure in that set remain uniformly polynomially equivalent.", "proof_summary": "Compare M_i(F_k) to M_k(F_k), use optimality to compare with M_k(F_l), and apply total-function polynomial bounds again.", "notes": "The completions must optimize a member of the stated set; this is not a claim about arbitrary completions. Uniform families and constants are required.", "status": "primary_statement_and_proof_route_checked" }, { "id": "completion-speedup-equivalence", "source": "HKMP26", "locator": "Lemma 50", "statement": "For a family of partial functions and a measure M uniformly polynomially equivalent to D on total functions: D(f) and native M(f) are polynomially equivalent iff minimum-completion M(f) and native M(f) are polynomially equivalent.", "proof_summary": "Observation 49 gives minimum-completion D=D; Observation 47 relates minimum-completion M to minimum-completion D. Substitute this equivalence on either side.", "notes": "This is a conditional family-level equivalence, not an unconditional bound D<=poly(native M) for partial functions.", "status": "primary_statement_and_proof_route_checked" }, { "id": "completion-domain-indicator", "source": "HKMP26", "locator": "Lemma 51", "statement": "If p is a globally [0,1]-bounded degree-d approximant to f on its promise and membership in the promise has a q-query Boolean algorithm, the zero-outside completion admits approximate degree O(d+q), with constants depending on the fixed error margins.", "proof_summary": "Construct a globally bounded constant-error acceptance polynomial a for the domain indicator. Multiply p*a. On the promise its error is at most epsilon+epsilon_prime; outside it is at most epsilon_prime. Degree is additive.", "notes": "The source states poly(d,q); its displayed product proof gives O(d+q) for fixed margins. Global boundedness of p is essential. An off-promise unbounded approximant cannot be substituted.", "status": "primary_statement_and_proof_route_checked" }, { "id": "completion-slice-degree", "source": "HKMP26", "locator": "Theorem 56", "statement": "For a function defined on the entire k-slice, a total completion has degree O(cubeBoundedAdeg(f)^2*bs(f)^4). In particular D(f)=O(Q(f)^10).", "proof_summary": "For a sufficiently balanced slice use the existing slice decision-tree theorem. Otherwise use the low-weight-domain certificate-query construction (Lemma 55); substitute k=O(bs^2).", "notes": "Domain is the entire slice, not an arbitrary subpromise inside it. Approximation is globally bounded. The native partial-function degree convention is not interchangeable.", "status": "primary_statement_and_proof_route_checked" }, { "id": "domain-D-bs-parameters", "source": "DFLLV22author", "locator": "Theorem 3.1", "statement": "D(f)<=beta^(-2)*max(Delta,n/M)*bs(f)^4.", "proof_summary": "Generalize the certificate-query procedure: shortening blocks gives small certificates, and query conflicts bound the number of certificate stages.", "notes": "Require M>0. If C<=M, or bs<=sqrt(beta*M), or approx_degree<=(beta*M/36)^(1/4), the source strengthens the prefactor to beta^(-2)*Delta. Parameter assumptions are explicit, not a uniform bs^4 theorem.", "status": "primary_statement_and_proof_route_checked" }, { "id": "domain-certificate-sensitivity", "source": "DFLLV22author", "locator": "Lemma 3.5", "statement": "C(f)<=min(beta_c^(-1)*s(f)*bs(f),beta^(-1)*bs(f)^2).", "proof_summary": "Choose maximal disjoint opposite-output changes and shorten them using the domain sensitivity ratios; their union is a certificate.", "notes": "The parameters beta and beta_c belong to the query domain; they are not functions of f.", "status": "primary_statement_and_proof_route_checked" }, { "id": "domain-witnessing-sensitivity", "source": "DFLLV22author", "locator": "Theorem 5.3", "statement": "degree(f)<=s(f)^2/alpha for an alpha-witnessing domain.", "proof_summary": "A nonzero top-degree pseudo-character coefficient yields an induced Boolean cube of dimension at least alpha*degree; Huang gives the square-root sensitivity lower bound.", "notes": "Alpha-witnessing is an additional representation-theoretic property. It is not implied by composability alone.", "status": "primary_statement_and_proof_route_checked" }, { "id": "natural-completion-Lipschitz", "source": "HKMP26", "locator": "Lemma 60 and Definition 59", "statement": "For a fixed globally bounded approximant p of degree d, a positive margin gamma on the entire cube gives approximate degree of sign(p) at most O(d/gamma) at fixed error. In HKMP26 Lemma 60 gamma=d^(-c), giving O(d^(c+1)).", "proof_summary": "If every off-promise point is within