Phase 3 citation and attribution review
STATUS: RIEMANN HYPOTHESIS OPEN
Review date: 2026-09-30. This is a bounded internal AI-agent publication review, not external peer review, a reconstruction of all cited proofs, or legal clearance.
Decision: no material citation blocker was found in the checked scope. The recent Connes-related inputs, local-RH distinctions, cyclicity sources and Priority 2 interpolation sources exist and support the limited uses recorded below. Conditional theorems have not been converted into an unconditional RH conclusion in the inspected public narratives. Retrieval limitations and historical metadata differences remain visible.
Scope and reproducible inventory
The review inspected docs/references.md, all 19 track pages, the critical linked source notes, docs/licensing.md, LICENSE-TEXT, LICENSE-CODE and CITATION.cff. The nineteenth page was added during review and is included in the final snapshot. The original research repository was not edited.
The citation inventory records 22 selected work/reference groups, exact source URLs, locators, the depth of each check and hashes of the inspected public track/reference/licensing files. Primary sources were read through the web tool on the review date. Publisher metadata, an author's manuscript, an abstract and a theorem passage are explicitly different levels of evidence.
Literal link discovery across 232 public Markdown files, together with the explicit URLs in data/references.json, found 402 distinct HTTP(S) URL tokens and 1,201 occurrences. The inherited bibliography has 111 statement records and 385 discovery entries. These numbers are a snapshot, not the count of verified mathematical works. Generated site files, audit files and Git internals were excluded from link discovery. Exact URL aliases such as arXiv PDF/abstract/HTML are not silently combined. URLs outside the selected checks are marked NOT_CHECKED_IN_THIS_REVIEW; no HTTP success code is inferred from a web-tool response.
The track pages mostly link to project reports rather than directly to external papers. The citation check therefore followed their critical report/source-note chains. Historical proof-claim records were not promoted to accepted inputs or exhaustively re-audited.
Critical primary-source checks
| Source and identity | Primary passage checked | Publication-use decision |
|---|---|---|
| Alain Connes and Walter D. van Suijlekom, Quadratic Forms, Real Zeros and Echoes of the Spectral Action (2025), arXiv:2511.23257v1; CMP 406, article 312, DOI 10.1007/s00220-025-05493-1. | Theorem 6.1; publisher metadata and abstract. The theorem uses the specified distribution form on trigonometric polynomials, lower-bounded essential selfadjointness and a simple isolated minimum with even eigenfunction. | Track 14 correctly keeps the actual Weil operator's evenness/simplicity obligation separate. The publisher confirms 18 November 2025 publication and a referee acknowledgement. The arXiv upload date is later and is not the journal publication date. |
| Alain Connes, Caterina Consani and Henri Moscovici, Zeta Spectral Triples, arXiv:2511.22755v1, 27 November 2025. | Lemmas 7.2–7.3 and §8. EMS contents also identify the chapter, authors and pp.39–75. | Known prolate/proxy estimates do not supply the two missing steps: actual even-simple ground and adequate comparison with that ground. The public normalized-comparison statement remains open. The project's separately recorded scalar convention is not presented as an extra theorem of the source. |
| Connes, Consani and Moscovici, Zeta zeros and prolate wave operators: Semilocal adelic operators, arXiv:2310.18423v2, 4 May 2024. | Theorem 4.6, Remark 4.2 and §4.8. | The finite-set Sonin-space isomorphism is a valid input in its stated spaces. It does not assert the uniform global ground comparison needed here. The main title on the abstract and subtitle in the body explain the slightly different recorded titles. |
| Connes and Consani, Spectral triples and ζ-cycles, publisher record, Enseign. Math. 69 (2023), 93–148, DOI 10.4171/LEM/1049. | Publisher identity and abstract/context were rechecked. | The known arithmetic/prolate framework is properly attributed. This publication review does not newly reconstruct the exact radical proof; the earlier scoped research audit remains the record for that derivation. |
| Daniel Bump, Kwok-Kwong Choi, Pär Kurlberg and Jeffrey Vaaler, A Local Riemann Hypothesis, I. Author-hosted text; author bibliography. | Theorem 1, printed p.3. | Hermite Mellin-polynomial zero location is a local statement. The available PDF is a preliminary 1996 manuscript; the bibliography identifies Mathematische Zeitschrift 233(1) (2000), 1–19. These dates/versions are not conflated. |
| Pär Kurlberg, A Local Riemann Hypothesis II, author-hosted text, Mathematische Zeitschrift 233(1) (2000), 21–37. | Theorem 4, printed p.14, and §5. | The quoted nonarchimedean theorem has odd residue characteristic. Complex-place failure is a separate result. Neither is a global theorem about all zeta zeros. |
| Bruno Sauvalle, On weak Mellin transforms, second degree characters and the Riemann hypothesis, publisher PDF, Acta Arith. 177.3 (2017), 219–275, DOI 10.4064/aa8240-7-2016. | First-page metadata and Theorem 4.6, printed pp.254–255. | The factorizable nondegenerate second-degree-character setting and unitary Hecke character are essential. Global weak Mellin zeros retain the Hecke-L zero contribution as well as the local factors. The linked project note does not treat local real-zero theorems as a proof of global RH. |
| Marcel de Jeu, Determinate multidimensional measures, the extended Carleman theorem and quasi-analytic weights, arXiv:math/0111019v2, revised 17 June 2002; Ann. Probab. 31 (2003), 1205–1227. | Theorem 2.3, p.5, equation (2.1); Theorem 1.1. | The finite positive all-moment measure and Carleman condition give polynomial density in Lp. The project's even projection and weighted multiplication are separate steps. Track 18 does not infer a form core, growing-order estimate or full-ground capture. |
| NIST DLMF 5.11.9, 25.9.3, 25.4.4. | Official Gamma, approximate-functional-equation and completion locators. | Standard unconditional inputs for Xi decay; no RH hypothesis is being added. These are reference formulas, not newly published 2026 results. |
| Aleksander Simonič and Valeriia V. Starichkova, Atkinson's formula for the mean square of ζ(s) with an explicit error term, arXiv:2105.06821v3, revised 13 December 2022; J. Number Theory 244 (2023), 111–168. | Theorem 3, §7, including large-height hypotheses. | The moment refinement is supplementary. The unconditional cyclicity conclusion is not represented as depending on this more precise asymptotic. |
The older selected entry points—Connes's 1998 trace-formula preprint, Connes–Consani–Marcolli's 2007 arithmetic quotient paper, Deligne's 1974 Weil I, Gillet–Soulé's 1990 Arithmetic intersection theory, Milne's 2008 course notes, Rodgers–Tao v5 and Burnol v6—were checked at the metadata/context or selected-statement depth specified in the JSON. This is deliberately weaker than verifying all their proofs. No arithmetic-purity or global positivity theorem was inferred merely from their titles or geometric terminology.
