# Revised mathematics reference audit

**The combined evidence supports 95 of the 100 sampled work identities: 88 matches and seven same-work minor bibliographic differences. Five remain unresolved; no genuine wrong-publication mismatch has been demonstrated.** Seven identity upgrades rely on operator-supplied source-reading notes, not new independent retrieval. Popa's date discrepancy is also resolved using a supplied catalogue reading.

For the ten frozen citation-context checks, three narrow attributions are supported by the supplied notes, three have qualified or version-limited support, and four remain unverifiable. None of the original theorem texts was independently inspected in this session. These are attribution judgments, not mathematical proof verification.

## Final classifications

| Bibliography category | Previous direct-check audit | Revised combined evidence |
|---|---:|---:|
| Match | 80 | 88 |
| Minor same-work difference | 8 | 7 |
| Demonstrated genuine mismatch | 0 | 0 |
| Unresolved | 12 | 5 |
| Total | 100 | 100 |

| Claim category | Final count | Evidence meaning |
|---|---:|---|
| Supported narrow attribution | 3 | R045, R046, R095; supported by supplied source-reading notes |
| Qualified or version-limited support | 3 | R020, R056, R074; restrictions or unverified implications/editions remain |
| Unverifiable | 4 | R004, R030, R068, R097 |
| Contradicted | 0 demonstrated | No retrieved or supplied passage establishes contradiction |
| Total | 10 | Same frozen contexts as before |

![Final classifications](math-audit-final-classifications.png)

The panels have different denominators and scales: 100 bibliography entries and ten claim checks. A zero dot denotes no demonstrated genuine mismatch. All six favorable claim assessments depend on supplied notes. The exact counts above, chart and evidence tables were checked for agreement.

## Evidence provenance

No new external publication or PDF retrieval was performed for this revision. The prior direct Crossref, DataCite and arXiv records, and secondary OpenAlex evidence, are preserved unchanged.

The direct audit still accounts for 88 work identities: its original 80 matches and eight same-work differences. The revision adds seven identities based on supplied notes. Of the final 88 matches, 80 retain their direct-only basis, seven depend on supplied identity notes, and one—Popa—combines a direct publisher deposit with a supplied historical-catalogue reading. The seven remaining minor differences retain their direct-record basis.

The supplement describes external operator readings and factual paraphrases, not expert peer review. Every supplied URL, source-document SHA-256, retrieval date, version and locator is preserved in the JSON and relevant CSV rows. **The source-document hashes are supplied identifiers, not checksums independently recomputed against PDFs in this session.** Only the attached JSON's checksum was computed here. Third-party PDFs are not embedded in these deliverables.

Supplement file SHA-256: 7129463ec56a84dfe90edd13e71c70fc838757b05ebe2e200035715625f548c6.

The bibliography CSV separates prior/direct classification, final classification, classification basis, direct original records, secondary records and supplementary notes. The claim CSV separates earlier verdicts and direct-session locators from new supplied-note locators and versions. The JSON retains the earlier summary and reasons as well as revised judgments.

## Bibliography changes

| ID | Final result and supplied locator | Remaining limits |
|---|---|---|
| R005 | **Match via notes.** [Milne course notes](https://www.jmilne.org/math/CourseNotes/ANT.pdf), PDF p.1: J. S. Milne, Algebraic Number Theory, v3.08, July 19, 2020. | Title-page facts are operator-reported, not independently read here. |
| R007 | **Identity match via notes.** [Erdős source](https://users.renyi.hu/~p_erdos/1958-18.pdf), PDF p.1: author and Euler-phi title match despite imperfect OCR. | Exact year, venue and page range remain unverified. A filename containing 1958 is not publication-date evidence. Version unspecified. |
| R009 | **Identity match via notes.** [Erdős source](https://users.renyi.hu/~p_erdos/1945-08.pdf), PDF p.1: author and title concerning Euler's phi function and related problems match. | February 9, 1945 is a received date, not a verified publication date or pagination. Version unspecified. |
| R058 | **Author-manuscript identity match via notes.** [Durrett manuscript](https://sites.math.duke.edu/~rtd/PTE/PTE5_011119.pdf), PDF p.1: Probability: Theory and Examples, Version 5, January 11, 2019. | Does not independently establish every detail of the commercially published fifth edition. |
| R076 | **Dissertation identity match via notes.** [Sørensen dissertation](https://di.ku.dk/forskning/Publikationer/tekniske_rapporter/tekniske-rapporter-1997/97-27.pdf), PDF pp.1–2: author, title, dissertation and University of Copenhagen. | The 1997/1998 distinction remains unresolved; source version unspecified. |
| R082 | **Match; historical date resolved via notes.** [Numdam item/BibTeX](https://www.numdam.org/item/AST_1995__232__187_0/): Popa, Astérisque 232 (1995), 187–202. | The direct Crossref deposit's later 2018 date is preserved. The supplied catalogue reading supports the cited 1995 historical issue; it is not a new direct DOI check. |
| R095 | **Preprint identity match via notes.** [Miyamoto RIMS1591](https://www.kurims.kyoto-u.ac.jp/preprint/file/RIMS1591.pdf), PDF pp.1–2: author and nearly circular planar convex domains title. | Publication date and the bibliography's specific revision date remain unverified. |
| R098 | **Identity and listed bibliographic fields supported via catalogue notes.** [EuDML 171213](https://eudml.org/doc/171213): Weck, title, Mathematische Zeitschrift 111 (1969), 387–398. | EuDML is a digital-library catalogue, not a publisher-deposited DOI record; this catalogue reading was externally supplied. |