r of a promised point and |p(x)-p(y)|<=L*r with L*r<=2/3-gamma, reverse triangle inequality gives the margin. Compose p with a univariate sign approximant.", "notes": "The completion is fixed by p, with sign(0)=+1. Require 0d have equal even- and odd-weight counts.", "proof_summary": "The parity difference is exactly the high-degree multilinear coefficient up to sign by Mobius inversion. All coefficients above d vanish iff degree<=d.", "notes": "The source writes degree=d using minimality of the starting exact degree to force the reverse inequality; without that premise the conclusion is only <=d.", "status": "primary_statement_and_proof_route_checked" }, { "id": "perturbation-Lemma71-audit", "source": "HKMP26", "locator": "Lemma 71, pp.33–34", "statement": "The source claims polynomial-cost completion from a domain-preserving Fourier coefficient perturbation with polynomial coefficient infinity norm and a full-cube inverse-polynomial margin.", "proof_summary": "It rescales p_Delta by M=||Delta||_infinity and applies a sign approximant. The displayed assertion |p_Delta(x)|/M<=2 is not justified by a bound on individual Fourier coefficients. Generally ||p_Delta||_infinity<=||p||_infinity+||Delta||_1<=1+s*||Delta||_infinity, introducing the support size s.", "notes": "Statement/proof pending. This identifies an unsupported proof step, not a counterexample to the final lemma. A sufficient repair is a polynomial bound on the FUNCTION sup norm relative to the margin; bounded coefficient infinity norm alone is not that bound. Kernel preservation holds only on the promised domain. Further issues to keep explicit: M may be zero; a polynomial degree of an arbitrary chosen approximant is not automatically polynomial in native approximate degree. The sparse-coefficient sufficient repair addresses these with explicit hypotheses.", "status": "statement_checked_proof_pending" }, { "id": "distributional-simulation-equivalence", "source": "GMNR26", "locator": "Theorem 3.3; Propositions 3.6 and 3.10", "statement": "For fixed p,q in (0,1), polynomial-overhead classical average-L1 simulation of quantum acceptance probabilities under product bias p is equivalent to simulation under the uniform q-density slice, and hence the uniform cube.", "proof_summary": "A short biased-input gadget simulates another bias; compose and coherently uncompute the gadget. Product/slice couplings, padding and quantum hybrid bounds control average error; fixing simulator randomness gives deterministic trees.", "notes": "A theorem equating conjectures, not a proof of simulation. Overheads depend on the biases; the theorem must not be specialized to arbitrarily vanishing bias without tracking those factors. Average L1 error is related to high-probability error by Markov and the bounded range.", "status": "primary_statement_and_proof_route_checked" }, { "id": "distributional-influence-equivalence", "source": "GMNR26", "locator": "Theorem 4.3; Corollary 4.8 and Proposition 4.12", "statement": "For fixed p,q in (0,1), the bounded-polynomial influential-coordinate criterion AA(mu_p) is equivalent to AA(nu_q).", "proof_summary": "Use the bias-conversion gadget and transfer variance/influence inequalities. Compare product and slice expectations for bounded low-degree polynomials and reduce each criterion to the other.", "notes": "The slice influence is distribution-specific; it is not the disjoint-transposition sensitivity in the general-domain catalogue. This equivalence does not resolve the conjecture or establish HKMP26 Conjecture 78.", "status": "primary_statement_and_proof_route_checked" }, { "id": "influence-normalization-and-slice-comparison", "source": "GMNR26", "locator": "Definition 2.5 and Lemma 2.6; AA14 before Conjecture 1.7; HKMP26 Definitions 21–22", "proof_sources": [ "AA14", "HKMP26" ], "statement": "Under uniform measure, AA14/GMNR26 flip influence equals 4 times the HKMP26 E[(D_i p)^2] normalization. On the uniform slice, swInf_(i,j)<=2*(Inf_i+Inf_j)<=4*max_i Inf_i.", "proof_summary": "The derivative normalization divides a flip difference by two. For a swap, insert the common intermediate off-slice neighbor and apply (a+b)^2<=2a^2+2b^2; swap invariance preserves the slice expectation.", "notes": "Bit flips can leave the slice. Changing off-slice values can change flip influence while preserving all slice values and swap influences. The new distributional equivalences use full-cube bounded polynomials, not arbitrary intrinsic slice extensions.", "status": "primary_statement_and_proof_route_checked" }, { "id": "perturbation-function-norm-repair", "source": "HKMP26", "locator": "Repository