Priority 2: conditioning attribution
Four entries were added only to the authorized finite-head subsection of references. The inherited statement records were left unchanged.
| Source | Exact use and boundary |
|---|---|
| Walter Gautschi, Norm Estimates for Inverses of Vandermonde Matrices, Numer. Math. 23 (1975), 337–347. | §3, p.339, (3.3), and §4, p.340, (4.1). Cardinal inverse and norm bound, with the transpose/norm convention explicitly recorded. The project supplies its square-node and Xi-weight application. |
| Pablo D. Brubeck, Yuji Nakatsukasa and Lloyd N. Trefethen, Vandermonde with Arnoldi, SIAM Review 63(2) (2021), 405–415, DOI 10.1137/19M130100X. | §4, p.409, (4.2)–(4.5), explains QR/orthogonal coordinates for the same polynomial. This is not an automatic physical-Gram stability result. The DOI and publication details are printed in the author-hosted publisher PDF. |
| DLMF §18.2. | Discrete orthogonality and Christoffel–Darboux conventions. Changing the metric or normalizing a head Gram cannot silently remove the physical lift cost. |
| Albert Cohen, Mark A. Davenport and Dany Leviatan, On the stability and accuracy of least squares approximations, arXiv:1111.4422v3. | Theorem 1, p.3, (1.2), assumes independent samples from its prescribed measure. It is not applied to the fixed Xi-weighted grid. The current unversioned source resolves to v3, revised 15 June 2018; the new English citation fixes that version. Other deterministic results in the paper are not claimed to be absent. |
The latest public narrative correctly separates raw-coordinate rank certificates from physical-Gram singular values. The source attribution supplies known tools, not the project's actual sharp-tail constants or finite interval certificate. No growing-order or RH conclusion is attached to this bibliographic check.
Nonblocking limitations and follow-up classification
- Weil's CERN entry: the cited record returned a bot-protection page. Its primary contents were not verified in this review. Do not label that specific link as freshly validated. This is an access limitation, not evidence that the citation is false; the precise criterion has other source records.
- Resolver access versus identity: Sauvalle's DOI/arXiv route and the DOI resolver for the Arnoldi paper did not return usable text through this tool. Publisher/author-hosted full texts did, and their headers confirm the bibliographic identities. The Connes–van Suijlekom DOI resolver also failed, but the direct Springer article succeeded. These are not established broken links.
- Version pinning: Burnol v6 is older than the listed v7; it is explicitly cited as v6 and is not called the latest version. Historical notes are not silently upgraded. Cohen–Davenport–Leviatan's old note remains candid about its then-unversioned URL; the new English citation is fixed to v3.
- Historical publication states: records for Zeta Spectral Triples include a preprint label and a later published-chapter/review-unconfirmed label. The publisher contents confirm the chapter; they do not prove individual peer review. Preserve the dated statement records rather than turning every publication string into a Boolean. These differences do not alter the missing-step conclusion.
- Inventory is not universal validation: discovery entries and old proof-claim citations are not accepted theorems. The 402-URL list includes unchecked historical and ancillary links. No exhaustive availability, metadata or proof claim is made.
None of these findings requires a guessed title, author, date or DOI replacement. The review found no material conflict between the checked primary statements and the scopes expressed in the current English track pages.
Licensing and citation-file consistency
The publication uses separate LICENSE-TEXT (CC BY 4.0) and LICENSE-CODE (MIT); absence of a single file named LICENSE is not a contradiction. The licensing page and CITATION.cff consistently use the authorized public display @ykbballer91, version 0.1.0, and both licenses. Cited scholars retain their scholarly names. The citation file deliberately omits date-released and describes a publication-review draft; its repository URL is not evidence that a release has occurred.
The inspected distribution contains no bundled third-party PDF. External links and bibliographic metadata do not transfer third-party copyright. This check did not clear every inherited quotation, establish ownership of every artifact or validate every CFF field against its schema; those are outside the bounded citation check. The visible policy applies licenses only to original material to the extent rights are held and excludes downloaded papers and dependency trees.
The final decision remains limited: the checked citations support publishing this record as an openly unfinished research log. They do not certify RH, the entire research archive, novelty, or external peer review.