# Mathematics reference audit

**Of 100 sampled bibliography entries, 88 could be matched to real works in original publication or repository records: 80 without a notable identity discrepancy and eight with minor, same-work bibliographic differences. No genuine wrong-publication mismatch was demonstrated. Twelve entries remain unresolved.** This is an identity audit with incomplete field coverage, not a finding that every bibliographic detail is correct.

For ten citation-context checks, original abstracts provided partial support in three cases; seven exact claims remained unverifiable. No claim received full-text confirmation. These results do not validate the papers’ proofs.

## Results

| Bibliography classification | Entries | Meaning |
|---|---:|---|
| Match | 80 | Original record supports work identity; ordinary formatting normalized |
| Minor difference | 8 | Same work, with a noticed date, wording, or label difference |
| Genuine mismatch | 0 demonstrated | No retrieved evidence established a wrong work or materially incompatible identifier |
| Unverified | 12 | Insufficient original-record evidence; not evidence of fabrication |
| **Total** | **100** | Five entries from each of 20 supplied papers |

The eight differences are **not eight proven citation errors**. Publisher records can use online-publication, edition, or digitization dates that differ from the date in a bibliography. The ledger preserves disagreements without automatically blaming the citing paper.

The CSV contains all 100 raw bibliography entries, classifications, publication-record evidence, source-file locations, and the ten claim assessments. The JSON also preserves source contexts, original metadata snapshots, bounded search results, access failures, and the sampling manifest.

## Sampling and scope

Input: sampled-math-papers.json. SHA-256: **ec9c84e53baaff4cb83c9a33d85380bf2d95bc3b8ad5b467e89b19212127b8ba**.

The attachment states repository https://github.com/openai/math and commit fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb. Those provenance statements come from the attachment; independent repository retrieval was blocked. All 221 embedded source-record checksums matched their supplied contents.

Bibliographies were reconstructed from LaTeX/BibTeX grouped by the 20 representative-paper directories. Citation commands were collected after removing LaTeX comments. Eight uncited BibTeX database entries were excluded. The frame contains **409 cited bibliography entries**; all included explicit bibitem entries were cited. No paper had duplicate bibliography keys. The PDFs were not compiled, so their rendered bibliographies were not independently validated.

Within each paper, rank entries by the hexadecimal SHA-256 digest of **20261008|bibliography|family_id|citation_key**, then select the five lowest ranks without replacement. Papers are ordered by family ID to assign R001–R100. The same work cited in different papers remains a distinct citation occurrence.

For claim checks, select the ten families with lowest hashes of **20261008|claim-family|family_id**. Within each, select the sampled cited entry with the lowest hash of **20261008|claim|family_id|citation_key**. Inspect its first occurrence in supplied source-record order, with up to 650 characters on each side. That occurrence need not be the first in the compiled paper. Offsets are Unicode character positions in comment-stripped source, not PDF pages.

The standalone script reproduced the exact 100 selected raw entries and all ten claim IDs. Equal allocation gives balanced paper coverage; these percentages are not weighted estimates for all 409 entries or the entire repository.

## Publication evidence

Primary evidence comprised **publisher-deposited Crossref records, original DataCite deposits, and official arXiv metadata**. OpenAlex supported discovery and secondary corroboration; it did not substitute for an original record.

For example, [Einsiedler–Katok–Lindenstrauss](https://doi.org/10.4007/annals.2006.164.513) and [Khayutin](https://doi.org/10.1215/00127094-2019-0016) have matching publisher-deposited titles and authors. Recent [Lin–Slavíková](https://arxiv.org/abs/2607.20206v1) and [Chang–Slote–Volberg–Zhang](https://arxiv.org/abs/2604.08095v2) preprints returned matching original arXiv records. Recent dates alone were not treated as suspicious.

Three supplied DOIs returned Crossref 404s but resolved in DataCite: [Cai–Liu](https://doi.org/10.4230/LIPIcs.ICALP.2020.23), [Poll](https://doi.org/10.6100/IR423044), and [Mull](https://doi.org/10.4230/LIPIcs.TYPES.2022.7). A lookup failure in one registry therefore did not become a mismatch.