An identity match is not certification of every field. Unresolved dates, editions or page ranges above remain explicit even when title/author identity is supported.

## Remaining minor differences

These are same-work record differences, not seven demonstrated citation errors. No new source notes resolve them.

| ID | Work | Preserved observation |
|---|---|---|
| R012 | [Milnor](https://doi.org/10.1515/9781400881819) | Cited 1968 versus publisher-record 1969-12-31 for the discovered edition; edition/date reconciliation unresolved. |
| R017 | [Teleman](https://doi.org/10.1007/s00222-011-0352-5) | Cited 2012 versus Crossref 2011-09-24, consistent with possible online-first/print dating. |
| R046 | [Jamneshan](https://doi.org/10.1017/etds.2022.43) | Cited 2023 versus Crossref 2022-06-21; supplied v4 theorem notes do not settle journal dating. |
| R052 | [Green–Tao](https://arxiv.org/abs/1002.2028v3) | Cited title has “an associated counting lemma”; arXiv omits “an.” Chapter details remain unconfirmed. |
| R078 | [Barthe–Coquand](https://doi.org/10.1017/S0956796803004726) | Cited 2006 versus Crossref 2003-09-16; same work, but print-date reconciliation unresolved. |
| R088 | [Hamilton](https://doi.org/10.4310/sdg.1993.v2.n1.a2) | Citation has “formation,” 1995; Crossref “formations,” 1993. Same work; historical dating unresolved. |
| R099 | [Nakamura–Uhlmann](https://doi.org/10.1007/s00222-002-0276-1) | Citation prefixes “Erratum:”; Crossref omits the label. Correction scope not assessed. |

## Five unresolved identities

| ID | Reference | Why it remains unresolved |
|---|---|---|
| R043 | Hutchinson, Fractals and self-similarity (1981) | DOI, author and year are corroborated, but the retrieved title fields are empty. No new note addresses it. |
| R050 | Derrien–Lesigne, pointwise polynomial ergodic theorem (1996) | No matching original publication record was retrieved; no supplementary note. |
| R057 | Wang, On a Conjecture of O'Donnell, ePrint 2020/002 | Original report was inaccessible; no supplementary note. |
| R069 | OpenAI, Sharp mass bounds for the two-dimensional O(4) model | Independent repository access was blocked; no supplementary note. |
| R072 | OpenAI, An explicit exact Hausdorff gauge for SLE | Independent repository access was blocked; no supplementary note. |

“Unresolved” describes evidence availability. It is not a finding of fabrication.

## Ten claim assessments

The judgments apply to the original frozen contexts. “Supported attribution” means the supplied paraphrase describes the specific source statement being attributed; it does not establish proof correctness or every subsequent inference. All versions and page numbers below are as supplied.