sufficient-condition repair of Lemma 71, using Lemma 27", "statement": "If q is a degree-d polynomial with gamma<=|q(x)|<=K on the entire cube, and sign(q) agrees with f on its promise, its sign completion has bounded approximate degree O(d*K/gamma) at fixed error.", "proof_summary": "Normalize by the FUNCTION bound K, then compose q/K with a univariate sign approximant on [-1,-gamma/K] union [gamma/K,1]. The degree multiplies by O(K/gamma), and the sign agrees with f on the promise.", "notes": "This is a repository sufficient condition, not a correction asserted by the authors. A coefficient bound gives only K<=||p||_infinity+||Delta||_1, so support dependence cannot be discarded without another argument.", "status": "repository_derivation_checked" }, { "id": "biased-natural-completion", "source": "HKMP26", "locator": "Lemma 66, pp.29–30", "statement": "Let beta=max(sqrt(p/(1-p)),sqrt((1-p)/p)), sigma=2*sqrt(p*(1-p)), and K=sum_{l=0}^{d-1} binom(n-1,l)*beta^(2l). If a degree-optimal bounded approximant has max_i sqrt(biasedCoordInfluence_i)<=sigma*eta/(2*sqrt(K)), its sign completion has approximate degree O(d^(c+1)).", "proof_summary": "In the biased orthonormal expansion, flipping coordinate i changes its basis factor by 2/sigma. Cauchy–Schwarz bounds the sum by sqrt(influence)*sqrt(K), yielding the Lipschitz bound eta; apply the global-margin sign construction.", "notes": "The influence here is the coefficient-squared normalization, not unscaled E_p[(D_i f)^2]. All n,d,p factors remain explicit. eta=(2/3-d^(-c))/r_C(Dom f), with positive numerator and nonzero radius.", "status": "primary_statement_and_proof_route_checked" }, { "id": "BD16-union-of-multislices", "source": "BD16", "locator": "Theorem 6", "statement": "For fixed alphabet size M and a coordinate-permutation-invariant promise X subseteq [M]^n, R(f)=O(Q(f)^(18*(M-1))). The function itself need not be permutation-invariant.", "proof_summary": "Control how the function behaves inside each orbit, then organize the orbits by their multiplicities. Distinguishing nearby orbit types is reduced to permutation inversion or counting and bounded using adversary arguments.", "notes": "The promise can be a union of multislices. M is a fixed alphabet parameter; this is not one universal exponent for growing M. Deterministic cost need not replace randomized cost for unions.", "status": "primary_statement_and_proof_route_checked" }, { "id": "perturbation-sparse-coefficient-repair", "source": "HKMP26", "locator": "Repository sufficient-condition derivation from Lemma 27; audit of Lemma 71", "statement": "Let p be full-cube bounded by 1, of degree at most d. Let q=p+sum_S Delta_S chi_S have degree at most d, s perturbed Fourier coordinates, coefficient bound M=max_S |Delta_S|, full-cube margin gamma>0, and correct sign on the promise. Then the sign completion has bounded approximate degree O(d*(1+s*M)/gamma) at fixed error.", "proof_summary": "Each character has absolute value one, so |q|<=1+sum_S |Delta_S|<=1+s*M. Normalize by K=1+s*M and compose with the sign approximant of Lemma 27. The margin becomes gamma/K and degrees multiply. This also handles M=0 without division by zero. If s,M,1/gamma are polynomial in d, completion approximate degree is polynomial in d.", "notes": "A sufficient repair, not an author correction. To conclude D(f)=poly(Q(f)), additionally require d polynomially bounded in the native approximate degree (hence in Q). Restrict a total completion decision tree to the promise and use the total-function D versus approximate-degree bound. No bound on s polynomial in d follows just from low degree: s can be sum_{j<=d} binomial(n,j). The kernel condition alone preserves only promised values.", "status": "repository_derivation_checked" }, { "id": "expanded-basis-majority-normalization-obstruction", "source": "HKMP26", "locator": "Repository example for the enlarged-basis discussion after Lemma 71", "proof_sources": [ "BdW02" ], "statement": "For odd n>=3, p=x_1 and q=sum_{i=1}^n x_i have degree 1. On the nonempty promise sum_{i=2}^n x_i=0, both equal f=x_1. The added coefficient perturbation has infinity norm 1 and q has full-cube margin 1, yet sign(q) has unbounded approximate degree as n grows.", "proof_summary": "All n signs sum to an odd integer, so |q|>=1 and ||q||_infinity=n. On the promise the added even-coordinate sum is zero. At a majority boundary, (n+1)/2 singleton flips change the output, so block sensitivity is at least (n+1)/2; the total-function approximate-degree lower bound gives Omega(sqrt(n)). Thus polynomial degree, coefficient norm