# Construction candidates from the uploaded repository catalog

## Provenance and limits

Repository: https://github.com/openai/math

Commit recorded in both source-pin.json and preprints-file-index.json: fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb.

Uploaded ZIP SHA-256, independently calculated: 150b3bfaee6b0d858334897253dad3f385b36512e2c34d8ae04f2b8fc6b05aef.

The ZIP contains eight members: CONTENTS.md, README.md, history.md, source-pin.json, catalog-pin-verification.json, preprints-file-index.json, repository-commits.json, and LICENSE. The file index contains 10,373 entries. README reports 719 manuscripts in 372 families. LICENSE is Apache License 2.0. The supplied verification JSON asserts that three catalog files match pinned Git blobs; this assertion was not independently authenticated against GitHub. The commit pin and file hashes below are taken from the uploaded catalog, not a fresh upstream retrieval.

The ZIP was read as an ordinary archive using Python zipfile; nothing was installed or executed from it. No paper text, graph edge list, numerical witness, or supporting verifier was included. Earlier public GitHub access failed with HTTP 403.

## Preferred retrieval candidate: product of simplices

Title: A product counterexample to the simplex maximum for projection-body volume.

Exact catalog quotation (CONTENTS.md, line 2179):

> The product of two ten-dimensional simplices has larger normalized projection-body volume than a twenty-dimensional simplex. This gives a counterexample to Brannen's proposed simplex maximum.

Pinned directory:
https://github.com/openai/math/tree/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-product-counterexample-to-the-simplex-maximum-for-projection-body-volume-September-24-2026

Relevant indexed files:

| Relative file | Bytes | Git blob SHA-1 |
|---|---:|---|
| README.md | 730 | b46aac76129559cd732d32f72e7db2bd1484daee |
| build/main.tex | 18985 | c32187535cddb5c38f28d332997dc0578cbad586 |
| build/references.tex | 1743 | b5ca002e7cd37b5cfbf11c9869d5e6bb92f6b701 |
| paper.pdf | 264039 | 1487c8bdba5b65c34084cb50bdb7f07f52c62b13 |

Why prioritize this: the catalog specifies a concrete geometric family and dimensions; the source is a single compact file. A canonical rational realization may permit an exact combinatorial volume comparison. This is a proposed method, not a computation performed or evidence that the inequality holds. First obtain build/main.tex to establish the precise normalization, definitions, construction, and theorem locator. Do not substitute a guessed normalization for the paper's claim.

## Alternative: Sidorenko graph

Title: A counterexample to Sidorenko's conjecture.

Exact catalog quotation (CONTENTS.md, line 3829):

> We disprove Sidorenko's conjecture with a bipartite graph on 35 vertices and 66 edges: its homomorphism density in some finite simple graph is smaller than the conjectured lower bound. The same connected graph also disproves the forcing conjecture: at one fixed density, asymptotically matching the edge and pattern densities does not imply quasirandomness.

Pinned directory:
https://github.com/openai/math/tree/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-counterexample-to-Sidorenkos-conjecture-September-23-2026

The index lists 15 files in this directory: a README, paper.pdf, and TeX/BibTeX source files. It lists no separate machine-readable graph data or checking program. That does not establish that an explicit witness is absent from the paper.

Useful indexed source: build/sections/introduction.tex, 18,486 bytes, Git blob SHA-1 32890ea14c106eeb0556da93226165ac93739371. The complete paper.pdf is 639,874 bytes, Git blob SHA-1 4855f35e8c778b8b591db993546d0607d62b5cdb.

For a supplied pattern H and a finite simple target G with n vertices and m edges, the standard density check would be t(H,G) < t(K2,G)^66. After verifying |V(H)|=35 and |E(H)|=66 and computing hom(H,G) exactly, this becomes the integer comparison hom(H,G) * n^97 < (2*m)^66. The catalog supplies neither H's edges nor G, and the target size and feasibility of exact counting are unknown. No such count was run.

## Status

Candidate identification is complete. Neither counterexample has been verified or refuted. A catalog abstract is not a precise theorem quotation from a paper. Obtain the selected paper/source at the pinned commit, authenticate its Git blob hash against the supplied index, preserve the actual witness, and only then independently implement and execute the exact check. Even a successful construction check would not verify the entire paper or its other claims.
