{ "schema_version": "public-reference-uses-v1", "status": "RIEMANN HYPOTHESIS OPEN", "prepared_on": "2026-09-30", "public_credit": "@ykbballer91", "record_policy": "Statement-level uses are retained separately even when they cite the same work. Verification strings, booleans and scoped prose are preserved as recorded; NO is never cast to a truthy Boolean. No blanket independent-verification claim.", "coverage": { "inherited_ledgers": 6, "inherited_statement_records": 108, "dated_cyclicity_uses": 3, "statement_records": 111, "additional_external_link_index": 385, "exhaustive_bibliography_claimed": false }, "records": [ { "id": "main:CMI", "source_ledger": "archive/literature/literature/sources.json", "source_record_id": "CMI", "title": "Riemann Hypothesis: official problem page", "authors": "Clay Mathematics Institute", "year": 2026, "external_urls": [ "https://www.claymath.org/millennium/riemann-hypothesis/" ], "claim_used": "RHの公式な未解決状態と定義", "exact_locator": "Unsolved / Riemann Hypothesis", "assumptions": "非自明零点。数値確認は定義を置き換えない", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "NO", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "CMI", "title": "Riemann Hypothesis: official problem page", "authors": "Clay Mathematics Institute", "year": 2026, "url": "https://www.claymath.org/millennium/riemann-hypothesis/", "claim_used": "RHの公式な未解決状態と定義", "exact_theorem": "Unsolved / Riemann Hypothesis", "assumptions": "非自明零点。数値確認は定義を置き換えない", "depends_on_RH": "NO", "peer_reviewed": "NO", "verified_independently": "NO", "notes": "2026-09-29閲覧。研究論文ではない。" } }, { "id": "main:RVM2022", "source_ledger": "archive/literature/literature/sources.json", "source_record_id": "RVM2022", "title": "Counting zeros of the Riemann zeta function", "authors": "Elchin Hasanalizade and Quanli Shen and Peng-Jie Wong", "year": 2022, "external_urls": [ "https://www-math.nsysu.edu.tw/~pjwong/stuff/CountingRiemannZeros.pdf" ], "claim_used": "零点計数 O(T log(T+2))", "exact_locator": "Sections 1–2, N_Q(T); 最適数値定数は使用しない", "assumptions": "全非自明零点を重複度込みで数える", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "RVM2022", "title": "Counting zeros of the Riemann zeta function", "authors": "Elchin Hasanalizade and Quanli Shen and Peng-Jie Wong", "year": 2022, "url": "https://www-math.nsysu.edu.tw/~pjwong/stuff/CountingRiemannZeros.pdf", "claim_used": "零点計数 O(T log(T+2))", "exact_theorem": "Sections 1–2, N_Q(T); 最適数値定数は使用しない", "assumptions": "全非自明零点を重複度込みで数える", "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO", "notes": "J. Number Theory 235, 219–241; DOI 10.1016/j.jnt.2021.06.032。BUILDERが原文照合。" } }, { "id": "main:Weil1952", "source_ledger": "archive/literature/literature/sources.json", "source_record_id": "Weil1952", "title": "Sur les formules explicites de la theorie des nombres premiers", "authors": "Andre Weil", "year": 1952, "external_urls": [ "https://cds.cern.ch/record/471308" ], "claim_used": "Weil判定法の原典追跡", "exact_locator": "原版の定理番号未照合。現代C_c∞版はSuzuki2023で照合", "assumptions": "原典のテスト空間は逐条未確認", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "NO", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "Weil1952", "title": "Sur les formules explicites de la theorie des nombres premiers", "authors": "Andre Weil", "year": 1952, "url": "https://cds.cern.ch/record/471308", "claim_used": "Weil判定法の原典追跡", "exact_theorem": "原版の定理番号未照合。現代C_c∞版はSuzuki2023で照合", "assumptions": "原典のテスト空間は逐条未確認", "depends_on_RH": "NO", "peer_reviewed": "NO", "verified_independently": "NO", "notes": "Lund supplement pp.252–265。書誌確認のみ。定理細部の根拠に単独使用しない。" } }, { "id": "main:Suzuki2023", "source_ledger": "archive/literature/literature/sources.json", "source_record_id": "Suzuki2023", "title": "Aspects of the screw function corresponding to the Riemann zeta function", "authors": "Masatoshi Suzuki", "year": 2023, "external_urls": [ "https://arxiv.org/pdf/2206.03682v4" ], "claim_used": "Weil基準の正確なC_c∞版", "exact_locator": "Section 3.2, equations (3.3)–(3.5)", "assumptions": "全complex compactly supported smooth test functions; multiplicities", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "Suzuki2023", "title": "Aspects of the screw function corresponding to the Riemann zeta function", "authors": "Masatoshi Suzuki", "year": 2023, "url": "https://arxiv.org/pdf/2206.03682v4", "claim_used": "Weil基準の正確なC_c∞版", "exact_theorem": "Section 3.2, equations (3.3)–(3.5)", "assumptions": "全complex compactly supported smooth test functions; multiplicities", "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO", "notes": "J. Lond. Math. Soc. 108, 1448–1487。PDF版はVersion of May 31, 2023（arXivヘッダ30 May 2023）。HTMLの2026本文Dateは参照せず版固定PDFを採用。" } }, { "id": "main:Suzuki2025", "source_ledger": "archive/literature/literature/sources.json", "source_record_id": "Suzuki2025", "title": "On the Hilbert space derived from the Weil distribution", "authors": "Masatoshi Suzuki", "year": 2025, "external_urls": [ "https://doi.org/10.4153/S0008414X25101739" ], "claim_used": "定義とWeil判定法の規約照合", "exact_locator": "Section 1 (1.1),(1.2); Section 3.1 (3.3)", "assumptions": "定義は無条件。Theorem 1.1のHilbert空間同型はRH仮定", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "Suzuki2025", "title": "On the Hilbert space derived from the Weil distribution", "authors": "Masatoshi Suzuki", "year": 2025, "url": "https://doi.org/10.4153/S0008414X25101739", "claim_used": "定義とWeil判定法の規約照合", "exact_theorem": "Section 1 (1.1),(1.2); Section 3.1 (3.3)", "assumptions": "定義は無条件。Theorem 1.1のHilbert空間同型はRH仮定", "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO", "notes": "RHを仮定するTheorem 1.1を無条件依存に採用しない。" } }, { "id": "main:Li1997", "source_ledger": "archive/literature/literature/sources.json", "source_record_id": "Li1997", "title": "The positivity of a sequence of numbers and the Riemann hypothesis", "authors": "Xian-Jin Li", "year": 1997, "external_urls": [ "https://doi.org/10.1006/jnth.1997.2137" ], "claim_used": "Li criterionのルート比較", "exact_locator": "λ_n≥0 for every nとRHの同値。全文の定理番号未照合", "assumptions": "ξから定義された全係数。有限個の計算は不足", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "Li1997", "title": "The positivity of a sequence of numbers and the Riemann hypothesis", "authors": "Xian-Jin Li", "year": 1997, "url": "https://doi.org/10.1006/jnth.1997.2137", "claim_used": "Li criterionのルート比較", "exact_theorem": "λ_n≥0 for every nとRHの同値。全文の定理番号未照合", "assumptions": "ξから定義された全係数。有限個の計算は不足", "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO", "notes": "出版社abstract照合のみ。主証明依存に採用しない。" } }, { "id": "main:BaezDuarte2002", "source_ledger": "archive/literature/literature/sources.json", "source_record_id": "BaezDuarte2002", "title": "A strengthening of the Nyman-Beurling criterion for the Riemann Hypothesis", "authors": "Luis Baez-Duarte", "year": 2002, "external_urls": [ "https://arxiv.org/pdf/math/0202141v2" ], "claim_used": "離散化されたL²近似基準", "exact_locator": "Theorem 1.1: χ_(0,1)∈closure span{fractional_part(1/(ax)): a∈N} iff RH", "assumptions": "H=L²(0,∞); closure in its norm", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "PREPRINT", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "BaezDuarte2002", "title": "A strengthening of the Nyman-Beurling criterion for the Riemann Hypothesis", "authors": "Luis Baez-Duarte", "year": 2002, "url": "https://arxiv.org/pdf/math/0202141v2", "claim_used": "離散化されたL²近似基準", "exact_theorem": "Theorem 1.1: χ_(0,1)∈closure span{fractional_part(1/(ax)): a∈N} iff RH", "assumptions": "H=L²(0,∞); closure in its norm", "depends_on_RH": "NO", "peer_reviewed": "PREPRINT", "verified_independently": "NO", "notes": "原稿v2本文確認。自然Möbius部分和のa.e.収束とH収束を区別。" } }, { "id": "main:ConreyLi1998", "source_ledger": "archive/literature/literature/sources.json", "source_record_id": "ConreyLi1998", "title": "A note on some positivity conditions related to zeta- and L-functions", "authors": "J. Brian Conrey and Xian-Jin Li", "year": 1998, "external_urls": [ "https://arxiv.org/pdf/math/9812166v1" ], "claim_used": "de Branges正値条件の障害調査", "exact_locator": "本文の条件と適用例を調査用に確認。今回その定理を使用しない", "assumptions": "特定のHilbert空間の正値条件。全de Branges手法の不可能性とは異なる", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "PREPRINT", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "ConreyLi1998", "title": "A note on some positivity conditions related to zeta- and L-functions", "authors": "J. Brian Conrey and Xian-Jin Li", "year": 1998, "url": "https://arxiv.org/pdf/math/9812166v1", "claim_used": "de Branges正値条件の障害調査", "exact_theorem": "本文の条件と適用例を調査用に確認。今回その定理を使用しない", "assumptions": "特定のHilbert空間の正値条件。全de Branges手法の不可能性とは異なる", "depends_on_RH": "NO", "peer_reviewed": "PREPRINT", "verified_independently": "NO", "notes": "arXiv版を固定。出版版メタデータの再確認は保留。" } }, { "id": "main:Connes1998", "source_ledger": "archive/literature/literature/sources.json", "source_record_id": "Connes1998", "title": "Trace formula in noncommutative geometry and the zeros of the Riemann zeta function", "authors": "Alain Connes", "year": 1998, "external_urls": [ "https://arxiv.org/abs/math/9811068v1" ], "claim_used": "Hilbert–Pólya / trace formulaのルート比較", "exact_locator": "abstractのspectral interpretationとtrace formulaへのreduction", "assumptions": "全trace formulaの成立を未証明条件として残す", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "PREPRINT", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "Connes1998", "title": "Trace formula in noncommutative geometry and the zeros of the Riemann zeta function", "authors": "Alain Connes", "year": 1998, "url": "https://arxiv.org/abs/math/9811068v1", "claim_used": "Hilbert–Pólya / trace formulaのルート比較", "exact_theorem": "abstractのspectral interpretationとtrace formulaへのreduction", "assumptions": "全trace formulaの成立を未証明条件として残す", "depends_on_RH": "NO", "peer_reviewed": "PREPRINT", "verified_independently": "NO", "notes": "本文88頁の全面監査なし。推論の依存に採用しない。" } }, { "id": "main:PRZZ2018", "source_ledger": "archive/literature/literature/sources.json", "source_record_id": "PRZZ2018", "title": "More than five-twelfths of the zeros of zeta are on the critical line", "authors": "Kyle Pratt and Nicolas Robles and Alexandru Zaharescu and Dirk Zeindler", "year": 2018, "external_urls": [ "https://arxiv.org/abs/1802.10521" ], "claim_used": "mollifier経路の過去の到達点", "exact_locator": "要旨の無条件割合改善。全文定理番号未照合", "assumptions": "全零点を重複度込みで数えた漸近割合", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "PREPRINT", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "PRZZ2018", "title": "More than five-twelfths of the zeros of zeta are on the critical line", "authors": "Kyle Pratt and Nicolas Robles and Alexandru Zaharescu and Dirk Zeindler", "year": 2018, "url": "https://arxiv.org/abs/1802.10521", "claim_used": "mollifier経路の過去の到達点", "exact_theorem": "要旨の無条件割合改善。全文定理番号未照合", "assumptions": "全零点を重複度込みで数えた漸近割合", "depends_on_RH": "NO", "peer_reviewed": "PREPRINT", "verified_independently": "NO", "notes": "最新記録という主張はしない。" } }, { "id": "main:GuthMaynard2024", "source_ledger": "archive/literature/literature/sources.json", "source_record_id": "GuthMaynard2024", "title": "New large value estimates for Dirichlet polynomials", "authors": "Larry Guth and James Maynard", "year": 2024, "external_urls": [ "https://arxiv.org/abs/2405.20552" ], "claim_used": "zero-density経路の進展", "exact_locator": "要旨: N(σ,T)≤T^(30(1−σ)/13+o(1))。精密σ域は未監査", "assumptions": "主定理を使用しないため、未確認の一様域は主張しない", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "PREPRINT", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "GuthMaynard2024", "title": "New large value estimates for Dirichlet polynomials", "authors": "Larry Guth and James Maynard", "year": 2024, "url": "https://arxiv.org/abs/2405.20552", "claim_used": "zero-density経路の進展", "exact_theorem": "要旨: N(σ,T)≤T^(30(1−σ)/13+o(1))。精密σ域は未監査", "assumptions": "主定理を使用しないため、未確認の一様域は主張しない", "depends_on_RH": "NO", "peer_reviewed": "PREPRINT", "verified_independently": "NO", "notes": "原稿要旨確認。2026出版情報は今回依存に不要。零点ゼロ個は従わない。 checkpoint04: v2 Theorem1.1をjoint-symbol適用限界として照合。大値の個数上界であり、全点非負性を保証しない。詳細joint-symbol-prior-art.md。" } }, { "id": "main:Lamzouri2026", "source_ledger": "archive/literature/literature/sources.json", "source_record_id": "Lamzouri2026", "title": "A new proof that more than 2/3 of the zeros of the Riemann zeta function are simple and on the critical line", "authors": "Youness Lamzouri", "year": 2026, "external_urls": [ "https://arxiv.org/html/2609.02882v1" ], "claim_used": "核最適化問題の出所と限定的barrier", "exact_locator": "Theorem 1.1; Proposition 2.1; Lemma 3.2; Remark 3.4", "assumptions": "Theorem 1.1は無条件の主張。核側はη real even, smooth, support(-1/2,1/2), ∫η²=1", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "PREPRINT", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "Lamzouri2026", "title": "A new proof that more than 2/3 of the zeros of the Riemann zeta function are simple and on the critical line", "authors": "Youness Lamzouri", "year": 2026, "url": "https://arxiv.org/html/2609.02882v1", "claim_used": "核最適化問題の出所と限定的barrier", "exact_theorem": "Theorem 1.1; Proposition 2.1; Lemma 3.2; Remark 3.4", "assumptions": "Theorem 1.1は無条件の主張。核側はη real even, smooth, support(-1/2,1/2), ∫η²=1", "depends_on_RH": "NO", "peer_reviewed": "PREPRINT", "verified_independently": "NO", "notes": "解析数論部分全体は独立未検証。