and inverse margin alone do not force the particular sign completion to have polynomial approximate degree in d.", "notes": "This uses the enlarged degree-one basis discussed after Lemma 71. It does not obey the lemma’s original restriction to perturbing only the sparsest approximant’s existing support: p=x_1 has support one. Nor does it refute D(f)=poly(Q(f)): on this promise f is computable with one query, and sign(p)=x_1 is an easy completion. It rigorously separates the sign-completion inference from the decision-complexity conclusion and from the fixed-support setting.", "status": "repository_derivation_checked" }, { "id": "linear-minimal-support-kernel-trivial", "source": "HKMP26", "locator": "Repository degree-one special case of the original-support setting of Definitions 68–69/Lemma 71", "statement": "Let p=a_empty+sum_i a_i x_i be full-cube bounded by one and support-minimal among bounded degree-at-most-one approximants to f with the fixed promise-error convention. On its nonzero Fourier support S, the evaluation matrix A_D(p) has trivial kernel.", "proof_summary": "For linear p, ||p||_infinity=|a_empty|+sum_i |a_i|. If nonzero Delta lies in the evaluation kernel, orient it so sum_{j in S} sign(a_j) Delta_j<=0. Before the first coefficient reaches zero, the l1 norm changes linearly with that nonpositive slope. At the first zero the polynomial remains bounded, has identical promised values and degree at most one, but smaller support: contradiction. Such a zero exists because some oriented coefficient moves toward zero; otherwise every sign(a_j)Delta_j would be nonnegative and at least one positive.", "notes": "Also applies when degree one is the optimal exact degree of the starting approximation problem; dropping to degree zero would contradict that optimality. Delta=0 must be treated without dividing by its norm. If p itself has full-cube positive margin, normalize by its function bound and apply sign approximation. This resolves the kernel issue for degree one, not for degree>=2. For higher degree the Fourier l1 norm need not equal the function sup norm, so this proof does not extend automatically.", "status": "repository_derivation_checked" }, { "id": "minimal-support-l1-kernel-trivial", "source": "HKMP26", "locator": "Repository arbitrary-degree sufficient condition for Definition 68 and Lemma 71", "statement": "If a support-minimal bounded degree-at-most-d approximant p has Fourier coefficient l1 norm at most one, its domain evaluation matrix on its nonzero support has trivial kernel. In particular this holds in any degree when the nonconstant Fourier characters in its support have F2-linearly independent incidence vectors.", "proof_summary": "A nonzero domain-kernel vector can be oriented to make the directional l1 slope nonpositive. Move to the first coefficient zero; up to that point signs stay fixed and l1 does not increase. The resulting polynomial stays globally bounded by its coefficient l1 norm, preserves all promised values and has no higher degree, but smaller support. This contradicts support minimality. For F2-independent character supports, every sign pattern of character values is attained on the cube; hence the function sup norm equals the coefficient l1 norm, including an optional constant term.", "notes": "This extends the degree-one argument without assuming degree one. Support minimality is essential. It gives a sufficient original-support case, not a solution when coefficient l1 norm exceeds one and Fourier supports have dependencies. A necessary condition for a nontrivial kernel in the support-minimal bounded setting is therefore l1 norm>1. A positive full-cube margin with Delta=0 is handled by normalizing the function bound, never by dividing by zero.", "status": "repository_derivation_checked" }, { "id": "fourier-closure-support-repair", "source": "HKMP26", "locator": "Repository character-rank sufficient condition for normalization repair", "statement": "If the supported nonconstant Fourier characters span F2-rank r, then at most 2^r distinct Fourier characters occur. Thus the coefficient repair has K<=1+2^r M, and approximate completion degree O(d*(1+2^r M)/gamma). In particular r=O(log d) suffices for polynomial overhead when M and 1/gamma are polynomial in d.", "proof_summary": "All characters in the support belong to the F2-span of r incidence vectors, which has exactly 2^r elements including zero. Sum their absolute coefficient perturbations to bound the function norm, then apply the indexed function-norm sign-approximation argument.", "notes": "This is a sufficient condition. A degree bound d