| ID | Revised judgment | Source reading, scope and outstanding limits |
|---|---|---|
| R020 | **Qualified support** | Iritani–Koto [arXiv:2307.03696v4](https://arxiv.org/abs/2307.03696v4), PDF pp.2,4; Theorem 1.1, Remarks 1.2/1.4/1.9, Corollary 1.8. Notes support the mirror construction, genus-zero scope and formal generic-semisimplicity equivalence. Theorem 1.1 uses rank at least two over a smooth projective base and globally generated dual; Remark 1.2 removes the dual-generation restriction by twisting without changing projectivization. Formal semisimplicity is not convergence. The downstream Virasoro consequence remains unverified. |
| R045 | **Supported narrow attribution via notes** | Bárány–Verma [arXiv:2507.05835v2](https://arxiv.org/abs/2507.05835v2), PDF pp.8,10–11. Notes state that weak exponential separation permits exact overlaps and Theorem 3.5 gives min(1, random-walk entropy/Lyapunov exponent), with general contraction ratios distinguished from the earlier homogeneous case. This supports the sampled statement, not its proof. |
| R046 | **Supported narrow attribution via notes** | Jamneshan [arXiv:2103.17167v4](https://arxiv.org/abs/2103.17167v4), PDF p.2, Theorem 1.1. Any group/probability algebra dynamical system; no ergodicity assumption, transfinite relatively compact tower and weakly mixing extension at the top. This supports the nonergodic structure-theory attribution. Adjacent joining, rational-time and pointwise-average assertions remain unverified and are not all attributed to this source. |
| R056 | **Version-limited support** | O'Donnell–Servedio [author draft](https://www.cs.columbia.edu/~rocco/Public/os-ccc.pdf), dated December 3, 2005; PDF pp.6–7, printed pp.5–6, Lemma 3 and Theorem 3. Notes support the signed singleton Fourier-sum bound by square root of expected fixed coordinates, then square root of log partition size. Interpreting the signed sum as total influence requires monotonicity. The final 2007 journal text and preservation of numbering there remain unverified. |
| R074 | **Qualified, version-limited support** | Zhan arXiv:1509.05015v3, document dated August 26, 2018; PDF p.13, Theorem 4.1 and equations 4.3–4.8. Notes support the identity for 0<κ<8 and conditional remaining-mass/killing-measure equations 4.3–4.5. The supplied URL is unversioned; the version assertion and hash are retained as supplied. The citing bibliography explicitly uses an author-hosted published article for numbering; equivalence to that edition, Proposition 2.2/equation (2.1), and subsequent past-mass derivations remain unverified. |
| R095 | **Supported narrow attribution via notes** | Miyamoto RIMS1591, PDF pp.1–2, Theorem A, Lemma B, Lemma 1.2. Notes identify an open bounded smooth domain and nonzero second Neumann eigenfunction, a strict diameter-squared/area bound near 1.378, and strict sqrt(λ₂)<j₀/diameter. Lemma 1.2 excludes interior critical points under those hypotheses, and symmetry is expressly unnecessary. Strict inequalities must be retained; no publication date or proof validity is inferred. |
| R004 | **Unverifiable** | Stark Lemmas 3 and 8: no supplied theorem passage. Zero-region constants, quadratic-subfield implication and nonnormal-field applicability remain unchecked. |
| R030 | **Unverifiable** | Kasteleyn: no supplied passage establishing polynomial-time exact counting of planar perfect matchings. Bibliographic identity is not claim verification. |
| R068 | **Unverifiable** | Osterwalder–Schrader Section IV.1: no supplied passage for the corrected reconstruction/linear-growth-condition attribution. |
| R097 | **Unverifiable** | Eskin/Cekić methodological relationship: no supplied passage. The claim also involves the separately cited Cekić paper. |

## Sampling and reproducibility

The original sample is unchanged: five entries per each of 20 papers from a source-reconstructed frame of 409 cited entries, after excluding eight uncited BibTeX database entries. All 221 embedded source checksums were verified in the original audit. No PDFs were compiled to establish the rendered bibliography.

Original input SHA-256: ec9c84e53baaff4cb83c9a33d85380bf2d95bc3b8ad5b467e89b19212127b8ba.
Supplied repository: https://github.com/openai/math; commit fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb. Repository provenance was not independently verified.

Selection ranks are SHA-256 of 20261008|bibliography|family_id|citation_key, taking the five lowest per family. Claim families use 20261008|claim-family|family_id; claim selection within those families uses 20261008|claim|family_id|citation_key. The first citation occurrence in supplied source-record order is retained. The existing reproduce_math_sample.py remains valid; the evidence JSON preserves the manifest and original reproducibility check.

The revised IDs, direct records, 100-row reference table, ten-row claim table and chart totals were checked for consistency. No new sampling was performed. Balanced five-per-paper percentages are not entry-weighted estimates for the frame or estimates for the whole repository.

Crossref's retrieved projection lacks volume/issue/pages and separate online/print dates; OpenAlex locators remain secondary. Original abstracts and externally supplied source readings answer different questions. The revised results strengthen identity and attribution evidence while leaving inaccessible editions, unspecified dates, uninspected implications and proof validity outside the verified scope.