kernel最適化だけを別途初等証明。" } }, { "id": "main:Wang2026", "source_ledger": "archive/literature/literature/sources.json", "source_record_id": "Wang2026", "title": "Simple critical zeros and distinct zeros of the Riemann zeta-function in short intervals", "authors": "Biao Wang", "year": 2026, "external_urls": [ "https://arxiv.org/html/2609.07918v1" ], "claim_used": "2026-09 の新着結果を把握", "exact_locator": "Theorem 1.1, equations (1.7),(1.8)", "assumptions": "fixed 0<θ<1, H=T^θ; liminf割合", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "PREPRINT", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "Wang2026", "title": "Simple critical zeros and distinct zeros of the Riemann zeta-function in short intervals", "authors": "Biao Wang", "year": 2026, "url": "https://arxiv.org/html/2609.07918v1", "claim_used": "2026-09 の新着結果を把握", "exact_theorem": "Theorem 1.1, equations (1.7),(1.8)", "assumptions": "fixed 0<θ<1, H=T^θ; liminf割合", "depends_on_RH": "NO", "peer_reviewed": "PREPRINT", "verified_independently": "NO", "notes": "原文の主張を確認。独立検証未済で主定理の依存にはしない。" } }, { "id": "main:Deligne1974", "source_ledger": "archive/literature/literature/sources.json", "source_record_id": "Deligne1974", "title": "La conjecture de Weil I", "authors": "Pierre Deligne", "year": 1974, "external_urls": [ "https://numdam.org/articles/10.1007/BF02684373/" ], "claim_used": "有限体上のRHとの類推範囲", "exact_locator": "有限体上の幾何学的Weil予想。本文定理番号は本探索で未照合", "assumptions": "有限体上の対象。数体のζへの対応構成を含まない", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "Deligne1974", "title": "La conjecture de Weil I", "authors": "Pierre Deligne", "year": 1974, "url": "https://numdam.org/articles/10.1007/BF02684373/", "claim_used": "有限体上のRHとの類推範囲", "exact_theorem": "有限体上の幾何学的Weil予想。本文定理番号は本探索で未照合", "assumptions": "有限体上の対象。数体のζへの対応構成を含まない", "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO", "notes": "Publ. Math. IHES 43, 273–307。今回ζへの推論に使用しない。" } }, { "id": "main:Montgomery1973", "source_ledger": "archive/literature/literature/sources.json", "source_record_id": "Montgomery1973", "title": "The pair correlation of zeros of the zeta function", "authors": "H. L. Montgomery", "year": 1973, "external_urls": [ "https://www-personal.umich.edu/~hlm/paircor1.pdf" ], "claim_used": "random matrix関連と仮定検査", "exact_locator": "Section I, Theorem / opening sentence", "assumptions": "RHを論文全体で仮定", "verification_as_recorded": { "depends_on_RH": "YES", "peer_reviewed": "YES", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "Montgomery1973", "title": "The pair correlation of zeros of the zeta function", "authors": "H. L. Montgomery", "year": 1973, "url": "https://www-personal.umich.edu/~hlm/paircor1.pdf", "claim_used": "random matrix関連と仮定検査", "exact_theorem": "Section I, Theorem / opening sentence", "assumptions": "RHを論文全体で仮定", "depends_on_RH": "YES", "peer_reviewed": "YES", "verified_independently": "NO", "notes": "Proc. Sympos. Pure Math. 24,181–193。最近の無条件版と混同しない。" } }, { "id": "main:CarrilloEtAl2019", "source_ledger": "archive/literature/literature/sources.json", "source_record_id": "CarrilloEtAl2019", "title": "Nonlinear aggregation-diffusion equations: radial symmetry and long time asymptotics", "authors": "J. A. Carrillo and S. Hittmeir and B. Volzone and Y. Yao", "year": 2019, "external_urls": [ "https://doi.org/10.1007/s00222-019-00898-x" ], "claim_used": "全実線密度エネルギーの先行研究照合", "exact_locator": "Section 3, Theorems 3.7 and 3.10", "assumptions": "d=1,m=2,M=1,W=2(|x|−1)へ特殊化。有限一次moment。核条件はノートで照合", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "CarrilloEtAl2019", "title": "Nonlinear aggregation-diffusion equations: radial symmetry and long time asymptotics", "authors": "J. A. Carrillo and S. Hittmeir and B. Volzone and Y. Yao", "year": 2019, "url": "https://doi.org/10.1007/s00222-019-00898-x", "claim_used": "全実線密度エネルギーの先行研究照合", "exact_theorem": "Section 3, Theorems 3.7 and 3.10", "assumptions": "d=1,m=2,M=1,W=2(|x|−1)へ特殊化。有限一次moment。核条件はノートで照合", "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO", "notes": "汎関数は今回のE−1と一致。cosineは既存Euler–Lagrangeから導出できる。原文に同定数が印刷されるとまでは主張しない。" } }, { "id": "main:Tschukin2017", "source_ledger": "archive/literature/literature/sources.json", "source_record_id": "Tschukin2017", "title": "Concepts of modeling surface energy anisotropy in phase-field approaches", "authors": "Tschukin and Silberzahn and Selzer and Amos and Schneider and Nestler", "year": 2017, "external_urls": [ "https://doi.org/10.1186/s40517-017-0077-9" ], "claim_used": "CDFエネルギーと既知sine profileの照合", "exact_locator": "Isotropic phase-field model, equations (6) and (7)", "assumptions": "double-obstacle potential, ε=2√2/π, γ=π/(2√2), φ=Fに特殊化", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "Tschukin2017", "title": "Concepts of modeling surface energy anisotropy in phase-field approaches", "authors": "Tschukin and Silberzahn and Selzer and Amos and Schneider and Nestler", "year": 2017, "url": "https://doi.org/10.1186/s40517-017-0077-9", "claim_used": "CDFエネルギーと既知sine profileの照合", "exact_theorem": "Isotropic phase-field model, equations (6) and (7)", "assumptions": "double-obstacle potential, ε=2√2/π, γ=π/(2√2), φ=Fに特殊化", "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO", "notes": "既知のprofileとの一致。CDF平方剰余の最古の原典としては指定しない。" } }, { "id": "main:DLMF59", "source_ledger": "archive/literature/literature/sources.json", "source_record_id": "DLMF59", "title": "DLMF 5.9 Integral Representations", "authors": "NIST Digital Library of Mathematical Functions", "year": 2026, "external_urls": [ "https://dlmf.nist.gov/5.9#E16" ], "claim_used": "独立有限行列のarchimedean積分とdigammaの規約", "exact_locator": "5.9.16; 5.7.6との部分分数照合", "assumptions": "Re(z)>0でのdigamma積分。区間L>0と整数frequencyに特殊化", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "OFFICIAL_REFERENCE", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "DLMF59", "title": "DLMF 5.9 Integral Representations", "authors": "NIST Digital Library of Mathematical Functions", "year": 2026, "url": "https://dlmf.nist.gov/5.9#E16", "claim_used": "独立有限行列のarchimedean積分とdigammaの規約", "exact_theorem": "5.9.16; 5.7.6との部分分数照合", "assumptions": "Re(z)>0でのdigamma積分。区間L>0と整数frequencyに特殊化", "depends_on_RH": "NO", "peer_reviewed": "OFFICIAL_REFERENCE", "verified_independently": "NO", "notes": "groskin-computation.mdの解析導出で使用。DLMFの全章を独立再証明したという意味ではない。" } }, { "id": "main:PythonFlint09", "source_ledger": "archive/literature/literature/sources.json", "source_record_id": "PythonFlint09", "title": "acb: complex numbers, python-flint 0.9.0 documentation", "authors": "python-flint developers", "year": 2026, "external_urls": [ "https://python-flint.readthedocs.io/en/latest/acb.html#acb.integral" ], "claim_used": "Arb認証求積のanalytic flagと返却ballの仕様", "exact_locator": "acb.integral; acb.sinc; digamma/polygamma API", "assumptions": "meromorphic integrandは極を含む場合nonfinite ball。有限返却ballを区間LDLへ渡す", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "OFFICIAL_DOCUMENTATION", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "PythonFlint09", "title": "acb: complex numbers, python-flint 0.9.0 documentation", "authors": "python-flint developers", "year": 2026, "url": "https://python-flint.readthedocs.io/en/latest/acb.html#acb.integral", "claim_used": "Arb認証求積のanalytic flagと返却ballの仕様", "exact_theorem": "acb.integral; acb.sinc; digamma/polygamma API", "assumptions": "meromorphic integrandは極を含む場合nonfinite ball。有限返却ballを区間LDLへ渡す", "depends_on_RH": "NO", "peer_reviewed": "OFFICIAL_DOCUMENTATION", "verified_independently": "NO", "notes": "原著のpython-flint0.8.0と今回0.9.0を区別。小例独立再現で実使用。ライブラリ内部の形式検証は未実施。" } }, { "id": "main:DLMF1054", "source_ledger": "archive/literature/literature/sources.json", "source_record_id": "DLMF1054", "title": "DLMF 10.54 Spherical Bessel Integral Representations", "authors": "NIST Digital Library of Mathematical Functions", "year": 2026, "external_urls": [ "https://dlmf.nist.gov/10.54" ], "claim_used": "Legendre Fourier規格化と無限tailの上界", "exact_locator": "10.54.1 and 10.54.2; power series 10.53.1", "assumptions": "nは非負整数。complex引数にはPoisson表示、実引数には絶対値1のcosを使用", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "OFFICIAL_REFERENCE", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "DLMF1054", "title": "DLMF 10.54 Spherical Bessel Integral Representations", "authors": "NIST Digital Library of Mathematical Functions", "year": 2026, "url": "https://dlmf.nist.gov/10.54", "claim_used": "Legendre Fourier規格化と無限tailの上界", "exact_theorem": "10.54.1 and 10.54.2; power series 10.53.1", "assumptions": "nは非負整数。complex引数にはPoisson表示、実引数には絶対値1のcosを使用", "depends_on_RH": "NO", "peer_reviewed": "OFFICIAL_REFERENCE", "verified_independently": "NO", "notes": "固定窓の独立certificateで使用。tailのn!・double factorial・Fourier位相を別途監査。" } }, { "id": "main:CC2021", "source_ledger": "archive/literature/literature/sources.json", "source_record_id": "CC2021", "title": "Weil positivity and trace formula, the archimedean place", "authors": "Alain Connes and Caterina Consani", "year": 2021, "external_urls": [ "https://link.springer.com/article/10.1007/s00029-021-00689-4" ], "claim_used": "限定domainを維持して引用可", "exact_locator": "Theorem 1; Theorem 11", "assumptions": "g in C_c^infinity(R_+^*), support [2^(-1/2),2^(1/2)], transform vanishes at i/2 and 0 for Theorem 1", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "CC2021", "title": "Weil positivity and trace formula, the archimedean place", "authors": "Alain Connes and Caterina Consani", "year": 2021, "url": "https://link.springer.com/article/10.1007/s00029-021-00689-4", "claim_used": "限定domainを維持して引用可", "exact_theorem": "Theorem 1; Theorem 11", "assumptions": "g in C_c^infinity(R_+^*), support [2^(-1/2),2^(1/2)], transform vanishes at i/2 and 0 for Theorem 1", "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO", "notes": " 今回未実施、第三者の独立計算再現は未調査" } }, { "id": "main:CC2023", "source_ledger": "archive/literature/literature/sources.json", "source_record_id": "CC2023", "title": "Spectral triples and zeta-cycles", "authors": "Alain Connes and Caterina Consani", "year": 2023, "external_urls": [ "https://ems.press/journals/lem/articles/11033001" ], "claim_used": "localized form/coreの先行文献", "exact_locator": "Section 2; Proposition 2.3 as restated in CCM2025 Proposition 3.4", "assumptions": "abstractのみ確認、主定理に不使用", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "CC2023", "title": "Spectral triples and zeta-cycles", "authors": "Alain Connes and Caterina Consani", "year": 2023, "url": "https://ems.press/journals/lem/articles/11033001", "claim_used": "localized form/coreの先行文献", "exact_theorem": "Section 2; Proposition 2.3 as restated in CCM2025 Proposition 3.4", "assumptions": "abstractのみ確認、主定理に不使用", "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO", "notes": " 今回未実施" } }, { "id": "main:CvS2025", "source_ledger": "archive/literature/literature/sources.json", "source_record_id": "CvS2025", "title": "Quadratic Forms, Real Zeros and Echoes of the Spectral Action", "authors": "Alain Connes and Walter D. van Suijlekom", "year": 2025, "external_urls": [ "https://arxiv.org/html/2511.23257v1" ], "claim_used": "条件付き固定窓無限次元定理。全窓仮定やxi同定を結論に加えない", "exact_locator": "Theorem 1.2 (informal overview); Theorem 6.1 (precise); Proposition 5.10", "assumptions": "real distribution on [0,L], quadratic form (6) on trigonometric polynomials; lower-bounded essentially self-adjoint operator; simple isolated lowest eigenvalue with reflection-even eigenfunction", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "CvS2025", "title": "Quadratic Forms, Real Zeros and Echoes of the Spectral Action", "authors": "Alain Connes and Walter D. van Suijlekom", "year": 2025, "url": "https://arxiv.org/html/2511.23257v1", "claim_used": "条件付き固定窓無限次元定理。全窓仮定やxi同定を結論に加えない", "exact_theorem": "Theorem 1.2 (informal overview); Theorem 6.1 (precise); Proposition 5.10", "assumptions": "real distribution on [0,L], quadratic form (6) on trigonometric polynomials; lower-bounded essentially self-adjoint operator; simple isolated lowest eigenvalue with reflection-even eigenfunction", "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO", "notes": " 今回逐行再構成は未実施" } }, { "id": "main:CCM2025", "source_ledger": "archive/literature/literature/sources.json", "source_record_id": "CCM2025", "title": "Zeta Spectral Triples", "authors": "Alain Connes and Caterina Consani and Henri Moscovici", "year": 2025, "external_urls": [ "https://arxiv.org/html/2511.22755v1" ], "claim_used": "core・下有界性・条件付き有限作用素。global positivityには未使用", "exact_locator": "Propositions 3.3 and 3.4; Theorem 3.6; Theorem 5.10; Section 8", "assumptions": "QW_lambda on L2([lambda^-1,lambda],du/u), form core E=span(V_n); finite theorem uses E_N and simple-even ground-state assumptions", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "PREPRINT", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "CCM2025", "title": "Zeta Spectral Triples", "authors": "Alain Connes and Caterina Consani and Henri Moscovici", "year": 2025, "url": "https://arxiv.org/html/2511.22755v1", "claim_used": "core・下有界性・条件付き有限作用素。global positivityには未使用", "exact_theorem": "Propositions 3.3 and 3.4; Theorem 3.6; Theorem 5.10; Section 8", "assumptions": "QW_lambda on L2([lambda^-1,lambda],du/u), form core E=span(V_n); finite theorem uses E_N and simple-even ground-state assumptions", "depends_on_RH": "NO", "peer_reviewed": "PREPRINT", "verified_independently": "NO", "notes": " 今回未実施。