does not itself bound r by O(log d), and support rank is distinct from the rank of the promise evaluation matrix. No necessity or full characterization is asserted.", "status": "repository_derivation_checked" }, { "id": "kernel-amplification-duality", "source": "HKMP26", "locator": "Repository diagnostic for fixed-support Lemma 71; finite LP duality", "statement": "For supported characters S, promise evaluation matrix A and full-cube evaluation vector v_x, define kappa=max_x max_{A Delta=0, ||Delta||_infinity<=1}|v_x^T Delta|. Then kappa=max_x min_y ||v_x-A^T y||_1. If ||p||_infinity<=1 and ||Delta||_infinity<=M, the perturbed function is bounded by 1+kappa*M.", "proof_summary": "The kernel-box feasible set is symmetric, so maximizing the absolute linear functional equals maximizing the positive functional. The LP dual of max{v_x^T Delta:A Delta=0,-1<=Delta<=1} minimizes ||v_x-A^T y||_1. Finite feasible bounded LP strong duality gives equality. Scale an actual kernel perturbation by M and apply the triangle inequality; M=0 is immediate. With margin gamma, sign composition gives degree O(d*(1+kappa*M)/gamma).", "notes": "kappa is a repository diagnostic, not a source-introduced complexity measure or a claimed polynomial bound. It obeys 0<=kappa<=|S| and kappa=0 iff ker A is trivial, since full-cube characters are linearly independent. The source assumptions do not presently give a verified polynomial bound on kappa for dependent supports. This isolates the missing normalization control without asserting the unrestricted lemma false. The kappa<=|S| bound assumes the uniform Walsh characters with values +/-1; it is not asserted unchanged for an unbounded biased orthonormal basis. A large kappa alone does not refute the lemma’s existential or query-complexity conclusion.", "status": "repository_derivation_checked" }, { "id": "support-elimination-boundary-obstruction", "source": "HKMP26", "locator": "Repository necessary condition for bounded support minimality", "statement": "For a support-minimal bounded approximant, any nonzero domain-kernel direction with finite first coefficient-zero time t_zero has maximal feasible segment endpoint t_feas0 and (-1-p(x))/h(x) for h(x)<0. If t_zero were in that feasible interval, promised values and degree would be preserved while a supported coefficient vanished, contradicting minimality. Immediate infeasibility at an already saturated point gives t_feas=0.", "notes": "A necessary obstruction, not a sufficient test of global support minimality. Directions need not have a finite coefficient-zero time in both orientations; the statement concerns orientations with finite such time. Hitting a boundary is not by itself proof that the segment cannot continue (a zero directional value there is irrelevant). The feasible interval must be used. No deduction that all dependent-support kernels vanish is made.", "status": "repository_derivation_checked" }, { "id": "permutation-product-witnessing", "source": "DFLLV26", "locator": "Repository tensor-product lemma from Theorem 5.5 and Section 5.3", "statement": "For a finite product of permutation domains with forward and inverse coordinate queries, the product spatial-degree filtration is 1/2-witnessing for disjoint-transposition sensitivity.", "proof_summary": "Each coordinate indicator monomial factors across permutation components, so V_d(product)=sum_{sum d_i<=d} tensor_i V_{d_i}. The source irreducible decomposition assigns each permutation isotypic component a degree depth and supplies pure-degree pseudo-characters spanning it. Products of those pseudo-characters span the product graded components. A product of degree depths summing to d induces a cube whose dimension is the sum of the factor dimensions, at least d/2. Neighbor changes from separate factors have disjoint supports. A Boolean function with nonzero top graded component correlates with one such product; source Lemma 5.2 then yields s>=sqrt(d/2).", "notes": "Uses the source permutation isotypic grading, not an unproved plethysm assertion for hypermatchings. Cube dimensions add and the parity signs multiply. This is a repository tensor argument, not a claim that the source explicitly states the product lemma.", "status": "repository_derivation_checked" }, { "id": "quantitative-kernel-line-slab-avoidance", "source": "HKMP26", "locator": "Repository bounded-radius refinement of Lemma 70", "statement": "Let h be a nonzero polynomial from a fixed-support domain-kernel vector u. Fix gamma>0. Suppose |p(x)|>=gamma whenever h(x)=0. Choose representatives I of distinct nonconstant affine pairs (p(x),h(x)), h(x)!