References and verification scope
STATUS: RIEMANN HYPOTHESIS OPEN
This publication credits @ykbballer91 for the project record. The authors of cited mathematical works retain their scholarly attribution. A citation is evidence for a particular statement under particular assumptions; it is not an endorsement of every claim in a paper.
The machine-readable reference uses preserve 108 statement-level records from six historical ledgers and add three uses from the dated cyclicity notes. The same work can appear more than once when a theorem, a conditional criterion and a heuristic discussion have different dependencies. Later prose citations also have an external-link discovery index with source locations. That index is not a list of verified theorems, and this is not an exhaustive bibliography.
Public primary sources for L0–L7 and their scope
These fixed versions support the 2026-10-09 update. Reading and comparison concern selected statements, not full reproofs or an exhaustive academic-novelty survey.
| Source and version | Checked statement and use |
|---|---|
| Jeffrey C. Lagarias, An Elementary Problem Equivalent to the Riemann Hypothesis, author version July 29, 2001. Primary author PDF, publication listing | Problem E / Theorem 1.1, RH equivalence. It does not supply the fixed finite rational comparison, L7 precision function or proof cost |
| Y. Choie, N. Lichiardopol, P. Moree, P. Solé, On Robin’s criterion for the Riemann hypothesis, JTNB 19 (2007), 357–372. Publication page, primary PDF | Section 5, Proposition 5.1, Lemmas 5.2–5.5: extremization by prime replacement and exponent ordering. Do not identify its family with L7's C without exponent ordering, or its comparison with fixed Lagarias |
| Andrew MacArevey, On the Lagarias Inequality and Superabundant Numbers, arXiv:2602.15905v2, revised 2026-02-19. Primary HTML | Section 2, Lemmas 2.1–2.7, Proposition 2.1, Corollary 2.1; Section 3, Theorem 3.1. Monotonicity of the normalized value and superabundance of a least counterexample. Fixed finite q and L7 precision/proof costs are separate |
| OpenAI, The Quasi-Riemann Hypothesis, 30 September 2026. 7/8 main manuscript, official Lean nonvanishing theorem | Official revision fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb. Main theorem, selected proof chain and settings checked. Scoped adoption as an external theorem retains strict Re(s)>7/8 and s≠1 in ordinary analysis |
| OpenAI, The Quasi-Riemann Hypothesis, October 5, 2026. 11/12 separate manuscript | Same fixed official revision. Local checks of mean-to-pointwise extraction, Mellin nonvanishing and selected cancellation/descent. Faithful Lean translation of the entire separate paper is not certified |
| Dave Goldblatt, external rechecking REPORT, raw logs | Revision 3f11b2203b76902f1b6b234115437302a94b79c3. External Lean/nanoda success logs and settings compared with target sources. Our own kernel/comparator not run; no external peer-review certification |
The external papers and formal proofs belong to their respective authors. They are not counted as our original proof of RH. Third-party full texts, libraries and caches are not redistributed. Methodology and reproduction limits explain the trust boundary.
How to read a verification status
- Publication status concerns the journal, preprint or reference source. It does not state that this project has verified the proof.
- Primary text checked means the indicated statement or passage was read. It does not mean every lemma in the paper was reconstructed.
- Scoped independent check describes a separate internal AI-agent derivation, audit or computation of a specified claim. It is not external peer review and does not certify unrelated claims.
- Conditional / RH-dependent is attached to the statement used, not flattened into a judgment about a whole paper. A valid conditional theorem remains conditional when applied here.
- NO, PARTIAL, booleans and longer scope notes are retained in their original types and wording under
verification_as_recorded. Missing fields remain missing or null. Text is never converted to a Boolean on the basis of whether it is nonempty.
The inherited bibliography includes bounded audits of proof claims and exploratory analogies. Such entries are historical context, not accepted RH inputs. The track reports state what survived each audit. No source was newly promoted to “independently verified” during publication editing.