本文の数値実験は著者によるもの" } }, { "id": "main:Suzuki2026v3", "source_ledger": "archive/literature/literature/sources.json", "source_record_id": "Suzuki2026v3", "title": "Weil's quadratic form via the screw function", "authors": "Masatoshi Suzuki", "year": 2026, "external_urls": [ "https://arxiv.org/abs/2606.09096" ], "claim_used": "局所・構成定理は検証候補。極限仮説を採用しない", "exact_locator": "Theorems 1.1, 1.3, 1.4, 1.5; Corollary 1.2; Corollary 1.6 (conditional limit)", "assumptions": "B_a=D*G_aD on H_0^1(-a,a); G_a on mean-zero L2; small-a positivity only; shifted space uses lambda1,N; dictionary real-even vectors; tail theorem C^{[-N,N]}, T>max(2pi N/log(c),7)", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "PREPRINT", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "Groskin2026v3", "title": "A finite Guinand-Weil dictionary and archimedean tail order for the truncated Weil quadratic form", "authors": "Akiva Groskin", "year": 2026, "url": "https://arxiv.org/abs/2607.02828", "claim_used": "有限辞書・archimedean tailの条件を独立監査。c13,N4のみArb認証再現。全c,N正値性ではない", "exact_theorem": "Theorem 2.5; Theorem 3.2; Corollary 3.3", "assumptions": "fixed c>1,N; dictionary real-even vectors; tail theorem C^{[-N,N]}, T>max(2pi N/log(c),7)", "depends_on_RH": "NO", "peer_reviewed": "PREPRINT", "verified_independently": "NO", "notes": " 2026-09-29限定監査: Theorem2.5実偶辞書とTheorem3.2主要tailを再導出。Lemma2.1係数2とCor3.3左端に修正注意。原著c13,N4再実行と独立Arb直接積分9次正定値認証が一致。原著401次証明書・論文全体の検証は未済。詳細proofs/audits/groskin-{dictionary,tail,computation}.md" } }, { "id": "main:Zhu2026v2", "source_ledger": "archive/literature/literature/sources.json", "source_record_id": "Zhu2026v2", "title": "Weil positivity in compact windows: a finite reduction, certified two-sided bounds, and a Landau-Widom decay law", "authors": "Xuefeng Zhu", "year": 2026, "external_urls": [ "https://arxiv.org/abs/2608.24827" ], "claim_used": "正しい有限還元の独立再導出に使用。原著L.8 head・simple-even主張は依存採用保留。", "exact_locator": "Theorem 1.2; Theorem 6.2; Corollary 6.3; Section 7 (withdrawal)", "assumptions": "positivity claimed for complex f supported in [-0.8,0.8]; corresponding autocorrelation support [-1.6,1.6]", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "PREPRINT", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "Zhu2026v2", "title": "Weil positivity in compact windows: a finite reduction, certified two-sided bounds, and a Landau-Widom decay law", "authors": "Xuefeng Zhu", "year": 2026, "url": "https://arxiv.org/abs/2608.24827", "claim_used": "正しい有限還元の独立再導出に使用。原著L.8 head・simple-even主張は依存採用保留。", "exact_theorem": "Theorem 1.2; Theorem 6.2; Corollary 6.3; Section 7 (withdrawal)", "assumptions": "positivity claimed for complex f supported in [-0.8,0.8]; corresponding autocorrelation support [-1.6,1.6]", "depends_on_RH": "NO", "peer_reviewed": "PREPRINT", "verified_independently": "NO", "notes": "Section 7 explicitly retracts an earlier L=1.19 positivity certificate claim due to wrong-direction prime-comb bound v2 PDF/TeX取得。§17のcode/matrix/factor/checksum一式の公開先を確認できず原著head未再現。一般entry boundを厳密反証、正しい係数でL.8 tail/couplingのみ独立修復。別のa=.5 fixed-window certificateは自作Arbで認証しT090に分離。 Theorem 1.3はRHを仮定するためこのNO分類から除外し、独立項目に分離。" } }, { "id": "main:KimEtAl2026v2", "source_ledger": "archive/literature/literature/sources.json", "source_record_id": "KimEtAl2026v2", "title": "A Numerical Realization of Suzuki's Weil-Quadratic-Form Operator: The Archimedean Spectral Law, its Universality, and an Operator Form of Weil's Positivity Criterion", "authors": "Taebong Kim and Youngsik Hong and Minsik Kim and Sunyoung Choi and Jaewon Jang and Junghoon Shin and Minseo Kim", "year": 2026, "external_urls": [ "https://arxiv.org/abs/2607.24830" ], "claim_used": "探索の参考のみ、主定理依存には使用しない", "exact_locator": "本文定理は未監査", "assumptions": "abstractのみ確認、主定理に不使用", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "PREPRINT", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "KimEtAl2026v2", "title": "A Numerical Realization of Suzuki's Weil-Quadratic-Form Operator: The Archimedean Spectral Law, its Universality, and an Operator Form of Weil's Positivity Criterion", "authors": "Taebong Kim and Youngsik Hong and Minsik Kim and Sunyoung Choi and Jaewon Jang and Junghoon Shin and Minseo Kim", "year": 2026, "url": "https://arxiv.org/abs/2607.24830", "claim_used": "探索の参考のみ、主定理依存には使用しない", "exact_theorem": "本文定理は未監査", "assumptions": "abstractのみ確認、主定理に不使用", "depends_on_RH": "NO", "peer_reviewed": "PREPRINT", "verified_independently": "NO", "notes": " 書誌・abstractだけ確認。本文定理監査・計算再現なし" } }, { "id": "main:Zhu2026Conditional", "source_ledger": "archive/literature/literature/sources.json", "source_record_id": "Zhu2026Conditional", "title": "Weil positivity in compact windows: conditional claim only", "authors": "Xuefeng Zhu", "year": 2026, "external_urls": [ "https://arxiv.org/html/2608.24827v2" ], "claim_used": "仮定検査のみ。依存不採用", "exact_locator": "Theorem 1.3", "assumptions": "RHを仮定", "verification_as_recorded": { "depends_on_RH": "YES", "peer_reviewed": "PREPRINT", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "Zhu2026Conditional", "title": "Weil positivity in compact windows: conditional claim only", "authors": "Xuefeng Zhu", "year": 2026, "url": "https://arxiv.org/html/2608.24827v2", "claim_used": "仮定検査のみ。依存不採用", "exact_theorem": "Theorem 1.3", "assumptions": "RHを仮定", "depends_on_RH": "YES", "peer_reviewed": "PREPRINT", "verified_independently": "NO", "notes": "同一論文の固定窓主張とは区別。" } }, { "id": "main:Suzuki2026Heuristic", "source_ledger": "archive/literature/literature/sources.json", "source_record_id": "Suzuki2026Heuristic", "title": "Weil quadratic form: heuristic discussion only", "authors": "Masatoshi Suzuki", "year": 2026, "external_urls": [ "https://arxiv.org/html/2606.09096v3" ], "claim_used": "仮定検査のみ。依存不採用", "exact_locator": "Section 7", "assumptions": "RH下のheuristic", "verification_as_recorded": { "depends_on_RH": "YES", "peer_reviewed": "PREPRINT", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "Suzuki2026Heuristic", "title": "Weil quadratic form: heuristic discussion only", "authors": "Masatoshi Suzuki", "year": 2026, "url": "https://arxiv.org/html/2606.09096v3", "claim_used": "仮定検査のみ。依存不採用", "exact_theorem": "Section 7", "assumptions": "RH下のheuristic", "depends_on_RH": "YES", "peer_reviewed": "PREPRINT", "verified_independently": "NO", "notes": "無条件の作用素構成定理とは区別。" } }, { "id": "main:CCLM2014", "source_ledger": "archive/literature/literature/sources.json", "source_record_id": "CCLM2014", "title": "Hilbert spaces and the pair correlation of zeros of the Riemann zeta-function", "authors": "Emanuel Carneiro and Vorrapan Chandee and Friedrich Littmann and Micah B. Milinovich", "year": 2014, "external_urls": [ "https://arxiv.org/pdf/1406.5462v1" ], "claim_used": "既知kernel最適化の先行結果", "exact_locator": "Section 3.5, Corollary 14", "assumptions": "Fourier support制限を持つextremal problem。zero correlation側のRH仮定と区別", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "PREPRINT", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "CCLM2014", "title": "Hilbert spaces and the pair correlation of zeros of the Riemann zeta-function", "authors": "Emanuel Carneiro and Vorrapan Chandee and Friedrich Littmann and Micah B. Milinovich", "year": 2014, "url": "https://arxiv.org/pdf/1406.5462v1", "claim_used": "既知kernel最適化の先行結果", "exact_theorem": "Section 3.5, Corollary 14", "assumptions": "Fourier support制限を持つextremal problem。zero correlation側のRH仮定と区別", "depends_on_RH": "NO", "peer_reviewed": "PREPRINT", "verified_independently": "NO", "notes": "BUILDERとDESTROYERが原文照合。最適化部分のみ関連。" } }, { "id": "main:MRT2019", "source_ledger": "archive/literature/literature/sources.json", "source_record_id": "MRT2019", "title": "Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges", "authors": "Kaisa Matomaki and Maksym Radziwill and Terence Tao", "year": 2019, "external_urls": [ "https://arxiv.org/html/1707.01315v3" ], "claim_used": "prime-power joint-symbolへの既存無条件cancellation評価の適用限界", "exact_locator": "Section 2.4, (29), Lemmas 2.6–2.7 and following application to Lambda", "assumptions": "固定saving exponentと十分大きい固定Bprime。log^Bprime(x)<=|t|<=x^Bprime。定数・閾値の依存を保持", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "MRT2019", "title": "Correlations of the von Mangoldt and higher divisor functions I. Long shift ranges", "authors": "Kaisa Matomaki and Maksym Radziwill and Terence Tao", "year": 2019, "url": "https://arxiv.org/html/1707.01315v3", "claim_used": "prime-power joint-symbolへの既存無条件cancellation評価の適用限界", "exact_theorem": "Section 2.4, (29), Lemmas 2.6–2.7 and following application to Lambda", "assumptions": "固定saving exponentと十分大きい固定Bprime。log^Bprime(x)<=|t|<=x^Bprime。定数・閾値の依存を保持", "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO", "notes": "Proc. Lond. Math. Soc.118 (2019),284–350; DOI10.1112/plms.12181。主張の一次資料照合のみ。全証明未再構成。詳細joint-symbol-prior-art.md。" } }, { "id": "weil:CC2021", "source_ledger": "archive/literature/literature/notes/weil-sources.json", "source_record_id": "CC2021", "title": "Weil positivity and trace formula, the archimedean place", "authors": [ "Alain Connes", "Caterina Consani" ], "year": 2021, "external_urls": [ "https://link.springer.com/article/10.1007/s00029-021-00689-4", "https://alainconnes.org/wp-content/uploads/Selecta.pdf", "https://doi.org/10.1007/s00029-021-00689-4" ], "claim_used": "g in C_c^infinity(R_+^*), support [2^(-1/2),2^(1/2)], transform vanishes at i/2 and 0 for Theorem 1", "exact_locator": [ "Theorem 1", "Theorem 11" ], "assumptions": "g in C_c^infinity(R_+^*), support [2^(-1/2),2^(1/2)], transform vanishes at i/2 and 0 for Theorem 1", "verification_as_recorded": { "peer_review": "査読誌掲載を出版社で確認", "independent_verification": "今回未実施、第三者の独立計算再現は未調査", "use_status": "限定domainを維持して引用可" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "CC2021", "authors": [ "Alain Connes", "Caterina Consani" ], "title": "Weil positivity and trace formula, the archimedean place", "year": 2021, "arxiv": "2006.13771", "doi": "10.1007/s00029-021-00689-4", "primary_urls": [ "https://link.springer.com/article/10.1007/s00029-021-00689-4", "https://alainconnes.org/wp-content/uploads/Selecta.pdf" ], "publication": "Selecta Mathematica 27, article 77; published 2021-07-29", "peer_review": "査読誌掲載を出版社で確認", "independent_verification": "今回未実施、第三者の独立計算再現は未調査", "locators": [ "Theorem 1", "Theorem 11" ], "domain": "g in C_c^infinity(R_+^*), support [2^(-1/2),2^(1/2)], transform vanishes at i/2 and 0 for Theorem 1", "use_status": "限定domainを維持して引用可" } }, { "id": "weil:CC2023", "source_ledger": "archive/literature/literature/notes/weil-sources.json", "source_record_id": "CC2023", "title": "Spectral triples and zeta-cycles", "authors": [ "Alain Connes", "Caterina Consani" ], "year": 2023, "external_urls": [ "https://ems.press/journals/lem/articles/11033001", "https://ems.press/content/serial-article-files/44477?nt=1", "https://doi.org/10.4171/LEM/1049" ], "claim_used": null, "exact_locator": [ "Section 2", "Proposition 2.3 as restated in CCM2025 Proposition 3.4" ], "assumptions": null, "verification_as_recorded": { "peer_review": "査読誌掲載を出版社で確認", "independent_verification": "今回未実施", "use_status": "localized form/coreの先行文献" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "CC2023", "authors": [ "Alain Connes", "Caterina Consani" ], "title": "Spectral triples and zeta-cycles", "year": 2023, "arxiv": "2106.01715", "doi": "10.4171/LEM/1049", "primary_urls": [ "https://ems.press/journals/lem/articles/11033001", "https://ems.press/content/serial-article-files/44477?nt=1" ], "publication": "Enseignement Mathematique 69 (2023), 93-148; 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lower-bounded essentially self-adjoint operator; simple isolated lowest eigenvalue with reflection-even eigenfunction", "verification_as_recorded": { "peer_review": "出版社の受理日・version of record・本文referee謝辞を確認", "independent_verification": "今回逐行再構成は未実施", "use_status": "条件付き固定窓無限次元定理。全窓仮定やxi同定を結論に加えない" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "CvS2025", "authors": [ "Alain Connes", "Walter D. van Suijlekom" ], "title": "Quadratic Forms, Real Zeros and Echoes of the Spectral Action", "year": 2025, "arxiv": "2511.23257v1", "doi": "10.1007/s00220-025-05493-1", "primary_urls": [ "https://arxiv.org/html/2511.23257v1", "https://link.springer.com/article/10.1007/s00220-025-05493-1" ], "publication": "Communications in Mathematical Physics 406, 312; accepted 2025-10-14, published 2025-11-18", "peer_review": "出版社の受理日・version of record・本文referee謝辞を確認", "independent_verification": "今回逐行再構成は未実施", "locators": [ "Theorem 1.2 (informal overview)", "Theorem 6.1 (precise)", "Proposition 5.10" ], "domain": "real distribution on [0,L], quadratic form (6) on trigonometric polynomials; 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EMS book chapter, pp.39-75, 2026, ISBN 978-3-98547-108-9", "peer_review": "書籍掲載確認。章の個別査読過程は未確認", "independent_verification": "今回未実施。本文の数値実験は著者によるもの", "locators": [ "Propositions 3.3 and 3.4", "Theorem 3.6", "Theorem 5.10", "Section 8" ], "domain": "QW_lambda on L2([lambda^-1,lambda],du/u), form core E=span(V_n); finite theorem uses E_N and simple-even ground-state assumptions", "missing": [ "全窓simple-even条件", "ground-state approximationとxi/zero limit同定" ], "use_status": "core・下有界性・条件付き有限作用素。