=0, and S=sum_{x in I} 1/|h(x)|. For any R>gamma*S there is t in [-R,R] with |p(x)+t*h(x)|>=gamma everywhere. If ||p||_infinity<=1, the resulting sign completion has approximate degree O(d*(1+R*||h||_infinity)/gamma).", "proof_summary": "For each representative, the forbidden t interval |p(x)+t*h(x)|gamma/epsilon to escape.", "proof_summary": "The slab is exactly [-gamma/epsilon,gamma/epsilon] when epsilon>0. The ratio can be arbitrarily large independently of the number or dimension of the slabs.", "notes": "An elementary obstruction to an inference, not a counterexample satisfying all original-support/minimality hypotheses of Lemma 71. No failure of the lemma’s final existential decision-complexity implication is claimed.", "status": "repository_derivation_checked" }, { "id": "verification-fractional-sharp-constants", "source": "Aar02", "locator": "Repository refinement of Definition 3/Lemma 5 sampling characterization", "statement": "Under expected-cost, rejecting-error-1/2 verification, FC/2<=RC<=(ln 2)*FC. The lower coefficient 1/2 is attained by a single bit. The upper coefficient ln 2 is asymptotically necessary over partial functions, using the promise {0^N} union {x:|x|=k}, with k->infinity and N=k^3.", "proof_summary": "For the upper bound independently query i with probability 1-2^(-w_i) from a fractional certificate. Missing disagreement probability is at most 1/2, and expected cost is at most (ln 2)FC. The lower bound uses query probabilities on the claimed input: twice those probabilities form a fractional certificate. For sharpness on the zero-or-weight-k promise, FC=N/k at zero and FC=1 at positive inputs. A verifier on zero has a random query set of size T; averaging a positive uniform k-subset gives miss probability g(T)=binom(N-T,k)/binom(N,k). Its piecewise-linear extension is convex, since the second finite differences are nonnegative, so Jensen and miss probability<=1/2 imply expected T at least the interpolated half-threshold. Uniform random query sets of the two adjacent sizes attain it. That threshold is (ln 2+o(1))*N/k when N=k^3. Positive claims cost at most 1/2 by querying one known 1 with probability 1/2.", "notes": "Constants are specific to the stored expected-query verification convention. This is a repository argument, not a printed sharp-constant theorem attributed to the source. The partial-promise sharpness does not establish sharpness of the upper coefficient on total functions or minimum completions. The linear upper inequalities themselves lift to minimum completions by choosing the right-hand minimizing completion.", "status": "repository_derivation_checked", "proof_audit": "independent_agent_mathematical_audit_passed_under_stored_conventions", "independent_audit_report": "docs/new-derivations-independent-audit.md" }, { "id": "minimal-support-bounded-interpolation-vertex", "source": "HKMP26", "locator": "Repository polyhedral analysis of fixed-support Definition 68/Lemma 71", "statement": "On the support S of a support-minimal bounded degree-at-most-d approximant p, the bounded exact-interpolation polytope P={a:A_D a=A_D a_p, |H a|<=1} is compact. Every vertex retains all S coordinates nonzero, and its active constraint matrix [A_D; H_T] has full column rank, where H_T are the saturated full-cube evaluations.", "proof_summary": "H is the full-cube evaluation matrix of distinct supported characters. Bounded function values bound every Fourier coefficient by one, so P is a compact nonempty polytope. A zero coefficient at any feasible point would produce a smaller-support bounded approximant with identical promised values and no higher degree, contradicting minimality. At a vertex, if the stacked domain and active-cube rows had a common nonzero null vector, sufficiently small positive and negative movements along it would preserve equality and all cube bounds; that would contradict extremality. Thus the stacked matrix has full column rank.", "notes": "This supplies only an augmented rank certificate, not a proof that A_D alone has full rank or that its kernel amplification is polynomially bounded. It explains the role of saturated off-promise constraints. It is not a constructed support-minimal counterexample to the unrestricted lemma. If p is degree-optimal and then support-minimal, a lower-degree feasible point would itself contradict optimality, so the same argument applies.", "status": "repository_derivation_checked", "proof_audit": "independent_agent_mathematical_audit_passed_under_stored_conventions", "independent_audit_report": "docs/new-derivations-independent-audit.md" }, { "id": "Aaronson-R0-RC-ndeg-logn-corrected-proof", "source": "Aar02", "locator": "Later 