{ "_public_export": { "status": "RIEMANN HYPOTHESIS OPEN", "record_type": "SANITIZED_HISTORICAL_SOURCE_SNAPSHOT", "source_path": "research/phase2_sources.json", "source_sha256": "f186ab603ae04820c5d37691d68fd7f7b6d2c5a0a5ecefb73c89e348cfdbf06e", "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": {} }, "scope": "Phase II targeted primary-source verification after internal construction; not exhaustive literature search", "novelty_claimed": false, "records": [ { "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": "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": "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": "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": "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": "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." } ] }