global positivityには未使用" } }, { "id": "weil:Suzuki2026v3", "source_ledger": "archive/literature/literature/notes/weil-sources.json", "source_record_id": "Suzuki2026v3", "title": "Weil's quadratic form via the screw function", "authors": [ "Masatoshi Suzuki" ], "year": 2026, "external_urls": [ "https://arxiv.org/abs/2606.09096", "https://arxiv.org/html/2606.09096v3", "https://doi.org/10.48550/arXiv.2606.09096" ], "claim_used": "B_a=D*G_aD on H_0^1(-a,a); G_a on mean-zero L2; small-a positivity only; shifted space uses lambda1,N; dictionary real-even vectors; tail theorem C^{[-N,N]}, T>max(2pi N/log(c),7)", "exact_locator": [ "Theorem 2.5", "Theorem 3.2", "Corollary 3.3" ], "assumptions": "fixed c>1,N; dictionary real-even vectors; tail theorem C^{[-N,N]}, T>max(2pi N/log(c),7)", "verification_as_recorded": { "peer_review": "preprint、査読掲載未確認", "independent_verification": "2026-09-29限定監査: Theorem2.5実偶辞書とTheorem3.2主要tailを再導出。Lemma2.1係数2とCor3.3左端に修正注意。原著c13,N4再実行と独立Arb直接積分9次正定値認証が一致。原著401次証明書・論文全体の検証は未済。詳細proofs/audits/groskin-{dictionary,tail,computation}.md", "use_status": "有限辞書・archimedean tailの条件を独立監査。c13,N4のみArb認証再現。全c,N正値性ではない" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "Groskin2026v3", "authors": [ "Akiva Groskin" ], "title": "A finite Guinand-Weil dictionary and archimedean tail order for the truncated Weil quadratic form", "year": 2026, "arxiv": "2607.02828v3", "doi": "10.48550/arXiv.2607.02828", "version_date": "2026-08-14", "primary_urls": [ "https://arxiv.org/abs/2607.02828", "https://arxiv.org/html/2607.02828v3" ], "peer_review": "preprint、査読掲載未確認", "independent_verification": "2026-09-29限定監査: Theorem2.5実偶辞書とTheorem3.2主要tailを再導出。Lemma2.1係数2とCor3.3左端に修正注意。原著c13,N4再実行と独立Arb直接積分9次正定値認証が一致。原著401次証明書・論文全体の検証は未済。詳細proofs/audits/groskin-{dictionary,tail,computation}.md", "locators": [ "Theorem 2.5", "Theorem 3.2", "Corollary 3.3" ], "domain": "fixed c>1,N; dictionary real-even vectors; tail theorem C^{[-N,N]}, T>max(2pi N/log(c),7)", "use_status": "有限辞書・archimedean tailの条件を独立監査。c13,N4のみArb認証再現。全c,N正値性ではない" } }, { "id": "weil:Zhu2026v2", "source_ledger": "archive/literature/literature/notes/weil-sources.json", "source_record_id": "Zhu2026v2", "title": "Weil positivity in compact windows: a finite reduction, certified two-sided bounds, and a Landau-Widom decay law", "authors": [ "Xuefeng Zhu" ], "year": 2026, "external_urls": [ "https://arxiv.org/abs/2608.24827", "https://arxiv.org/html/2608.24827v2", "https://doi.org/10.48550/arXiv.2608.24827" ], "claim_used": "positivity claimed for complex f supported in [-0.8,0.8]; corresponding autocorrelation support [-1.6,1.6]", "exact_locator": [ "Theorems 1.1, 1.2, 1.3, 1.4", "Theorem 6.2", "Corollary 6.3", "Section 7" ], "assumptions": "positivity claimed for complex f supported in [-0.8,0.8]; corresponding autocorrelation support [-1.6,1.6]", "verification_as_recorded": { "peer_review": "preprint、査読掲載未確認", "independent_verification": "v2 PDF/TeX取得。§17のcode/matrix/factor/checksum一式の公開先を確認できず原著head未再現。一般entry boundを厳密反証、正しい係数でL.8 tail/couplingのみ独立修復。別のa=.5 fixed-window certificateは自作Arbで認証しT090に分離。", "use_status": "正しい有限還元の独立再導出に使用。原著L.8 head・simple-even主張は依存採用保留。" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "Zhu2026v2", "authors": [ "Xuefeng Zhu" ], "title": "Weil positivity in compact windows: a finite reduction, certified two-sided bounds, and a Landau-Widom decay law", "year": 2026, "arxiv": "2608.24827v2", "doi": "10.48550/arXiv.2608.24827", "version_date": "2026-09-02", "primary_urls": [ "https://arxiv.org/abs/2608.24827", "https://arxiv.org/html/2608.24827v2" ], "peer_review": "preprint、査読掲載未確認", "independent_verification": "v2 PDF/TeX取得。§17のcode/matrix/factor/checksum一式の公開先を確認できず原著head未再現。一般entry boundを厳密反証、正しい係数でL.8 tail/couplingのみ独立修復。別のa=.5 fixed-window certificateは自作Arbで認証しT090に分離。", "locators": [ "Theorems 1.1, 1.2, 1.3, 1.4", "Theorem 6.2", "Corollary 6.3", "Section 7" ], "domain": "positivity claimed for complex f supported in [-0.8,0.8]; corresponding autocorrelation support [-1.6,1.6]", "version_warning": "Section 7 explicitly retracts an earlier L=1.19 positivity certificate claim due to wrong-direction prime-comb bound", "use_status": "正しい有限還元の独立再導出に使用。原著L.8 head・simple-even主張は依存採用保留。" } }, { "id": "weil:KimEtAl2026v2", "source_ledger": "archive/literature/literature/notes/weil-sources.json", "source_record_id": "KimEtAl2026v2", "title": "A Numerical Realization of Suzuki's Weil-Quadratic-Form Operator: The Archimedean Spectral Law, its Universality, and an Operator Form of Weil's Positivity Criterion", "authors": [ "Taebong Kim", "Youngsik Hong", "Minsik Kim", "Sunyoung Choi", "Jaewon Jang", "Junghoon Shin", "Minseo Kim" ], "year": 2026, "external_urls": [ "https://arxiv.org/abs/2607.24830", "https://doi.org/10.48550/arXiv.2607.24830" ], "claim_used": null, "exact_locator": null, "assumptions": null, "verification_as_recorded": { "peer_review": "preprint、査読掲載未確認", "independent_verification": "書誌・abstractだけ確認。本文定理監査・計算再現なし", "use_status": "探索の参考のみ、主定理依存には使用しない" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "KimEtAl2026v2", "authors": [ "Taebong Kim", "Youngsik Hong", "Minsik Kim", "Sunyoung Choi", "Jaewon Jang", "Junghoon Shin", "Minseo Kim" ], "title": "A Numerical Realization of Suzuki's Weil-Quadratic-Form Operator: The Archimedean Spectral Law, its Universality, and an Operator Form of Weil's Positivity Criterion", "year": 2026, "arxiv": "2607.24830v2", "doi": "10.48550/arXiv.2607.24830", "version_date": "2026-07-29", "primary_urls": [ "https://arxiv.org/abs/2607.24830" ], "peer_review": "preprint、査読掲載未確認", "independent_verification": "書誌・abstractだけ確認。本文定理監査・計算再現なし", "use_status": "探索の参考のみ、主定理依存には使用しない" } }, { "id": "structural:ST-Riemann1859", "source_ledger": "archive/literature/research/structural_sources.json", "source_record_id": "ST-Riemann1859", "title": "Über die Anzahl der Primzahlen unter einer gegebenen Grösse", "authors": "Bernhard Riemann; 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Wilkins", "year": 1859, "external_urls": [ "https://www.claymath.org/wp-content/uploads/2023/04/Wilkins-translation.pdf" ], "claim_used": "center_scale: 完成関数の関数等式・中心化の根拠", "exact_locator": "本文の完成関数と関数等式の導出（番号なし）", "assumptions": "ξ の標準的完成。集合の対称性から各零点の固定性は従わない", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "HISTORICAL_PUBLICATION", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "ST-Riemann1859", "title": "Über die Anzahl der Primzahlen unter einer gegebenen Grösse", "authors": "Bernhard Riemann; English translation by David R. Wilkins", "year": 1859, "url": "https://www.claymath.org/wp-content/uploads/2023/04/Wilkins-translation.pdf", "claim_used": "center_scale: 完成関数の関数等式・中心化の根拠", "exact_theorem": "本文の完成関数と関数等式の導出（番号なし）", "assumptions": "ξ の標準的完成。集合の対称性から各零点の固定性は従わない", "depends_on_RH": "NO", "peer_reviewed": "HISTORICAL_PUBLICATION", "verified_independently": "NO", "notes": "翻訳された原論文。現代の査読手続とは区別。対称性の反例はノート内の直接計算。" } }, { "id": "structural:ST-BerryKeating1999", "source_ledger": "archive/literature/research/structural_sources.json", "source_record_id": "ST-BerryKeating1999", "title": "The Riemann Zeros and Eigenvalue Asymptotics", "authors": "M. V. Berry; J. P. Keating", "year": 1999, "external_urls": [ "https://epubs.siam.org/doi/abs/10.1137/S0036144598347497" ], "claim_used": "center_scale: 裸の尺度作用とζ零点同定を区別する先行研究", "exact_locator": "Abstract の未知の Hermitian operator と XP 提案。証明定理として使用しない", "assumptions": "数学的なスペクトル対応の提案という範囲。統計の全主張を無条件として採用しない", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "ST-BerryKeating1999", "title": "The Riemann Zeros and Eigenvalue Asymptotics", "authors": "M. V. Berry; J. P. Keating", "year": 1999, "url": "https://epubs.siam.org/doi/abs/10.1137/S0036144598347497", "claim_used": "center_scale: 裸の尺度作用とζ零点同定を区別する先行研究", "exact_theorem": "Abstract の未知の Hermitian operator と XP 提案。証明定理として使用しない", "assumptions": "数学的なスペクトル対応の提案という範囲。統計の全主張を無条件として採用しない", "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO", "notes": "SIAM Review 41, 236–266。数学的 prior art のみ。物理的定数・添付物理資料は証明入力に不使用。" } }, { "id": "structural:ST-RodgersTao2018", "source_ledger": "archive/literature/research/structural_sources.json", "source_record_id": "ST-RodgersTao2018", "title": "The De Bruijn-Newman constant is non-negative", "authors": "Brad Rodgers; Terence Tao", "year": 2018, "external_urls": [ "https://arxiv.org/pdf/1801.05914v5" ], "claim_used": "center_scale: 特定のtheta核の熱変形と一般の正偶核を区別", "exact_locator": "Introduction の H_t と Λ の定義; Theorem 1: Λ≥0; RH iff Λ≤0", "assumptions": "論文で定義された特定の Φ と H_t。任意の非負偶核ではない", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "ST-RodgersTao2018", "title": "The De Bruijn-Newman constant is non-negative", "authors": "Brad Rodgers; Terence Tao", "year": 2018, "url": "https://arxiv.org/pdf/1801.05914v5", "claim_used": "center_scale: 特定のtheta核の熱変形と一般の正偶核を区別", "exact_theorem": "Introduction の H_t と Λ の定義; Theorem 1: Λ≥0; RH iff Λ≤0", "assumptions": "論文で定義された特定の Φ と H_t。任意の非負偶核ではない", "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO", "notes": "v5=2021-07-03、出版版に§8修正を加えた版とarXivが明記。反証法内での仮定と定理の無条件性を区別。全文再検証なし。" } }, { "id": "structural:ST-TwamleyMilburn2006", "source_ledger": "archive/literature/research/structural_sources.json", "source_record_id": "ST-TwamleyMilburn2006", "title": "The Quantum Mellin transform", "authors": "J. Twamley; G. J. Milburn", "year": 2006, "external_urls": [ "https://arxiv.org/pdf/quant-ph/0702107v1" ], "claim_used": "center_scale: log座標・半密度の数学的変換式の先行例", "exact_locator": "Equations (16), (25), (33)（ノート作成担当者の原文照合位置）", "assumptions": "半直線の L²(dx) と指定の log 座標。自己共役domainはノート内で別途設定", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "ST-TwamleyMilburn2006", "title": "The Quantum Mellin transform", "authors": "J. Twamley; G. J. Milburn", "year": 2006, "url": "https://arxiv.org/pdf/quant-ph/0702107v1", "claim_used": "center_scale: log座標・半密度の数学的変換式の先行例", "exact_theorem": "Equations (16), (25), (33)（ノート作成担当者の原文照合位置）", "assumptions": "半直線の L²(dx) と指定の log 座標。自己共役domainはノート内で別途設定", "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO", "notes": "New J. Phys. 8, 328 (2006), DOI10.1088/1367-2630/8/12/328。arXiv投稿2007。数学的変換式の prior art のみ、実験・物理定数は不使用。" } }, { "id": "structural:ST-RVM2022", "source_ledger": "archive/literature/research/structural_sources.json", "source_record_id": "ST-RVM2022", "title": "Counting zeros of the Riemann zeta function", "authors": "Elchin Hasanalizade; Quanli Shen; Peng-Jie Wong", "year": 2022, "external_urls": [ "https://arxiv.org/abs/2107.06506" ], "claim_used": "center_scale: ζ零点計数と固定区間一次微分作用素の線形成長の差", "exact_locator": "Sections 1–2 の N(T) と Riemann–von Mangoldt 主項; 最適数値定数は使用しない", "assumptions": "全非自明零点を重複度込みで数える。臨界線上零点だけの計数ではない", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "ST-RVM2022", "title": "Counting zeros of the Riemann zeta function", "authors": "Elchin Hasanalizade; Quanli Shen; Peng-Jie Wong", "year": 2022, "url": "https://arxiv.org/abs/2107.06506", "claim_used": "center_scale: ζ零点計数と固定区間一次微分作用素の線形成長の差", "exact_theorem": "Sections 1–2 の N(T) と Riemann–von Mangoldt 主項; 最適数値定数は使用しない", "assumptions": "全非自明零点を重複度込みで数える。臨界線上零点だけの計数ではない", "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO", "notes": "J. Number Theory 235,219–241, DOI10.1016/j.jnt.2021.06.032。主台帳RVM2022と重複。" } }, { "id": "structural:ST-CC2021", "source_ledger": "archive/literature/research/structural_sources.json", "source_record_id": "ST-CC2021", "title": "Weil positivity and trace formula, the archimedean place", "authors": "Alain Connes; Caterina Consani", "year": 2021, "external_urls": [ "https://alainconnes.org/wp-content/uploads/Selecta.pdf" ], "claim_used": "center_scale / variational_closure: 尺度作用・半密度・Weil規約と局所結果の境界", "exact_locator": "Equations (5),(17), Appendix A, Appendix B; 限定domainのTheorem 1", "assumptions": "全Weil明示公式と局所圧縮の補正を保持。Theorem1のsupport等の前件を外して全窓へ拡張しない", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "ST-CC2021", "title": "Weil positivity and trace formula, the archimedean place", "authors": "Alain Connes; Caterina Consani", "year": 2021, "url": "https://alainconnes.org/wp-content/uploads/Selecta.pdf", "claim_used": "center_scale / variational_closure: 尺度作用・半密度・Weil規約と局所結果の境界", "exact_theorem": "Equations (5),(17), Appendix A, Appendix B; 限定domainのTheorem 1", "assumptions": "全Weil明示公式と局所圧縮の補正を保持。Theorem1のsupport等の前件を外して全窓へ拡張しない", "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO", "notes": "Selecta Math.27 (2021), DOI10.1007/s00029-021-00689-4。全素数・全supportの正値性を導く定理としては使用しない。" } }, { "id": "structural:ST-CC2023", "source_ledger": "archive/literature/research/structural_sources.json", "source_record_id": "ST-CC2023", "title": "Spectral triples and ζ-cycles", "authors": "Alain Connes; Caterina Consani", "year": 2023, "external_urls": [ "https://ems.press/content/serial-article-files/44477?nt=1" ], "claim_used": "prime_jump_obstruction / accumulation_spectrum: 素数平行移動の明示公式、局所形式coreの先行例", "exact_locator": "§2.1 (2.6)–(2.12), Proposition2.1; Proposition2.3（CCM2025 Prop3.4にも再掲）", "assumptions": "固定乗法窓、Haar測度、supportにより有限となる素数冪和。