12-page author manuscript, Theorem 10 pp. 6–7, footnote 3 p. 6; old preprint pp. 5–6", "statement": "For total n-bit f, R0(f)=O(RC(f)*ndeg(f)*log(2+n)), with expected classical query costs and the source one-sided nondeterministic polynomial degree.", "proof_summary": "At each restriction choose a consistent zero input and query its randomized certificate. Every maxonomial of a nondeterministic representative contains a block whose flip changes that zero input to a one input. Coupling forces the verifier to hit that block with probability at least one half. The shrinking-monomial potential of Lemma 9 bounds the expected number of rounds by O(ndeg(f) log(2+n)); each round costs O(RC(f)) in expectation. Stop when the restricted function is constant.", "notes": "Use the later proof, not the old auxiliary assertion that resetting a sampled coordinate set toward a zero input makes every other input zero with probability at least one half. The source expressly calls a previous claim false and superfluous, without identifying that claim in the correction footnote; the reset assertion is isolated here by direct comparison. Retain n and the mixed product. Minimizing the factors over unrelated completions is invalid. Does not produce a new parameter-free completion exponent.", "status": "later_author_manuscript_statement_and_proof_route_checked", "read_snapshot": "AarAuthor" }, { "id": "Tal-fbs-bs-sensitivity-fractional-packing-constant", "source": "Tal13", "supporting_sources": [ "CL12" ], "locator": "Original ECCC TR12-163, Section 5.1.2 pp. 13–14; CL12 introduction and related work pp. 2–5", "statement": "For every nonconstant total Boolean function, fbs(f)<=bs(f)*(s(f)-1+1/s(f)). The constant function case has all three measures zero and is handled separately.", "proof_summary": "At a fixed input x retain the inclusion-minimal sensitive blocks. Their fractional matching number is fbs_x and integral matching number is bs_x. Every block B has size at most s(f), because at x^B each coordinate in B is sensitive by minimality. Pad shorter edges with fresh private vertices to uniform size s(f); neither matching optimum changes. Apply the uniform-hypergraph fractional matching bound nu*<=(s-1+1/s)nu and maximize over x. Tal prints the resulting Boolean-function inequality.", "notes": "Tal cites Chan–Lau for the packing bound; the later Chan–Lau journal-layout manuscript explicitly credits the older unweighted theorem to Zoltán Füredi, Maximum degree and fractional matchings in uniform hypergraphs, Combinatorica 1(2):155–162 (1981), and the weighted extension to Füredi, Kahn and Seymour, On the fractional matching polytope of a hypergraph, Combinatorica 13(2):167–180 (1993). Those older original texts have not been inspected here. This improves the mixed-product constant, not a recorded single-parameter exponent. Do not minimize bs and s separately across completions. Projective-plane sharpness at a single input does not establish global Boolean-function sharpness.", "status": "original_Boolean_statement_checked_underlying_original_priority_source_pending" }, { "id": "fractional-bs-quadratic-subtractive-completion", "source": "Tal13", "locator": "Repository consequence of original Section 5.1.2 and LP duality; minimum-completion lifting", "statement": "Put g(0)=0 and g(b)=b^2-b+1 for integers b>=1. For total f, fbs(f)=FC(f)<=g(bs(f)); for arbitrary partial f, completion_fbs(f)=completion_FC(f)<=g(completion_bs(f)).", "proof_summary": "For nonconstant total f, 1<=s<=b=bs and t-1+1/t is increasing on [1,infinity), so the original mixed packing bound gives fbs<=b*(b-1+1/b)=b^2-b+1. Fractional packing/covering LP duality gives FC=fbs. For completions choose a single total extension F minimizing bs. Minimum fbs is at most fbs(F), and apply the total bound to F. When b=0, this extension is constant and both minima are zero.", "notes": "This retains lower-order terms beyond the quadratic upper exponent. It does not change the count of matching completion exponents. Separate minima are never multiplied. A separate agent checked the original Tal statement as well as the deduction; this is not an independent proof audit of the earlier hypergraph theorem. No optimality or novelty claim.", "status": "repository_derivation_checked", "proof_audit": "independent_agent_original_statement_algebra_and_completion_lift_passed" } ], "literature_searches": [ { "id": "BK24-forward-targeted-2026-10-17", "date": "2026-10-17", "queries": [ "total search exact quantum 2026", "search problems approximate degree