The Crossref connector omits volume, issue, pagination, and separate online/print dates. OpenAlex sometimes supplied locators, marked as secondary. Missing fields were not assumed correct. ArXiv can establish work identity without independently confirming attached journal or chapter details.

## Minor differences

| ID | Work and original record | Observed difference |
|---|---|---|
| R012 | [Milnor, Singular Points of Complex Hypersurfaces](https://doi.org/10.1515/9781400881819) | Citation: 1968. Retrieved publisher record for the discovered edition: 1969-12-31. Same author, title, publisher and AM-61 designation; exact edition/date unresolved. |
| R017 | [Teleman, 2D semi-simple field theories](https://doi.org/10.1007/s00222-011-0352-5) | Citation: 2012. Crossref: 2011-09-24. Compatible with online-first versus print dating; both date types were not returned. |
| R046 | [Jamneshan, Furstenberg–Zimmer structure theory](https://doi.org/10.1017/etds.2022.43) | Citation: 2023. Crossref: 2022-06-21. Same work and DOI; revised arXiv v4 also matches. |
| R052 | [Green–Tao, arithmetic regularity lemma](https://arxiv.org/abs/1002.2028v3) | Citation says “an associated counting lemma”; arXiv omits “an.” Same work/authors. Chapter details were not independently established. |
| R078 | [Barthe–Coquand, non-normalizing pure type systems](https://doi.org/10.1017/S0956796803004726) | Citation: 2006. Crossref: 2003-09-16. Same work; secondary volume/pages agree. Actual print-date reconciliation unresolved. |
| R082 | [Popa, free-independent sequences](https://doi.org/10.24033/ast.318) | Citation: 1995. Crossref: 2018-11-06. Same work/author/venue; later date may concern digitization or an edition. Historical issue inaccessible. |
| R088 | [Hamilton, Ricci-flow singularities](https://doi.org/10.4310/sdg.1993.v2.n1.a2) | Citation: “formation,” 1995. Crossref: “formations,” 1993. Same work; publication versus volume/conference dating unresolved. |
| R099 | [Nakamura–Uhlmann, elasticity correction](https://doi.org/10.1007/s00222-002-0276-1) | Citation prefixes “Erratum:”; Crossref omits that label. Same DOI/authors/date; secondary pages agree. Correction scope was not assessed. |

Routine capitalization, punctuation, initials, journal abbreviations and equivalent mathematical typography were normalized. A truncated DOI generated during R063 extraction was repaired before classification; that was a parser artifact, not a bibliography defect.

## Unresolved references

| ID | Citation | Why unresolved |
|---|---|---|
| R005 | Milne, Algebraic Number Theory, v3.08 (2020) | Author-site access denied; exact version not established |
| R007 | Erdős, Some remarks on Euler’s φ function (1958) | Original access denied; bounded searches did not establish an original record |
| R009 | Erdős, Some remarks on Euler’s φ function and some related problems (1945) | Same limitation |
| R043 | Hutchinson, Fractals and self-similarity (1981) | DOI exists and author/year agree, but retrieved Crossref and OpenAlex titles are empty; identity only partially corroborated |
| R050 | Derrien–Lesigne, pointwise polynomial ergodic theorem (1996) | No matching original record retrieved |
| R057 | Wang, On a Conjecture of O’Donnell, ePrint 2020/002 | Original ePrint access denied |
| R058 | Durrett, Probability: Theory and Examples, fifth edition (2019) | Author-hosted version inaccessible; exact edition not established |
| R069 | OpenAI, Sharp mass bounds for the two-dimensional O(4) model (2026) | Independent repository retrieval blocked |
| R072 | OpenAI, An explicit exact Hausdorff gauge for SLE (2026) | Independent repository retrieval blocked |
| R076 | Sørensen, Normalization in λ-Calculus and Type Theory (1997; revised 1998) | University-hosted original access denied |
| R095 | Miyamoto, nearly circular planar convex domains, RIMS1591 (2007) | Original inaccessible; a related 2009 article does not establish this exact preprint |
| R098 | Weck, exterior elasticity problems (1969) | EuDML access denied; discovery did not establish an original record |

Unsuccessful searches and inaccessible records do not demonstrate fabrication or nonexistence.