全実線作用素和の収束ではない", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "ST-CC2023", "title": "Spectral triples and ζ-cycles", "authors": "Alain Connes; Caterina Consani", "year": 2023, "url": "https://ems.press/content/serial-article-files/44477?nt=1", "claim_used": "prime_jump_obstruction / accumulation_spectrum: 素数平行移動の明示公式、局所形式coreの先行例", "exact_theorem": "§2.1 (2.6)–(2.12), Proposition2.1; Proposition2.3（CCM2025 Prop3.4にも再掲）", "assumptions": "固定乗法窓、Haar測度、supportにより有限となる素数冪和。全実線作用素和の収束ではない", "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO", "notes": "L’Enseignement Mathématique69,93–148, DOI10.4171/LEM/1049。prime jump の発散障害はノート内で独立導出。" } }, { "id": "structural:ST-Connes1998Local", "source_ledger": "archive/literature/research/structural_sources.json", "source_record_id": "ST-Connes1998Local", "title": "Trace formula in noncommutative geometry and the zeros of the Riemann zeta function", "authors": "Alain Connes", "year": 1998, "external_urls": [ "https://arxiv.org/pdf/math/9811068v1" ], "claim_used": "center_scale / accumulation_spectrum: critical zerosのスペクトル実現と半局所trace", "exact_locator": "III Theorem1 / Corollary2; VII Theorem4", "assumptions": "IIIはδ>1の重み付き商空間で既に臨界線上の零点のみ、重複度制限あり。VIIは無限素点を含む有限Sとcompact support Schwartz h", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "ST-Connes1998Local", "title": "Trace formula in noncommutative geometry and the zeros of the Riemann zeta function", "authors": "Alain Connes", "year": 1998, "url": "https://arxiv.org/pdf/math/9811068v1", "claim_used": "center_scale / accumulation_spectrum: critical zerosのスペクトル実現と半局所trace", "exact_theorem": "III Theorem1 / Corollary2; VII Theorem4", "assumptions": "IIIはδ>1の重み付き商空間で既に臨界線上の零点のみ、重複度制限あり。VIIは無限素点を含む有限Sとcompact support Schwartz h", "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO", "notes": "Selecta Math.5(1999),29–106, DOI10.1007/s000290050042。arXiv v1の番号。全零点捕捉・全表現unitaryとは主張しない。" } }, { "id": "structural:ST-Connes1998GlobalCriterion", "source_ledger": "archive/literature/research/structural_sources.json", "source_record_id": "ST-Connes1998GlobalCriterion", "title": "Trace formula in noncommutative geometry and the zeros of the Riemann zeta function — global criterion", "authors": "Alain Connes", "year": 1998, "external_urls": [ "https://arxiv.org/pdf/math/9811068v1" ], "claim_used": "accumulation_spectrum: 大域traceの未証明箇所とRH同値の区別", "exact_locator": "VIII Theorem5; 数体対応の説明 pp.45–47", "assumptions": "Theorem5の明記された体は正標数。全Hecke L関数のRHと指定の大域trace公式の同値。数体では異なる近似射影の説明が必要", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "ST-Connes1998GlobalCriterion", "title": "Trace formula in noncommutative geometry and the zeros of the Riemann zeta function — global criterion", "authors": "Alain Connes", "year": 1998, "url": "https://arxiv.org/pdf/math/9811068v1", "claim_used": "accumulation_spectrum: 大域traceの未証明箇所とRH同値の区別", "exact_theorem": "VIII Theorem5; 数体対応の説明 pp.45–47", "assumptions": "Theorem5の明記された体は正標数。全Hecke L関数のRHと指定の大域trace公式の同値。数体では異なる近似射影の説明が必要", "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO", "notes": "同値定理の主張はRHを前提としない。大域trace公式の真を無条件入力にしていない。正標数の定理文を数体へそのまま引用しない。" } }, { "id": "structural:ST-CCM2025Core", "source_ledger": "archive/literature/research/structural_sources.json", "source_record_id": "ST-CCM2025Core", "title": "Zeta Spectral Triples — localized form and core", "authors": "Alain Connes; Caterina Consani; Henri Moscovici", "year": 2025, "external_urls": [ "https://arxiv.org/html/2511.22755v1" ], "claim_used": "accumulation_spectrum: exact restrictionsの固定窓極限", "exact_locator": "Propositions3.3–3.4; Theorem3.6", "assumptions": "λ>1、L²([λ⁻¹,λ],du/u)、E=span(V_n)がform core、有限E_Nへの厳密制限", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "PUBLISHED_CHAPTER_REVIEW_UNCONFIRMED", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "ST-CCM2025Core", "title": "Zeta Spectral Triples — localized form and core", "authors": "Alain Connes; Caterina Consani; Henri Moscovici", "year": 2025, "url": "https://arxiv.org/html/2511.22755v1", "claim_used": "accumulation_spectrum: exact restrictionsの固定窓極限", "exact_theorem": "Propositions3.3–3.4; Theorem3.6", "assumptions": "λ>1、L²([λ⁻¹,λ],du/u)、E=span(V_n)がform core、有限E_Nへの厳密制限", "depends_on_RH": "NO", "peer_reviewed": "PUBLISHED_CHAPTER_REVIEW_UNCONFIRMED", "verified_independently": "NO", "notes": "EMS公刊章 DOI10.4171/ELM/37/3。固定窓の最小値は有限制限最小値の極限。全NのPSDは別前件。" } }, { "id": "structural:ST-CCM2025Conditional", "source_ledger": "archive/literature/research/structural_sources.json", "source_record_id": "ST-CCM2025Conditional", "title": "Zeta Spectral Triples — finite selfadjoint construction and missing steps", "authors": "Alain Connes; Caterina Consani; Henri Moscovici", "year": 2025, "external_urls": [ "https://arxiv.org/html/2511.22755v1" ], "claim_used": "accumulation_spectrum: 条件付き自己共役構成と未証明のζ同定", "exact_locator": "Theorem5.10; Section8", "assumptions": "有限最低固有値ε_Nが単純、ξが偶、δ_Nξ=1。商計量はQW−ε_N I。全窓simple/evenと真の最低固有関数近似は§8の未完手順", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "PUBLISHED_CHAPTER_REVIEW_UNCONFIRMED", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "ST-CCM2025Conditional", "title": "Zeta Spectral Triples — finite selfadjoint construction and missing steps", "authors": "Alain Connes; Caterina Consani; Henri Moscovici", "year": 2025, "url": "https://arxiv.org/html/2511.22755v1", "claim_used": "accumulation_spectrum: 条件付き自己共役構成と未証明のζ同定", "exact_theorem": "Theorem5.10; Section8", "assumptions": "有限最低固有値ε_Nが単純、ξが偶、δ_Nξ=1。商計量はQW−ε_N I。全窓simple/evenと真の最低固有関数近似は§8の未完手順", "depends_on_RH": "NO", "peer_reviewed": "PUBLISHED_CHAPTER_REVIEW_UNCONFIRMED", "verified_independently": "NO", "notes": "RH仮定の定理ではないが、明示された条件付き。ε_Nを引いた正性は元形式のPSDを示さない。数値一致を全零点同定としない。" } }, { "id": "structural:ST-CvS2025", "source_ledger": "archive/literature/research/structural_sources.json", "source_record_id": "ST-CvS2025", "title": "Quadratic Forms, Real Zeros and Echoes of the Spectral Action", "authors": "Alain Connes; Walter D. van Suijlekom", "year": 2025, "external_urls": [ "https://arxiv.org/html/2511.23257v1" ], "claim_used": "accumulation_spectrum: 条件付き固定窓のFourier実零点定理", "exact_locator": "Theorem6.1（正確な定理文）; Proposition5.10", "assumptions": "実分布Dから式(6)で三角多項式上の形式。半有界かつ本質的自己共役、最下点が孤立単純固有値、固有関数が区間反転で偶", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "ST-CvS2025", "title": "Quadratic Forms, Real Zeros and Echoes of the Spectral Action", "authors": "Alain Connes; Walter D. van Suijlekom", "year": 2025, "url": "https://arxiv.org/html/2511.23257v1", "claim_used": "accumulation_spectrum: 条件付き固定窓のFourier実零点定理", "exact_theorem": "Theorem6.1（正確な定理文）; Proposition5.10", "assumptions": "実分布Dから式(6)で三角多項式上の形式。半有界かつ本質的自己共役、最下点が孤立単純固有値、固有関数が区間反転で偶", "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO", "notes": "Commun.Math.Phys.406:312, DOI10.1007/s00220-025-05493-1。operator coreの仮定を単なるform coreと置換しない。" } }, { "id": "structural:ST-Suzuki2026Local", "source_ledger": "archive/literature/research/structural_sources.json", "source_record_id": "ST-Suzuki2026Local", "title": "Weil’s quadratic form via the screw function — local statements", "authors": "Masatoshi Suzuki", "year": 2026, "external_urls": [ "https://arxiv.org/html/2606.09096v3" ], "claim_used": "全4ノートのWeil明示公式・局所Friedrichs実現・小窓正値性", "exact_locator": "§1.1; Theorem1.1; Theorem1.4; §2.4", "assumptions": "固定a>0でB_a=D*G_aD、Friedrichs拡張。正値・最低固有値単純性の無条件結果は十分小さいaに限定", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "PREPRINT", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "ST-Suzuki2026Local", "title": "Weil’s quadratic form via the screw function — local statements", "authors": "Masatoshi Suzuki", "year": 2026, "url": "https://arxiv.org/html/2606.09096v3", "claim_used": "全4ノートのWeil明示公式・局所Friedrichs実現・小窓正値性", "exact_theorem": "§1.1; Theorem1.1; Theorem1.4; §2.4", "assumptions": "固定a>0でB_a=D*G_aD、Friedrichs拡張。正値・最低固有値単純性の無条件結果は十分小さいaに限定", "depends_on_RH": "NO", "peer_reviewed": "PREPRINT", "verified_independently": "NO", "notes": "v3 2026-09-23。全窓正値性や§7のRH下heuristicは使用statementに含めない。" } }, { "id": "structural:ST-Suzuki2026Limit", "source_ledger": "archive/literature/research/structural_sources.json", "source_record_id": "ST-Suzuki2026Limit", "title": "Weil’s quadratic form via the screw function — conditional limit criterion", "authors": "Masatoshi Suzuki", "year": 2026, "external_urls": [ "https://arxiv.org/html/2606.09096v3" ], "claim_used": "center_scale: 自己共役性以外に必要な全零点同定条件の具体例", "exact_locator": "Corollary1.6, equation(1.12)", "assumptions": "十分大きいaでλ(a)<λ_a, θ(a), 上半平面でanalyticなφ(a,z)を選び、e^φ Wがξ/(ξ+ξ′)へ上半平面compact上一様収束", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "PREPRINT", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "ST-Suzuki2026Limit", "title": "Weil’s quadratic form via the screw function — conditional limit criterion", "authors": "Masatoshi Suzuki", "year": 2026, "url": "https://arxiv.org/html/2606.09096v3", "claim_used": "center_scale: 自己共役性以外に必要な全零点同定条件の具体例", "exact_theorem": "Corollary1.6, equation(1.12)", "assumptions": "十分大きいaでλ(a)<λ_a, θ(a), 上半平面でanalyticなφ(a,z)を選び、e^φ Wがξ/(ξ+ξ′)へ上半平面compact上一様収束", "depends_on_RH": "NO", "peer_reviewed": "PREPRINT", "verified_independently": "NO", "notes": "含意『極限条件⇒RH』はRHを仮定しない。ただし極限条件は未証明。旧版の全平面/z²ξ/ξ′と混在させない。" } }, { "id": "structural:ST-Burnol2004", "source_ledger": "archive/literature/research/structural_sources.json", "source_record_id": "ST-Burnol2004", "title": "Two complete and minimal systems associated with the zeros of the Riemann zeta function", "authors": "Jean-François Burnol", "year": 2004, "external_urls": [ "https://arxiv.org/pdf/math/0203120v6" ], "claim_used": "accumulation_spectrum: Sonine/Mellinの解析性・評価ベクトル・完全性の限界", "exact_locator": "Theorem2.1; Proposition2.2; Theorem3.1; Theorem7.7; Section8", "assumptions": "K_a: fとcosine変換が(0,a)で0、L_a:ともに定数。全複素評価点。零点は全非自明零点を重複度付きで使用", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "ST-Burnol2004", "title": "Two complete and minimal systems associated with the zeros of the Riemann zeta function", "authors": "Jean-François Burnol", "year": 2004, "url": "https://arxiv.org/pdf/math/0203120v6", "claim_used": "accumulation_spectrum: Sonine/Mellinの解析性・評価ベクトル・完全性の限界", "exact_theorem": "Theorem2.1; Proposition2.2; Theorem3.1; Theorem7.7; Section8", "assumptions": "K_a: fとcosine変換が(0,a)で0、L_a:ともに定数。全複素評価点。零点は全非自明零点を重複度付きで使用", "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO", "notes": "JTNB16,65–94, DOI10.5802/jtnb.434。一般Sonine関数は任意追加零点を許す。完全系・主要零点密度からRHは従わない。" } }, { "id": "structural:ST-NakamuraSuzuki2023Euler", "source_ledger": "archive/literature/research/structural_sources.json", "source_record_id": "ST-NakamuraSuzuki2023Euler", "title": "On infinitely divisible distributions related to the Riemann hypothesis — Euler distribution", "authors": "Takashi Nakamura; Masatoshi Suzuki", "year": 2023, "external_urls": [ "https://arxiv.org/pdf/2306.08317v1" ], "claim_used": "accumulation_spectrum: ζ(σ+it)/ζ(σ)のcompound Poisson表示", "exact_locator": "Section1, Euler distribution representation following equation(1.7)", "assumptions": "σ>1。Lévy測度Σ_{p,r}p^(−rσ)/r δ_(−rlogp)。この項目はTheorem1.2のRH下零点Lévy測度とは別", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "ST-NakamuraSuzuki2023Euler", "title": "On infinitely divisible distributions related to the Riemann hypothesis — Euler distribution", "authors": "Takashi Nakamura; Masatoshi Suzuki", "year": 2023, "url": "https://arxiv.org/pdf/2306.08317v1", "claim_used": "accumulation_spectrum: ζ(σ+it)/ζ(σ)のcompound Poisson表示", "exact_theorem": "Section1, Euler distribution representation following equation(1.7)", "assumptions": "σ>1。Lévy測度Σ_{p,r}p^(−rσ)/r δ_(−rlogp)。この項目はTheorem1.2のRH下零点Lévy測度とは別", "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO", "notes": "Statistics&Probability Letters, DOI10.1016/j.spl.2023.109889。σ↓1で弱収束が失われる説明はノート内の解析。" } }, { "id": "structural:ST-Nakamura2015Characteristic", "source_ledger": "archive/literature/research/structural_sources.json", "source_record_id": "ST-Nakamura2015Characteristic", "title": "A complete Riemann zeta distribution and the Riemann hypothesis — characteristic functions", "authors": "Takashi Nakamura", "year": 2015, "external_urls": [ "https://arxiv.org/pdf/1504.03438" ], "claim_used": "accumulation_spectrum: 正測度表示自体ではRHに足りない例", "exact_locator": "Theorem1.1; proof Section2.1", "assumptions": "全実σでΞ_σ(t)=ξ(σ−it)/ξ(σ)。完成ξを使用。裸ζとは異なる", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "ST-Nakamura2015Characteristic", "title": "A complete Riemann zeta distribution and the Riemann hypothesis — characteristic functions", "authors": "Takashi Nakamura", "year": 2015, "url": "https://arxiv.org/pdf/1504.03438", "claim_used": "accumulation_spectrum: 正測度表示自体ではRHに足りない例", "exact_theorem": "Theorem1.1; proof Section2.1", "assumptions": "全実σでΞ_σ(t)=ξ(σ−it)/ξ(σ)。完成ξを使用。裸ζとは異なる", "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO", "notes": "Bernoulli21,604–617。PDFは出版物の電子再録。特性関数性と零点位置を区別。" } }, { "id": "structural:ST-Nakamura2015PretendedCriterion", "source_ledger": "archive/literature/research/structural_sources.json", "source_record_id": "ST-Nakamura2015PretendedCriterion", "title": "A complete Riemann zeta distribution and the Riemann hypothesis — pretended infinite divisibility criterion", "authors": "Takashi Nakamura", "year": 2015, "external_urls": [ "https://arxiv.org/pdf/1504.03438" ], "claim_used": "accumulation_spectrum: 特性関数性より強い別条件がRH同値であること", "exact_locator": "Theorem1.2; definition in Section1.3", "assumptions": "全σ∈(1/2,1)でΞ_σがpretended-infinitely divisible。