adversary 2025 2026", "Separations in query complexity for total search problems 2025 2026 exact degree" ], "primary_hits": [ "BK24", "AGL26", "ZPP26" ], "conclusion": "No resolving original source for BK24 Open Problems 1–3 was identified in these targeted searches. Their present status remains unverified; this is not evidence of openness or an exhaustive citation search." }, { "id": "conjecture-status-revisions-2026-10-17", "date": "2026-10-17", "queries": [ "exact approximate monomial sparsity conjecture restriction trees 2025 2026", "extremal Forrelation conjecture lower bound 2026 revision", "total search exact quantum adversary degree open problems 2025 2026" ], "primary_hits": [ "Sparse25", "DMbasis25", "CCM26", "CDM25" ], "conclusion": "Literal sparsity conjecture refuted; log-scale progress entered. Forrelation v2 is nonadaptive only. No resolving primary source for remaining BK24 questions, D versus bs quadratic conjecture or pure input-independent monomial AND conjecture was identified; this is not certification of current openness." }, { "id": "search-status-2026-10-17-followup", "date": "2026-10-17", "queries": [ "total search exact quantum randomized separation 2025 2026", "total search adversary approximate degree conical 2026", "Separations in query complexity for total search problems publication" ], "primary_hits": [ "BK24", "HKMP26final" ], "conclusion": "BK24 arXiv page lists only v1. MFCS 2026 cites that preprint but does not resolve its three search questions. A search hit at ECCC TR24-172 is a different paper, Quantum Communication Advantage in TFNP; it is not a BK24 revision. No resolving primary source identified; present openness remains unverified." }, { "id": "relevant-variable-status-2026-10-17", "date": "2026-10-17", "queries": [ "Wellens 4.3935 relevant variables", "Conjecture 3.17 Wellens", "relevant variables sensitivity 4^s constant bound" ], "primary_hits": [ "Wel22" ], "conclusion": "The original paper itself improves its introductory constant to 4.3935 in Theorem 3.13. No resolving primary source for Conjecture 3.17 identified; current status remains unverified." }, { "id": "broader-domain-conjectures-2026-10-17", "date": "2026-10-17", "queries": [ "symmetric group robust low degree 2025 Boolean", "perfect matching low degree robust Boolean 2026", "2603.29256 correction", "Distributional Variants Aaronson Ambainis conjecture 2026" ], "primary_hits": [ "DFLLV22author", "HKMP26", "AA14", "GMNR26" ], "conclusion": "The 2026 distributional paper proves equivalences and still states the influence conjecture, not a resolution. No primary resolution identified for HKMP26 Conjectures 72/78, robust low degree on S_n or perfect matchings, or the general q-analog framework; the records retain formulation/current-status caveats." }, { "id": "perturbation-correction-followup-2026-10-17", "date": "2026-10-17", "queries": [ "\"2603.29256\" \"perturbation\"", "\"Complexity measures on the symmetric group\" \"robust\" conjecture" ], "primary_hits": [ "DFLLV21", "DFLLV26" ], "conclusion": "No primary-source correction of HKMP26 Lemma 71 identified. This limited search does not establish absence of a correction or present openness. Restricted perturbation formulation remains unverified." }, { "id": "higher-support-normalization-followup", "date": "2026-10-17", "queries": [ "\"2603.29256\" \"Lemma 71\"", "\"From Promises to Totality\" \"Conjecture 72\"" ], "primary_hits": [], "conclusion": "No primary correction located in these narrow searches. No conclusion about absence of a correction or current openness is drawn. Source-level proof-pending status remains." }, { "id": "lambda-partite-lift-priority-search", "date": "2026-10-17", "queries": [ "hypermatchings sensitivity decision tree complexity permutation lift", "\"partite\" \"sensitivity\" \"complexity measures\"" ], "primary_hits": [ "DFLLV26" ], "conclusion": "No primary publication of this exact product-lift bound located in this limited search. It is recorded as a repository derivation without a novelty/priority claim. Source representation-theoretic limitations for direct hypermatching witnesses are not claimed resolved." }, { "id": "verification-sharp-constants-priority-check", "date": "2026-10-17", "queries": [ "\"randomized certificate\" \"ln 2\"", "\"fractional certificate\" \"ln 2\"", "\"randomized certificate\" \"fractional\" Aaronson constants" ], "primary_hits": [ "Aar02", "Tal13" ], "conclusion": "Original Tal Claim 5.5 checked: printed constant-factor equivalence, not the claimed optimal upper coefficient. Limited search does not establish priority." } ] }