Selected mathematical entry points
| Topic | Primary or official source | Exact use and boundary |
|---|---|---|
| Weil criterion and explicit formula | Weil, 1952; Connes, trace formula | Positivity criterion and arithmetic trace framework; the global positivity hypothesis must still be supplied. |
| Arithmetic quotient and Frobenius comparison | Connes–Consani–Marcolli, math/0703392v1; Deligne, Weil I | Exact domains, traces and multiplicities; finite-field purity is not automatically a number-field theorem. |
| Polarization and heights | Gillet–Soulé; Milne, Abelian Varieties | The positive structures are used only on their stated geometric spaces. No identity with the full zeta-zero space is inferred. |
| Heat deformation | Rodgers–Tao, 1801.05914v5 | Nonnegativity of the de Bruijn–Newman constant; this does not prove the missing upper bound at time zero. |
| Localized Weil form and arithmetic radical | Connes–Consani, Spectral triples and ζ-cycles | The known arithmetic radical is explicitly attributed; it does not by itself select a unique finite ground state. |
| Real-zero theorem and prolate proxy | Connes–van Suijlekom, 2511.23257v1; Connes–Consani–Moscovici, 2511.22755v1 | Conditional ground-state hypotheses, known proxy convergence, and the missing normalized comparison are separate in the local-to-global theorem stack. |
| Finite structure used in CORE-S | Connes–Consani–Moscovici, 2511.22755v1, §5.1–5.2, Lemmas 5.1–5.2 | The divided-difference and commutator identities are known inputs. The derived simplicity result does not prove eventual cofinal parity ordering. |
| Sonin spaces | Burnol, math/0203120v6 | Completion and minimality results in their stated spaces; the hierarchy audit does not substitute this chain for the actual projected Weil energy. |
| Moment density | de Jeu, math/0111019v2, Theorem 2.3, p.5 | Carleman implies polynomial L² density for the finite positive measure used in cyclicity. Theorem 5.1's Stieltjes statement is kept distinct. |
| Exponential decay of Xi | DLMF 5.11.9, 25.9.3, 25.4.4 | Standard Gamma decay, an unconditional zeta estimate and completion conventions. No RH input. |
| Supplementary moments | Simonič–Starichkova, 2105.06821v3 | A mean-square error estimate used for a moment asymptotic; the cyclicity theorem does not depend on that refinement. |
Finite-head interpolation and conditioning
The Priority 2 update uses the following established tools. These prose citations supplement the historical statement ledger; they do not change its recorded verification statuses.
| Source | Passage checked and scope |
|---|---|
| Walter Gautschi, Norm Estimates for Inverses of Vandermonde Matrices, Numerische Mathematik 23 (1975), 337–347. Author-hosted paper. | §3, p.339, equation (3.3), gives the cardinal-polynomial inverse; §4, p.340, equation (4.1), gives its norm bound. The paper places nodes in columns, so its infinity-norm formula becomes a 1-norm formula for the transposed convention used here. |
| Pablo D. Brubeck, Yuji Nakatsukasa and Lloyd N. Trefethen, Vandermonde with Arnoldi, SIAM Review 63(2) (2021), 405–415. Author-hosted paper, DOI:10.1137/19M130100X. | §4, p.409, equations (4.2)–(4.5), relate monomial and discrete orthogonal coordinates through QR. The polynomial span is preserved; this does not by itself control the physical function norm or the sharp-support perturbation. |
| NIST Digital Library of Mathematical Functions, §18.2.3, §18.2.5, §18.2.12–13. | Discrete orthogonality, its norm, and the Christoffel–Darboux kernel. Normalizing the head Gram does not identify it with the global or support-restricted physical Gram. |
| Albert Cohen, Mark A. Davenport and Dany Leviatan, On the stability and accuracy of least squares approximations, arXiv:1111.4422v3, revised 15 June 2018; first submitted 2011. | Theorem 1, p.3, equation (1.2), is a probabilistic Gram estimate with independent samples from the prescribed measure. That theorem is not a certificate for this project's deterministic Xi-weighted Fourier grid. Other sampling results in the paper are not being ruled out by this applicability check. |
The conditioning note records the exact matrix conventions and the distinction between raw coefficients, head norm and physical norm. Reading these passages is a scoped citation check, not a new verification of every proof in the cited works.
Where the exact statement-level records live
- Initial theorem and computation ledger.
- Weil frontier source notes.
- Structural heuristic source uses.
- Generator and boundary sources.
- Frobenius comparison sources.
- Polarization and intersection sources.
For later tracks, the canonical report and its supporting note provide theorem numbers, hypotheses, versions and failed applicability checks. Particularly relevant are the local-to-global stack, the Sonin audit and the cyclicity source note.
Rights and retrieval
Only project commentary and bibliographic metadata are reproduced here. Third-party full texts, cached PDFs/HTML/TeX and ancillary software are excluded. External links point to the relevant publisher, repository or official reference; their content and rights are outside this publication. Source retrieval dates and versions are preserved where the research recorded them. No unrecorded access date, peer-review outcome or new author identity is supplied.