符号付きLévy測度を許し、通常のLévy可積分条件を課さないという同論文独自の定義", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "ST-Nakamura2015PretendedCriterion", "title": "A complete Riemann zeta distribution and the Riemann hypothesis — pretended infinite divisibility criterion", "authors": "Takashi Nakamura", "year": 2015, "url": "https://arxiv.org/pdf/1504.03438", "claim_used": "accumulation_spectrum: 特性関数性より強い別条件がRH同値であること", "exact_theorem": "Theorem1.2; definition in Section1.3", "assumptions": "全σ∈(1/2,1)でΞ_σがpretended-infinitely divisible。符号付きLévy測度を許し、通常のLévy可積分条件を課さないという同論文独自の定義", "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO", "notes": "同値命題自体はRH無仮定。RH⇒の明示測度表示を無条件に使用しない。通常のinfinitely divisibleと同一視禁止。Theorem1.4はσ>1でnot infinitely divisible but quasiと述べる。" } }, { "id": "structural:ST-PinchoverTintarev2004", "source_ledger": "archive/literature/research/structural_sources.json", "source_record_id": "ST-PinchoverTintarev2004", "title": "A ground state alternative for singular Schrödinger operators", "authors": "Yehuda Pinchover; Kyril Tintarev", "year": 2004, "external_urls": [ "https://arxiv.org/pdf/math/0411658v1" ], "claim_used": "variational_closure: ground-state identityの適用条件を検査", "exact_locator": "Lemma2.4, equation(2.6); Theorem1.4の非負性前件", "assumptions": "指定の局所楕円作用素L=−div(A∇)+V。Lemma2.4は有界C¹領域D, Lψ=0, v∈H¹(D;R), vψ|∂D=0。Theorem1.4は初めから形式非負", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "UNCONFIRMED_FOR_THIS_AUDIT", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "ST-PinchoverTintarev2004", "title": "A ground state alternative for singular Schrödinger operators", "authors": "Yehuda Pinchover; Kyril Tintarev", "year": 2004, "url": "https://arxiv.org/pdf/math/0411658v1", "claim_used": "variational_closure: ground-state identityの適用条件を検査", "exact_theorem": "Lemma2.4, equation(2.6); Theorem1.4の非負性前件", "assumptions": "指定の局所楕円作用素L=−div(A∇)+V。Lemma2.4は有界C¹領域D, Lψ=0, v∈H¹(D;R), vψ|∂D=0。Theorem1.4は初めから形式非負", "depends_on_RH": "NO", "peer_reviewed": "UNCONFIRMED_FOR_THIS_AUDIT", "verified_independently": "NO", "notes": "arXiv header2004と本文date2018が異なる。今回は査読出版の確認を追加していない。一般Weil形式への移植・逆向きの正値性導出は禁止。" } }, { "id": "structural:ST-Groskin2026Tail", "source_ledger": "archive/literature/research/structural_sources.json", "source_record_id": "ST-Groskin2026Tail", "title": "A finite Guinand–Weil dictionary and archimedean tail order for the truncated Weil quadratic form", "authors": "Akiva Groskin", "year": 2026, "external_urls": [ "https://arxiv.org/html/2607.02828v3" ], "claim_used": "variational_closure: 特定有限行列のarchimedean cutoffと他の極限を分離", "exact_locator": "Theorem3.2; Corollary3.3（定義の照合用Theorem2.5）", "assumptions": "固定c>1,N、T>max(2πN/log(c),7)、指定の有限周波数行列。dictionaryは実偶ベクトル", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "PREPRINT", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "ST-Groskin2026Tail", "title": "A finite Guinand–Weil dictionary and archimedean tail order for the truncated Weil quadratic form", "authors": "Akiva Groskin", "year": 2026, "url": "https://arxiv.org/html/2607.02828v3", "claim_used": "variational_closure: 特定有限行列のarchimedean cutoffと他の極限を分離", "exact_theorem": "Theorem3.2; Corollary3.3（定義の照合用Theorem2.5）", "assumptions": "固定c>1,N、T>max(2πN/log(c),7)、指定の有限周波数行列。dictionaryは実偶ベクトル", "depends_on_RH": "NO", "peer_reviewed": "PREPRINT", "verified_independently": "NO", "notes": "v3 2026-08-14。checkpoint02で辞書・tailの限定独立監査、版固定package39/39checksum確認、c13,N4の原著再実行と独立Arb直接積分を実施。論文全体・401次認証未再現のためverified欄はNOを維持。全c,N正値性・support単調性ではない。詳細proofs/audits/groskin-computation.md" } }, { "id": "structural:ST-Zhu2026Window", "source_ledger": "archive/literature/research/structural_sources.json", "source_record_id": "ST-Zhu2026Window", "title": "Weil positivity in compact windows: a finite reduction, certified two-sided bounds, and a Landau–Widom decay law", "authors": "Xuefeng Zhu", "year": 2026, "external_urls": [ "https://arxiv.org/html/2608.24827v2" ], "claim_used": "variational_closure: 有限blockと残りのcouplingを共に評価する設計の参考", "exact_locator": "Theorem1.2; Theorem6.2; Corollary6.3; Section7 withdrawal", "assumptions": "指定された固定窓の複素試験関数と認証誤差の前件。局所数値主張は独立再検証していない", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "PREPRINT", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "ST-Zhu2026Window", "title": "Weil positivity in compact windows: a finite reduction, certified two-sided bounds, and a Landau–Widom decay law", "authors": "Xuefeng Zhu", "year": 2026, "url": "https://arxiv.org/html/2608.24827v2", "claim_used": "variational_closure: 有限blockと残りのcouplingを共に評価する設計の参考", "exact_theorem": "Theorem1.2; Theorem6.2; Corollary6.3; Section7 withdrawal", "assumptions": "指定された固定窓の複素試験関数と認証誤差の前件。局所数値主張は独立再検証していない", "depends_on_RH": "NO", "peer_reviewed": "PREPRINT", "verified_independently": "NO", "notes": "checkpoint03: v2原本の取得後も補足code/行列/factorの公開先未確認。一般entry boundにT/π欠落を厳密反証しtail/couplingは独立修復。原著L.8 head未認証。自作a=.5全mode認証は別ノートT090。全窓・RH主張に使用しない。" } }, { "id": "structural:ST-DLMF25", "source_ledger": "archive/literature/research/structural_sources.json", "source_record_id": "ST-DLMF25", "title": "DLMF §25.11 Hurwitz Zeta Function", "authors": "NIST Digital Library of Mathematical Functions", "year": 2026, "external_urls": [ "https://dlmf.nist.gov/25.11#E43" ], "claim_used": "Euler境界の数値プローブに用いるEuler–Maclaurin近似", "exact_locator": "§25.11(iii), equation 25.11.43", "assumptions": "固定s、正の大きいa。実装N=1000、B2〜B8、binary64、区間保証なし", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "OFFICIAL_REFERENCE", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "ST-DLMF25", "title": "DLMF §25.11 Hurwitz Zeta Function", "authors": "NIST Digital Library of Mathematical Functions", "year": 2026, "url": "https://dlmf.nist.gov/25.11#E43", "claim_used": "Euler境界の数値プローブに用いるEuler–Maclaurin近似", "exact_theorem": "§25.11(iii), equation 25.11.43", "assumptions": "固定s、正の大きいa。実装N=1000、B2〜B8、binary64、区間保証なし", "depends_on_RH": "NO", "peer_reviewed": "OFFICIAL_REFERENCE", "verified_independently": "NO", "notes": "RHの解析入力でなく数値実装の参照。ζ(2)のsanity error約1.8e−15は他点の誤差証明ではない。" } }, { "id": "structural:ST-CeroneDragomir2009", "source_ledger": "archive/literature/research/structural_sources.json", "source_record_id": "ST-CeroneDragomir2009", "title": "Some convexity properties of Dirichlet series with positive terms", "authors": "Pietro Cerone and Sever S. Dragomir", "year": 2009, "external_urls": [ "https://rgmia.org/papers/v8n4/IDSLCNew.pdf" ], "claim_used": "固定積imbalanceの正係数Dirichlet級数・対数凸性の既知性", "exact_locator": "2005-10-25公開稿 §1 (1.5), §3 Proposition4, Remark5", "assumptions": "実パラメータで正係数級数が収束する範囲。零点への等号移行なし", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "ST-CeroneDragomir2009", "title": "Some convexity properties of Dirichlet series with positive terms", "authors": "Pietro Cerone and Sever S. Dragomir", "year": 2009, "url": "https://rgmia.org/papers/v8n4/IDSLCNew.pdf", "claim_used": "固定積imbalanceの正係数Dirichlet級数・対数凸性の既知性", "exact_theorem": "2005-10-25公開稿 §1 (1.5), §3 Proposition4, Remark5", "assumptions": "実パラメータで正係数級数が収束する範囲。零点への等号移行なし", "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO", "notes": "Math.Nachr.282(2009),964–975。DOI10.1002/mana.200610783。該当公式のみ照合、全論文未検証。逆数凹性の結果は不採用。" } }, { "id": "structural:ST-Nielsen2022", "source_ledger": "archive/literature/research/structural_sources.json", "source_record_id": "ST-Nielsen2022", "title": "A note on some information-theoretic divergences between Zeta distributions", "authors": "Frank Nielsen", "year": 2022, "external_urls": [ "https://arxiv.org/html/2104.10548v3" ], "claim_used": "prime-power imbalance aggregateの半分が既知Bhattacharyya距離・Jensen差に一致", "exact_locator": "Table1; Theorem1直前のBhattacharyya coefficient and Jensen divergence", "assumptions": "両方のζ分布パラメータ>1。実確率分布。解析接続後には適用しない", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "UNCONFIRMED_FOR_THIS_AUDIT", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "ST-Nielsen2022", "title": "A note on some information-theoretic divergences between Zeta distributions", "authors": "Frank Nielsen", "year": 2022, "url": "https://arxiv.org/html/2104.10548v3", "claim_used": "prime-power imbalance aggregateの半分が既知Bhattacharyya距離・Jensen差に一致", "exact_theorem": "Table1; Theorem1直前のBhattacharyya coefficient and Jensen divergence", "assumptions": "両方のζ分布パラメータ>1。実確率分布。解析接続後には適用しない", "depends_on_RH": "NO", "peer_reviewed": "UNCONFIRMED_FOR_THIS_AUDIT", "verified_independently": "NO", "notes": "一次公式をLITERATUREとrootが照合。全論文の正確性は未検証。直接代数で今回の特殊化を再導出。imbalance-prior-art.md参照。" } }, { "id": "structural:ST-AndersonTrapp1975", "source_ledger": "archive/literature/research/structural_sources.json", "source_record_id": "ST-AndersonTrapp1975", "title": "Shorted Operators. II", "authors": "W. N. Anderson, Jr.; G. E. Trapp", "year": 1975, "external_urls": [ "https://epubs.siam.org/doi/10.1137/0128007" ], "claim_used": "support-propagation: shorted operator の既知性と ambient positivity 前件の確認", "exact_locator": "出版社 Abstract の定義と存在主張のみ。本文の定理番号は未照合", "assumptions": "Hilbert 空間上の正作用素 A と固定閉部分空間 S。0≤X≤A、Ran(X)⊂S の最大化。未知の Weil 作用素の正値性を前提にしない", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "ST-AndersonTrapp1975", "title": "Shorted Operators. II", "authors": "W. N. Anderson, Jr.; G. E. Trapp", "year": 1975, "url": "https://epubs.siam.org/doi/10.1137/0128007", "claim_used": "support-propagation: shorted operator の既知性と ambient positivity 前件の確認", "exact_theorem": "出版社 Abstract の定義と存在主張のみ。本文の定理番号は未照合", "assumptions": "Hilbert 空間上の正作用素 A と固定閉部分空間 S。0≤X≤A、Ran(X)⊂S の最大化。未知の Weil 作用素の正値性を前提にしない", "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO", "notes": "SIAM J. Appl. Math.28(1975),60–71, DOI10.1137/0128007。出版社 abstract のみ閲読、全文未確認。ノートの可逆 A に対する Schur 平方完成は直接代数であり、本論文の未読定理に帰属させない。" } }, { "id": "structural:ST-AlpayEtAl2010", "source_ledger": "archive/literature/research/structural_sources.json", "source_record_id": "ST-AlpayEtAl2010", "title": "Krein systems and canonical systems on a finite interval: accelerants with a jump discontinuity at the origin and continuous potentials", "authors": "D. Alpay; I. Gohberg; M. A. Kaashoek; L. Lerer; A. L. Sakhnovich", "year": 2010, "external_urls": [ "https://arxiv.org/pdf/0912.4444v1" ], "claim_used": "support-propagation: accelerant/resolvent から canonical system を構成する際の全区間可逆性と、逆構成後の核同定の義務", "exact_locator": "Theorems1.1–1.2; equations(1.1)–(1.2),(1.10)–(1.13); p.6、(2.10)直前の全区間可逆性と T_T の strict positivity の同値", "assumptions": "Hermitian 行列核 k は[-T,T]で連続（原点の jump 可）。直接構成では T_τ=I−∫_0^τk(x−s)ds が全0<τ≤Tで可逆。逆構成は連続 Dirac potential v を入力", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "ST-AlpayEtAl2010", "title": "Krein systems and canonical systems on a finite interval: accelerants with a jump discontinuity at the origin and continuous potentials", "authors": "D. Alpay; I. Gohberg; M. A. Kaashoek; L. Lerer; A. L. Sakhnovich", "year": 2010, "url": "https://arxiv.org/pdf/0912.4444v1", "claim_used": "support-propagation: accelerant/resolvent から canonical system を構成する際の全区間可逆性と、逆構成後の核同定の義務", "exact_theorem": "Theorems1.1–1.2; equations(1.1)–(1.2),(1.10)–(1.13); p.6、(2.10)直前の全区間可逆性と T_T の strict positivity の同値", "assumptions": "Hermitian 行列核 k は[-T,T]で連続（原点の jump 可）。直接構成では T_τ=I−∫_0^τk(x−s)ds が全0<τ≤Tで可逆。逆構成は連続 Dirac potential v を入力", "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO", "notes": "IEOT68(2010),115–150, DOI10.1007/s00020-010-1803-x、出版社公刊情報照合。arXiv v1(2009-12-22)の該当本文・定理文を閲読。全証明未検証。逆構成の一意 accelerant と算術的 Weil 核の一致は未証明であり、小区間可逆性から全区間可逆性を結論しない。" } }, { "id": "structural:ST-Suzuki2021Canonical", "source_ledger": "archive/literature/research/structural_sources.json", "source_record_id": "ST-Suzuki2021Canonical", "title": "Hamiltonians arising from L-functions in the Selberg class", "authors": "Masatoshi Suzuki", "year": 2021, "external_urls": [ "https://arxiv.org/html/1606.05726v3" ], "claim_used": "support-propagation: Fredholm canonical family の局所構成と、全域 determinant 非消滅・endpoint 極限を含む RH 同値条件", "exact_locator": "Theorems2.1–2.4; equations(2.5)–(2.13); §3(K1)–(K5); §5.1冒頭の条件を引き継ぐ Proposition5.1", "assumptions": "L∈S_R、ω>0、ν∈Z_>0、νωd_L>1。局所構成は±1が K[t] の固有値でない範囲。Thm2.4 は ω_n↓0、全t≥0でdet(I±K_n[t])≠0、全z∈C_+でJ_n(t;z,z)→0を要求。Prop5.1は E∈HBbar 条件付き", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "ST-Suzuki2021Canonical", "title": "Hamiltonians arising from L-functions in the Selberg class", "authors": "Masatoshi Suzuki", "year": 2021, "url": "https://arxiv.org/html/1606.05726v3", "claim_used": "support-propagation: Fredholm canonical family の局所構成と、全域 determinant 非消滅・endpoint 極限を含む RH 同値条件", "exact_theorem": "Theorems2.1–2.4; equations(2.5)–(2.13); §3(K1)–(K5); §5.1冒頭の条件を引き継ぐ Proposition5.1", "assumptions": "L∈S_R、ω>0、ν∈Z_>0、νωd_L>1。局所構成は±1が K[t] の固有値でない範囲。Thm2.4 は ω_n↓0、全t≥0でdet(I±K_n[t])≠0、全z∈C_+でJ_n(t;z,z)→0を要求。Prop5.1は E∈HBbar 条件付き", "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO", "notes": "JFA281(2021),109116, DOI10.1016/j.jfa.2021.109116。公刊本文取得は今回失敗し、番号・条件はarXiv v3(2020-10-01)に固定して該当箇所閲読。全証明未検証。同値定理自体はRH無仮定だが、その前件を証明済みとしない。K[t]はE^sharp/EのHankel核で、Weil窓作用素との同一視は別義務。