## Ten nearby claims

**Three checks have partial, abstract-only support; seven are unverifiable. None is classified as fully supported or contradicted.** Selection was fixed before verification. Exact citing contexts and inspected abstracts are in the JSON.

| ID | Nearby attribution | Assessment |
|---|---|---|
| R020 | Iritani–Koto construct a mirror theorem and quantum D-module decomposition for arbitrary projective bundles; nearby text invokes semisimplicity and a Virasoro consequence. | **Partial support.** The [abstract](https://arxiv.org/abs/2307.03696v4) specifies a “not necessarily split vector bundle” and the direct-sum decomposition. It mentions semisimplicity applications, but not the precise equivalence or downstream Virasoro conclusion. |
| R074 | For 0<κ<8, integrated Green-weighted two-sided radial SLE laws give natural-length-biased chordal SLE; theorem/equation locators support a remaining-mass argument. | **Partial support.** The [abstract](https://arxiv.org/abs/1509.05015v3) states the integrated-law identity and parameter range. Theorem 4.1, Proposition 2.2, equations and the derived past-mass identity were not inspected. |
| R046 | Jamneshan supplies general Furstenberg–Zimmer structure theory in a paragraph involving nonergodic systems. | **Partial support.** The [abstract](https://arxiv.org/abs/2103.17167v4) removes countability/separability restrictions and states structure theory “in full generality.” Detailed nonergodic decomposition and joining assertions were not checked. Nearby pointwise convergence is separately attributed to other sources. |
| R045 | Bárány–Verma Theorem 3.5 gives the entropy-rate dimension formula allowing exact collisions under weak exponential separation. | **Unverifiable.** The [abstract](https://arxiv.org/abs/2507.05835v2) discusses separation and dimensions, but not this exact theorem or collision allowance. Topic overlap is insufficient. |
| R004 | Stark Lemmas 3 and 8 give specified zero-region bounds and force a quadratic subfield, including for nonnormal fields. | **Unverifiable.** [Paper identity](https://doi.org/10.1007/BF01405166) matches; lemmas, constants and hypotheses were not inspected. |
| R056 | O’Donnell–Servedio Theorem 3 and Lemma 3 bound the signed singleton sum by a square root of expected fixed coordinates. | **Unverifiable.** [Identity](https://doi.org/10.1137/060669309) matches; theorem text unavailable. |
| R068 | Osterwalder–Schrader’s corrected reconstruction uses the linear growth condition in Section IV.1. | **Unverifiable.** [Identity](https://doi.org/10.1007/BF01608978) matches; Section IV.1 uninspected. |
| R030 | Kasteleyn’s Pfaffian methods give polynomial-time exact counting for planar graphs. | **Unverifiable from retrieved evidence.** [Publication identity](https://doi.org/10.1063/1.1703953) matches, but no passage was available to check the attribution. |
| R095 | Miyamoto Lemma 1.2 and Theorem A exclude interior critical points under a spectral-diameter condition. | **Unverifiable.** Neither the exact preprint identity nor theorem text was established. |
| R097 | Cekić’s holomorphic-direction parameter argument follows Eskin’s Yang–Mills analysis. | **Unverifiable.** [Eskin’s publication](https://doi.org/10.1007/s002200100522) matches, but the methodological relationship was not inspected and also depends on the separately cited Cekić paper. |

## Evidence limits and reuse

Requested direct access to author and publisher sites was denied. ArXiv and GitHub were listed as allowed domains, but actual direct retrieval was blocked by the session’s address policy. No alternate route bypassed those decisions. Available original metadata and abstracts do not replace inaccessible theorem text.

The evidence supports the narrow conclusion that **most sampled references identify real works**, with no demonstrated wrong-work mismatch. It does not establish that unresolved references are false, that all dates/page ranges are accurate, or that sources support all nearby claims.

To regenerate the sample, run:

    python reproduce_math_sample.py sampled-math-papers.json > sample-manifest.json

The offline script verifies the exact input checksum. It reproduces selection, not the audit judgments or live database state; the JSON preserves the retrieved state used for those judgments.
