{ "_public_export": { "status": "RIEMANN HYPOTHESIS OPEN", "record_type": "SANITIZED_HISTORICAL_SOURCE_SNAPSHOT", "source_path": "research/structural_sources.json", "source_sha256": "c08411033fe108382a3155782ce343bdbe5ea73ee7c8a460bdbc3c4ac62ea60e", "exported_on": "2026-09-30", "public_credit": "@ykbballer91", "license": "CC-BY-4.0", "ai_assisted": true, "original_fields_are_historical": true, "checks_rerun_in_public_export": false, "scope": "Historical claims and verification records retain their original scope. Priority 1 cyclicity is even L2 only; full-ground capture and RH remain open.", "references": { "literature/notes/spiral-scale-prior-art.md": { "public_path": "archive/literature/literature/notes/spiral-scale-prior-art.md", "status": "EXPORTED" }, "literature/notes/support-propagation-prior-art.md": { "public_path": "archive/literature/literature/notes/support-propagation-prior-art.md", "status": "EXPORTED" }, "proofs/audits/groskin-computation.md": { "public_path": "archive/audits/proofs/audits/groskin-computation.md", "status": "EXPORTED" }, "research/notes/accumulation_spectrum.md": { "public_path": "archive/reports/research/notes/accumulation_spectrum.md", "status": "EXPORTED" }, "research/notes/center_scale.md": { "public_path": "archive/reports/research/notes/center_scale.md", "status": "EXPORTED" }, "research/notes/prime_jump_obstruction.md": { "public_path": "archive/reports/research/notes/prime_jump_obstruction.md", "status": "EXPORTED" }, "research/notes/variational_closure.md": { "public_path": "archive/reports/research/notes/variational_closure.md", "status": "EXPORTED" } } }, "checked_at": "2026-09-29", "scope": [ "research/notes/center_scale.md", "research/notes/accumulation_spectrum.md", "research/notes/variational_closure.md", "research/notes/prime_jump_obstruction.md", "literature/notes/support-propagation-prior-art.md", "literature/notes/spiral-scale-prior-art.md" ], "exhaustive": false, "classification_note": "RH?=NO は登録したstatementがRHを前提としないという意味。未証明条件付きの含意・RHとの同値定理を含み、前件の真を認証しない。independently verified?=NO は原文照合・査読掲載と全文の独立証明検証を区別する。物理添付は除外。", "records": [ { "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": "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": "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": "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": "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": "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": "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": "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": "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": "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": "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": "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": "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": "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": "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": "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": "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": "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": "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": "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": "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": "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": "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": "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": "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": "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": "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": "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": "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": "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": "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": "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": "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": "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。" } ] }