2026 screw-form 論文とは別。" } }, { "id": "structural:ST-KreinLanger2014", "source_ledger": "archive/literature/research/structural_sources.json", "source_record_id": "ST-KreinLanger2014", "title": "Continuation of Hermitian Positive Definite Functions and Related Questions", "authors": "M. G. Krein; H. Langer", "year": 2014, "external_urls": [ "https://link.springer.com/article/10.1007/s00020-013-2091-z" ], "claim_used": "support-propagation: 正定値関数の延長問題と canonical systems/strings の関係の先行文献", "exact_locator": "出版社 Abstract と書誌のみ。個別定理・条件は未照合であり証明入力にしない", "assumptions": "本記録は研究対象の帰属のみ。延長存在定理の正確な仮定を未確認のまま Weil 核へ適用しない", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "ST-KreinLanger2014", "title": "Continuation of Hermitian Positive Definite Functions and Related Questions", "authors": "M. G. Krein; H. Langer", "year": 2014, "url": "https://link.springer.com/article/10.1007/s00020-013-2091-z", "claim_used": "support-propagation: 正定値関数の延長問題と canonical systems/strings の関係の先行文献", "exact_theorem": "出版社 Abstract と書誌のみ。個別定理・条件は未照合であり証明入力にしない", "assumptions": "本記録は研究対象の帰属のみ。延長存在定理の正確な仮定を未確認のまま Weil 核へ適用しない", "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO", "notes": "IEOT78(2014),1–69, DOI10.1007/s00020-013-2091-z。出版社 abstract のみ閲読、全文未確認。ある正の延長の存在と、素数冪 source が指定する延長との一致を区別する。" } }, { "id": "structural:ST-LapidusMaier1995", "source_ledger": "archive/literature/research/structural_sources.json", "source_record_id": "ST-LapidusMaier1995", "title": "The Riemann Hypothesis and Inverse Spectral Problems for Fractal Strings", "authors": "Michel L. Lapidus; Helmut Maier", "year": 1995, "external_urls": [ "https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms/52.1.15" ], "claim_used": "spiral-scale: fixed D の逆スペクトル問題と縦線無零点の同値、D=1/2での失敗", "exact_locator": "Theorems2.3–2.4; §2(2.8)–(2.9)のweak converse", "assumptions": "D∈(0,1)、有限総長のfractal string、Dirichlet周波数。原著のweak converseは計数誤差 O(x^D/(log x)^(1+ε)), ε>0。全該当stringを量化", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "ST-LapidusMaier1995", "title": "The Riemann Hypothesis and Inverse Spectral Problems for Fractal Strings", "authors": "Michel L. Lapidus; Helmut Maier", "year": 1995, "url": "https://londmathsoc.onlinelibrary.wiley.com/doi/10.1112/jlms/52.1.15", "claim_used": "spiral-scale: fixed D の逆スペクトル問題と縦線無零点の同値、D=1/2での失敗", "exact_theorem": "Theorems2.3–2.4; §2(2.8)–(2.9)のweak converse", "assumptions": "D∈(0,1)、有限総長のfractal string、Dirichlet周波数。原著のweak converseは計数誤差 O(x^D/(log x)^(1+ε)), ε>0。全該当stringを量化", "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO", "notes": "JLMS52(1995),15–34, DOI10.1112/jlms/52.1.15。出版社PDF取得失敗、著者本人アップロードの原論文全文(ResearchGate) §§2–3を閲読、OCR崩れをLapidus2015本人解説と照合。全文未検証。全D≠1/2での成功がRH同値。o(x^D)版を1995原著の定理に読み込まない。" } }, { "id": "structural:ST-Lapidus2015Sound", "source_ledger": "archive/literature/research/structural_sources.json", "source_record_id": "ST-Lapidus2015Sound", "title": "The Sound of Fractal Strings and the Riemann Hypothesis", "authors": "Michel L. Lapidus", "year": 2015, "external_urls": [ "https://arxiv.org/pdf/1505.01548v1" ], "claim_used": "spiral-scale: 逆問題の誤差版境界、Cantor string のexact self-similarityと複素次元", "exact_locator": "§6 Remark13(b), Theorems14–15, Corollary16; §7 equations(33)–(37)", "assumptions": "ζ_nu=ζ ζ_Lは正規化周波数 n/ell_j と絶対収束域。Cantor stringは長さ3^(−n−1)、重複度2^n。o(x^D)改善版はHilberdink未公刊結果に帰属", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "UNCONFIRMED_FOR_THIS_AUDIT", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "ST-Lapidus2015Sound", "title": "The Sound of Fractal Strings and the Riemann Hypothesis", "authors": "Michel L. Lapidus", "year": 2015, "url": "https://arxiv.org/pdf/1505.01548v1", "claim_used": "spiral-scale: 逆問題の誤差版境界、Cantor string のexact self-similarityと複素次元", "exact_theorem": "§6 Remark13(b), Theorems14–15, Corollary16; §7 equations(33)–(37)", "assumptions": "ζ_nu=ζ ζ_Lは正規化周波数 n/ell_j と絶対収束域。Cantor stringは長さ3^(−n−1)、重複度2^n。o(x^D)改善版はHilberdink未公刊結果に帰属", "depends_on_RH": "NO", "peer_reviewed": "UNCONFIRMED_FOR_THIS_AUDIT", "verified_independently": "NO", "notes": "著者自身の研究の解説一次資料。該当本文閲読、全証明・公刊査読状況未確認。幾何zetaのpolesとRiemann zetaのzerosを区別。一般self-similarityから実部1/2を導かない。" } }, { "id": "structural:ST-HerichiLapidus2013", "source_ledger": "archive/literature/research/structural_sources.json", "source_record_id": "ST-HerichiLapidus2013", "title": "Fractal Complex Dimensions, Riemann Hypothesis and Invertibility of the Spectral Operator", "authors": "Hafedh Herichi; Michel L. Lapidus", "year": 2013, "external_urls": [ "https://arxiv.org/pdf/1210.0882v3" ], "claim_used": "spiral-scale: weighted shiftとζ(∂_c)のnormal functional calculus、quasi-invertibilityのRH同値式", "exact_locator": "Theorems3.7,3.9,3.15,3.17; Definition5.3; Theorems5.9,6.1,6.8; AppendixB", "assumptions": "H_c=L²(R,e^(−2cu)du)、所定weighted Sobolev domain。平行移動和の有界作用素定理はc>1。quasi-invertibilityは各高さ切断での可逆性。通常可逆性は0がζ(c+iR)の閉包にないこと", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "PUBLISHED_CHAPTER_REVIEW_UNCONFIRMED", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "ST-HerichiLapidus2013", "title": "Fractal Complex Dimensions, Riemann Hypothesis and Invertibility of the Spectral Operator", "authors": "Hafedh Herichi; Michel L. Lapidus", "year": 2013, "url": "https://arxiv.org/pdf/1210.0882v3", "claim_used": "spiral-scale: weighted shiftとζ(∂_c)のnormal functional calculus、quasi-invertibilityのRH同値式", "exact_theorem": "Theorems3.7,3.9,3.15,3.17; Definition5.3; Theorems5.9,6.1,6.8; AppendixB", "assumptions": "H_c=L²(R,e^(−2cu)du)、所定weighted Sobolev domain。平行移動和の有界作用素定理はc>1。quasi-invertibilityは各高さ切断での可逆性。通常可逆性は0がζ(c+iR)の閉包にないこと", "depends_on_RH": "NO", "peer_reviewed": "PUBLISHED_CHAPTER_REVIEW_UNCONFIRMED", "verified_independently": "NO", "notes": "Contemporary Mathematics600(2013)、v3=2013-02-05。著者本人による研究解説で各定理は先行memoir[HerLa1]に帰属。該当定理・条件のみ閲読、全証明未検証。normal≠self-adjoint、quasi≠通常可逆、切断≠有限次元行列。全c≠1/2のquasi可逆性を未証明前件として保持。" } }, { "id": "structural:ST-Burnol2001Causality", "source_ledger": "archive/literature/research/structural_sources.json", "source_record_id": "ST-Burnol2001Causality", "title": "An adelic causality problem related to abelian L-functions", "authors": "Jean-François Burnol", "year": 2001, "external_urls": [ "https://arxiv.org/pdf/math/0001013v3" ], "claim_used": "spiral-scale: unitary尺度作用とRH同値なscattering causalityの区別", "exact_locator": "Definitions1.1,1.4; Lemma1.5; Theorem1.7; Introduction; ζへの特殊化は§2", "assumptions": "global field K、idele class group、Poisson–Tate map E、指定support class S_≤1からD_±を定義。D_-⊥D_+ iff 全abelian L-functions of KのRH", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "ST-Burnol2001Causality", "title": "An adelic causality problem related to abelian L-functions", "authors": "Jean-François Burnol", "year": 2001, "url": "https://arxiv.org/pdf/math/0001013v3", "claim_used": "spiral-scale: unitary尺度作用とRH同値なscattering causalityの区別", "exact_theorem": "Definitions1.1,1.4; Lemma1.5; Theorem1.7; Introduction; ζへの特殊化は§2", "assumptions": "global field K、idele class group、Poisson–Tate map E、指定support class S_≤1からD_±を定義。D_-⊥D_+ iff 全abelian L-functions of KのRH", "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO", "notes": "J.Number Theory87(2001),253–269、最終稿v3=2000-09-14。該当本文・定理文を照合、全文未検証。unitarity等は構成済みだがcausalityは未証明のRH同値条件。全abelian Lの量化をζのみに無断縮約しない。off-line zerosのscattering polesを自己共役generatorの固有値と混同しない。" } }, { "id": "structural:ST-Burnol2008Dilation", "source_ledger": "archive/literature/research/structural_sources.json", "source_record_id": "ST-Burnol2008Dilation", "title": "On some bound and scattering states associated with the cosine kernel", "authors": "Jean-François Burnol", "year": 2008, "external_urls": [ "https://arxiv.org/pdf/0801.0530v2" ], "claim_used": "spiral-scale: log(a) Dirac systemとdilation generatorの厳密なunitary同定の先行例", "exact_locator": "Definition5; Theorems7–8; equations(13)–(16); Introduction最終段落", "assumptions": "cosine kernel 2cos(2πxy)、a>0、u=log a、Fredholm determinantからμ。所定Sonine/Dirac空間とdomain", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "UNCONFIRMED_FOR_THIS_AUDIT", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "ST-Burnol2008Dilation", "title": "On some bound and scattering states associated with the cosine kernel", "authors": "Jean-François Burnol", "year": 2008, "url": "https://arxiv.org/pdf/0801.0530v2", "claim_used": "spiral-scale: log(a) Dirac systemとdilation generatorの厳密なunitary同定の先行例", "exact_theorem": "Definition5; Theorems7–8; equations(13)–(16); Introduction最終段落", "assumptions": "cosine kernel 2cos(2πxy)、a>0、u=log a、Fredholm determinantからμ。所定Sonine/Dirac空間とdomain", "depends_on_RH": "NO", "peer_reviewed": "UNCONFIRMED_FOR_THIS_AUDIT", "verified_independently": "NO", "notes": "該当本文閲読、全文・公刊査読状況未確認。H_0=−i(x d/dx+1/2)と実スペクトルの同定はζ零点のexact identificationではない。原著IntroductionはRiemann零点をexactに得る問題を扱わないと明記。" } }, { "id": "structural:ST-Zworski2017Resonances", "source_ledger": "archive/literature/research/structural_sources.json", "source_record_id": "ST-Zworski2017Resonances", "title": "Mathematical study of scattering resonances", "authors": "Maciej Zworski", "year": 2017, "external_urls": [ "https://link.springer.com/article/10.1007/s13373-017-0099-4" ], "claim_used": "spiral-scale: 自己共役L²スペクトルと解析接続resonancesの定義域の違い", "exact_locator": "§§2.1–2.2 Theorem2, Definitions1–2, equations(2.20)–(2.24); §2.6", "assumptions": "R³上V∈L∞_comp。自己共役性には実V、domain H²。resolvent継続はL²_comp→L²_loc、complex scaling後は非自己共役", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "ST-Zworski2017Resonances", "title": "Mathematical study of scattering resonances", "authors": "Maciej Zworski", "year": 2017, "url": "https://link.springer.com/article/10.1007/s13373-017-0099-4", "claim_used": "spiral-scale: 自己共役L²スペクトルと解析接続resonancesの定義域の違い", "exact_theorem": "§§2.1–2.2 Theorem2, Definitions1–2, equations(2.20)–(2.24); §2.6", "assumptions": "R³上V∈L∞_comp。自己共役性には実V、domain H²。resolvent継続はL²_comp→L²_loc、complex scaling後は非自己共役", "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "NO", "notes": "Bull.Math.Sci.7(2017),1–85, DOI10.1007/s13373-017-0099-4。出版社の該当数学本文を閲読、全文未検証。物理的定数・解釈は証明に不使用。共鳴は元のL²固有値ではなくmeromorphic continuationのpoles。" } }, { "id": "phase2:P2-C-MUELLER", "source_ledger": "archive/literature/research/phase2_sources.json", "source_record_id": "P2-C-MUELLER", "title": "A Spectral Interpretation of the Zeros of the Constant Term of Certain Eisenstein Series", "authors": "Werner Müller", "year": 2007, "external_urls": [ "https://webdoc.sub.gwdg.de/ebook/serien/e/sfb611/331.pdf" ], "claim_used": "Modular Eisenstein constant term and cut-off Laplacian, not Riemann RH", "exact_locator": "Introduction (0.1), (0.2), Theorems 0.1–0.2; Section 1 closed-form construction", "assumptions": "Modular arithmetic surface; cut-off parameter in stated range; spectral parameter differs from zeta argument by factor 2; center normalization excluded from application", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "Author preprint version inspected", "verified_independently": "PARTIAL: boundary coefficient and direct consequences C1–C7; not entire scattering theory" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "P2-C-MUELLER", "title": "A Spectral Interpretation of the Zeros of the Constant Term of Certain Eisenstein Series", "authors": "Werner Müller", "year": 2007, "url": "https://webdoc.sub.gwdg.de/ebook/serien/e/sfb611/331.pdf", "claim_used": "Modular Eisenstein constant term and cut-off Laplacian, not Riemann RH", "exact_theorem": "Introduction (0.1), (0.2), Theorems 0.1–0.2; Section 1 closed-form construction", "assumptions": "Modular arithmetic surface; cut-off parameter in stated range; spectral parameter differs from zeta argument by factor 2; center normalization excluded from application", "depends_on_RH": "NO", "peer_reviewed": "Author preprint version inspected", "verified_independently": "PARTIAL: boundary coefficient and direct consequences C1–C7; not entire scattering theory", "notes": "Primary research manuscript. Root and two independent agents audited arithmetic, cancellation, domains and limits. Center s=1/2 special normalization was not used." } }, { "id": "phase2:P2-C-LS", "source_ledger": "archive/literature/research/phase2_sources.json", "source_record_id": "P2-C-LS", "title": "The Riemann hypothesis for certain integrals of Eisenstein series", "authors": "Jeffrey C. Lagarias; Masatoshi Suzuki", "year": 2006, "external_urls": [ "https://arxiv.org/abs/math/0412039" ], "claim_used": "Prior occurrence and scope of different constant-term zero theorem", "exact_locator": "Constant-term zero theorem for y>=1, with possible real exceptions", "assumptions": "Specific Eisenstein integral or constant-term family; not the Riemann xi zero set", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "YES, Journal of Number Theory 118, 98–122", "verified_independently": "PARTIAL: coefficient independently derived; original zero theorem not reproved" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "P2-C-LS", "title": "The Riemann hypothesis for certain integrals of Eisenstein series", "authors": "Jeffrey C. Lagarias; Masatoshi Suzuki", "year": 2006, "url": "https://arxiv.org/abs/math/0412039", "claim_used": "Prior occurrence and scope of different constant-term zero theorem", "exact_theorem": "Constant-term zero theorem for y>=1, with possible real exceptions", "assumptions": "Specific Eisenstein integral or constant-term family; not the Riemann xi zero set", "depends_on_RH": "NO", "peer_reviewed": "YES, Journal of Number Theory 118, 98–122", "verified_independently": "PARTIAL: coefficient independently derived; original zero theorem not reproved", "notes": "Abstract and author note https://www.math.titech.ac.jp/top/~msuzuki/shizuoka_2004_2.pdf equations (1)–(4) checked; no claim of exhaustive paper audit." } }, { "id": "phase2:P2-A-RT", "source_ledger": "archive/literature/research/phase2_sources.json", "source_record_id": "P2-A-RT", "title": "The de Bruijn–Newman constant is non-negative", "authors": "Brad Rodgers; Terence Tao", "year": 2020, "external_urls": [ "https://arxiv.org/html/1801.05914v3" ], "claim_used": "Known threshold existence and nonnegative lower bound; all negative repo times have nonreal zeros", "exact_locator": "Section 1 equations (1)–(4), Theorem 1.1; classical upper bound cited there", "assumptions": "Actual Riemann kernel; normalization H_RT(t,z)=H_repo(t/4,z/2)/8", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "YES, Forum of Mathematics Pi 8 (2020), e6", "verified_independently": "NO for Theorem 1.1; normalization and local compactification independently checked" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "P2-A-RT", "title": "The de Bruijn–Newman constant is non-negative", "authors": "Brad Rodgers; Terence Tao", "year": 2020, "url": "https://arxiv.org/html/1801.05914v3", "claim_used": "Known threshold existence and nonnegative lower bound; all negative repo times have nonreal zeros", "exact_theorem": "Section 1 equations (1)–(4), Theorem 1.1; classical upper bound cited there", "assumptions": "Actual Riemann kernel; normalization H_RT(t,z)=H_repo(t/4,z/2)/8", "depends_on_RH": "NO", "peer_reviewed": "YES, Forum of Mathematics Pi 8 (2020), e6", "verified_independently": "NO for Theorem 1.1; normalization and local compactification independently checked", "notes": "Known input only. No new proof of threshold nonnegativity; Lambda=0 remains RH-equivalent." } }, { "id": "phase2:P2-B-CE", "source_ledger": "archive/literature/research/phase2_sources.json", "source_record_id": "P2-B-CE", "title": "The Laguerre inequality and the distribution of zeros of entire functions", "authors": "George Csordas; Alain Escassut", "year": 2005, "external_urls": [ "https://www.numdam.org/item/AMBP_2005__12_2_331_0.pdf" ], "claim_used": "Prior occurrence of complex Laguerre/Herglotz criterion", "exact_locator": "Definition 1.1; Theorem 2.3, equation (2.4), printed p.334", "assumptions": "Real entire function of specified product/growth class", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "PARTIAL: actual xi Herglotz iff RH directly rederived using paired product" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "P2-B-CE", "title": "The Laguerre inequality and the distribution of zeros of entire functions", "authors": "George Csordas; Alain Escassut", "year": 2005, "url": "https://www.numdam.org/item/AMBP_2005__12_2_331_0.pdf", "claim_used": "Prior occurrence of complex Laguerre/Herglotz criterion", "exact_theorem": "Definition 1.1; Theorem 2.3, equation (2.4), printed p.334", "assumptions": "Real entire function of specified product/growth class", "depends_on_RH": "NO", "peer_reviewed": "YES", "verified_independently": "PARTIAL: actual xi Herglotz iff RH directly rederived using paired product", "notes": "No use as unconditional positivity of actual xi; cycle18 prior obstruction is acknowledged." } }, { "id": "phase2:P2-B-EKT", "source_ledger": "archive/literature/research/phase2_sources.json", "source_record_id": "P2-B-EKT", "title": "Spectral asymptotics for canonical systems", "authors": "Jonathan Eckhardt; Aleksey Kostenko; Gerald Teschl", "year": 2018, "external_urls": [ "https://www.mat.univie.ac.at/~gerald/ftp/articles/AsymCS.pdf" ], "claim_used": "Conditions of canonical inverse theorem and convergence dictionary", "exact_locator": "Section 2.1, Theorem 2.2, Proposition 2.3", "assumptions": "Herglotz input / positive trace-normalized Hamiltonians already supplied", "verification_as_recorded": { "depends_on_RH": "NO", "peer_reviewed": "YES, J. Reine Angew. Math. 736, 285–315", "verified_independently": "NO for inverse theorem; signs and conditional Riccati/model formulas checked" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "P2-B-EKT", "title": "Spectral asymptotics for canonical systems", "authors": "Jonathan Eckhardt; Aleksey Kostenko; Gerald Teschl", "year": 2018, "url": "https://www.mat.univie.ac.at/~gerald/ftp/articles/AsymCS.pdf", "claim_used": "Conditions of canonical inverse theorem and convergence dictionary", "exact_theorem": "Section 2.1, Theorem 2.2, Proposition 2.3", "assumptions": "Herglotz input / positive trace-normalized Hamiltonians already supplied", "depends_on_RH": "NO", "peer_reviewed": "YES, J. Reine Angew. Math. 736, 285–315", "verified_independently": "NO for inverse theorem; signs and conditional Riccati/model formulas checked", "notes": "Does not turn an arbitrary prime germ into Herglotz input; no arithmetic Hamiltonian constructed." } }, { "id": "phase2:P2-D-RIEMANN", "source_ledger": "archive/literature/research/phase2_sources.json", "source_record_id": "P2-D-RIEMANN", "title": "On the Number of Primes Less Than a Given Magnitude", "authors": "Bernhard Riemann; English translation David R. Wilkins", "year": 1859, "external_urls": [ "https://www.claymath.org/wp-content/uploads/2023/04/Wilkins-translation.pdf" ], "claim_used": "Original theta/Mellin/functional-equation reconstruction", "exact_locator": "Translation text p.3 / PDF p.4 and following page", "assumptions": "Standard full Gaussian theta sum and analytic continuation", "verification_as_recorded": { "depends_on_RH": "NO for reconstruction", "peer_reviewed": "Historical primary publication, English translation inspected", "verified_independently": "PARTIAL: lattice Mellin formula and modular domain calculations rederived" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "P2-D-RIEMANN", "title": "On the Number of Primes Less Than a Given Magnitude", "authors": "Bernhard Riemann; English translation David R. Wilkins", "year": 1859, "url": "https://www.claymath.org/wp-content/uploads/2023/04/Wilkins-translation.pdf", "claim_used": "Original theta/Mellin/functional-equation reconstruction", "exact_theorem": "Translation text p.3 / PDF p.4 and following page", "assumptions": "Standard full Gaussian theta sum and analytic continuation", "depends_on_RH": "NO for reconstruction", "peer_reviewed": "Historical primary publication, English translation inspected", "verified_independently": "PARTIAL: lattice Mellin formula and modular domain calculations rederived", "notes": "DLMF 20.7.32, 25.4.3–4, 25.5.13–14 used to check modern conventions. No claim that modularity proves RH." } }, { "id": "phase3:P3-DE74", "source_ledger": "archive/literature/research/phase3_sources.json", "source_record_id": "P3-DE74", "title": "La conjecture de Weil I", "authors": [ "Pierre Deligne" ], "year": 1974, "external_urls": [ "https://www.numdam.org/item/PMIHES_1974__43__273_0.pdf" ], "claim_used": "Trace-to-determinant and tensor/pole purity mechanism", "exact_locator": "1.5.1/1.5.4; Thms3.2,5.10,6.2; Lemma7.1; 7.3", "assumptions": "Smooth projective main theorem; local sheaf assumptions separately stated", "verification_as_recorded": { "depends_on_RH": false, "peer_reviewed": true, "verified_independently": "Relevant primary statements and proof dependencies checked; not full proof reconstruction" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "P3-DE74", "title": "La conjecture de Weil I", "authors": [ "Pierre Deligne" ], "year": 1974, "url": "https://www.numdam.org/item/PMIHES_1974__43__273_0.pdf", "exact_theorem": "1.5.1/1.5.4; Thms3.2,5.10,6.2; Lemma7.1; 7.3", "claim_used": "Trace-to-determinant and tensor/pole purity mechanism", "assumptions": "Smooth projective main theorem; local sheaf assumptions separately stated", "depends_on_RH": false, "peer_reviewed": true, "verified_independently": "Relevant primary statements and proof dependencies checked; not full proof reconstruction", "notes": "Source theorem status is distinguished from any unproved positivity conjecture in the research notes." } }, { "id": "phase3:P3-GR64", "source_ledger": "archive/literature/research/phase3_sources.json", "source_record_id": "P3-GR64", "title": "Formule de Lefschetz et rationalité des fonctions L", "authors": [ "Alexander Grothendieck" ], "year": 1966, "external_urls": [ "https://www.numdam.org/item/SB_1964-1966__9__41_0.pdf" ], "claim_used": "Cohomological trace/determinant spine", "exact_locator": "Bourbaki279, Thm5.1, §§3–6", "assumptions": "Finite type over finite field; constructible l-adic context; lecture Dec1964", "verification_as_recorded": { "depends_on_RH": false, "peer_reviewed": true, "verified_independently": "Relevant primary statements and proof dependencies checked; not full proof reconstruction" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "P3-GR64", "title": "Formule de Lefschetz et rationalité des fonctions L", "authors": [ "Alexander Grothendieck" ], "year": 1966, "url": "https://www.numdam.org/item/SB_1964-1966__9__41_0.pdf", "exact_theorem": "Bourbaki279, Thm5.1, §§3–6", "claim_used": "Cohomological trace/determinant spine", "assumptions": "Finite type over finite field; constructible l-adic context; lecture Dec1964", "depends_on_RH": false, "peer_reviewed": true, "verified_independently": "Relevant primary statements and proof dependencies checked; not full proof reconstruction", "notes": "Source theorem status is distinguished from any unproved positivity conjecture in the research notes." } }, { "id": "phase3:P3-MILNE", "source_ledger": "archive/literature/research/phase3_sources.json", "source_record_id": "P3-MILNE", "title": "Abelian Varieties", "authors": [ "James S. Milne" ], "year": null, "external_urls": [ "https://www.jmilne.org/math/CourseNotes/AV.pdf" ], "claim_used": "Rosati positivity and same arithmetic Frobenius adjoint identity", "exact_locator": "I §14 Thm14.3; II §1 Lemmas1.2–1.3", "assumptions": "Polarized abelian varieties; author textbook notes", "verification_as_recorded": { "depends_on_RH": false, "peer_reviewed": false, "verified_independently": "Relevant primary statements and proof dependencies checked; not full proof reconstruction" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "P3-MILNE", "title": "Abelian Varieties", "authors": [ "James S. Milne" ], "year": null, "url": "https://www.jmilne.org/math/CourseNotes/AV.pdf", "exact_theorem": "I §14 Thm14.3; II §1 Lemmas1.2–1.3", "claim_used": "Rosati positivity and same arithmetic Frobenius adjoint identity", "assumptions": "Polarized abelian varieties; author textbook notes", "depends_on_RH": false, "peer_reviewed": false, "verified_independently": "Relevant primary statements and proof dependencies checked; not full proof reconstruction", "notes": "Source theorem status is distinguished from any unproved positivity conjecture in the research notes." } }, { "id": "phase3:P3-OORT", "source_ledger": "archive/literature/research/phase3_sources.json", "source_record_id": "P3-OORT", "title": "Abelian varieties over finite fields", "authors": [ "Frans Oort" ], "year": null, "external_urls": [ "https://math.nyu.edu/~tschinke/books/finite-fields/final/05_oort.pdf" ], "claim_used": "Curve Jacobian positivity and Frobenius", "exact_locator": "§3 Props3.3–3.4", "assumptions": "The printed simple-case proposition separated from general FV argument", "verification_as_recorded": { "depends_on_RH": false, "peer_reviewed": false, "verified_independently": "Relevant primary statements and proof dependencies checked; not full proof reconstruction" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "P3-OORT", "title": "Abelian varieties over finite fields", "authors": [ "Frans Oort" ], "year": null, "url": "https://math.nyu.edu/~tschinke/books/finite-fields/final/05_oort.pdf", "exact_theorem": "§3 Props3.3–3.4", "claim_used": "Curve Jacobian positivity and Frobenius", "assumptions": "The printed simple-case proposition separated from general FV argument", "depends_on_RH": false, "peer_reviewed": false, "verified_independently": "Relevant primary statements and proof dependencies checked; not full proof reconstruction", "notes": "Source theorem status is distinguished from any unproved positivity conjecture in the research notes." } }, { "id": "phase3:P3-KED", "source_ledger": "archive/literature/research/phase3_sources.json", "source_record_id": "P3-KED", "title": "RH for abelian varieties", "authors": [ "Kiran S. Kedlaya" ], "year": null, "external_urls": [ "https://kskedlaya.org/weil-cohom/chapter-6.html" ], "claim_used": "Faithful Tate-module realization", "exact_locator": "6.0.4–6.0.5", "assumptions": "Author course notes", "verification_as_recorded": { "depends_on_RH": false, "peer_reviewed": false, "verified_independently": "Relevant primary statements and proof dependencies checked; not full proof reconstruction" }, "publication_editorial_status": "INHERITED_STATEMENT_RECORD_NOT_REVERIFIED_DURING_PUBLICATION", "original_record": { "id": "P3-KED", "title": "RH for abelian varieties", "authors": [ "Kiran S. 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