# Exact check: product counterexample for projection-body volume

**Result: the explicit construction survives an independent exact-arithmetic check.** For the supplied integer-vertex polytope, the normalized projection-body volume exceeds the standard 20-simplex value by exactly 83845/5505024, approximately 1.5230633%. This is a check of Theorem 1's particular witness, conditional on the standard geometric identities stated below; it is not verification of the entire paper.

## Source and precise claim

OpenAI, *A product counterexample to the simplex maximum for projection-body volume*, September 24, 2026. Repository commit: `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`.

[Immutable TeX source](https://github.com/openai/math/blob/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-product-counterexample-to-the-simplex-maximum-for-projection-body-volume-September-24-2026/build/main.tex#L90).

Theorem 1, source lines 90–100 (mathematical notation typeset from the TeX):

> Let \(T_{10}=\operatorname{conv}(0,e_1,\ldots,e_{10})\subset\mathbb R^{10}\), where the \(e_i\) are the coordinate unit vectors, and let \(K=T_{10}\times T_{10}\subset\mathbb R^{20}\). Then
> \[
> \frac{R_{20}(K)}{c_{20}}
> =\frac{121\binom{20}{10}}{21\cdot2^{20}}
> =\frac{22\,355\,476}{22\,020\,096}>1.
> \]
> Thus \(K\) violates [the proposed bound] in dimension twenty.

The bracketed words replace the source's equation cross-reference. Source lines 43–58 define the projection body by its support function
\[
h_{\Pi P}(u)=\operatorname{vol}_{d-1}(\operatorname{proj}_{u^\perp}P),\qquad
R_d(P)=\frac{|\Pi P|}{|P|^{d-1}},\qquad c_d=\frac{(d+1)d^d}{d!}.
\]
The proposed bound is \(|\Pi P|\le c_d|P|^{d-1}\). It concerns the projection body itself, not its polar.

I read the supplied TeX. The PDF was authenticated by its bytes, not separately interpreted. Recomputed SHA-256, Git blob SHA-1 (including the Git blob header), and lengths agree for all four paper files with both the supplied provenance and the separately uploaded catalog. In particular:

- TeX Git blob: `c32187535cddb5c38f28d332997dc0578cbad586`.
- TeX SHA-256: `5302b6298402cf1665e5747766b53055ecad84a5027b1657811731389879f0e0`.
- Paper ZIP SHA-256: `e937e168b67ed3b60b7d5a1ae7b9a36f43d40e67639f5f15ea672225937fbe68`.
- Catalog ZIP SHA-256: `150b3bfaee6b0d858334897253dad3f385b36512e2c34d8ae04f2b8fc6b05aef`.

This establishes agreement between the two supplied snapshots and the bytes checked. The association with the upstream commit is supplied provenance, not independently authenticated by a successful GitHub request.

## Independent reconstruction

The checker uses only Python's standard library, arbitrary-precision integers, and `fractions.Fraction`. It executes no supplied paper code. The paper's claimed ratio and simplex constant are used only as final assertions, after the geometric computation.

1. Generate the 121 vertices \((a,b)\), where each of \(a,b\) belongs to \(\{0,e_1,\ldots,e_{10}\}\). Save all coordinates.
2. Use the inequalities \(x_i\ge0\), \(y_i\ge0\), \(\sum_i x_i\le1\), and \(\sum_i y_i\le1\). Check every vertex against them, full dimension, strict interior of the vertex centroid, affine rank 19 for every facet, and rank 20 for the active normals at every vertex. Each of the 22 facets contains 110 vertices.
3. Recover each facet's two simplex factors from its incident vertices. Confirm their Cartesian product equals the incident-vertex set. Compute their Gram determinants and dimensions. The squared facet area follows from the simplex Gram formula and orthogonal product measure; dividing by the squared primitive-normal length gives a rational square. Its exact square root gives the area-normal coefficient. No floating-point normals or convex-hull tolerances are used.
4. Compute body volume by summing pyramids from the interior centroid to the facets.
5. Form the projection-body generators from the resulting area-normal vectors and enumerate **every** maximal determinant: 231 for K and 21 for the benchmark T20. Sum their absolute, correctly scaled values. Every determinant is evaluated by both fraction-free Bareiss elimination and rational Gaussian elimination; they agree.
6. Independently construct and process the 21 vertices of T20, then compare the two computed normalized volumes.

**Completeness argument outside the code:** the inequalities describe exactly the convex hull of the listed vertices. Given a feasible pair, use barycentric weights \(\lambda=(1-\sum x_i,x_1,\ldots,x_{10})\) and \(\mu=(1-\sum y_i,y_1,\ldots,y_{10})\). The nonnegative weights \(\lambda_i\mu_j\) sum to one and express the pair as a convex combination of the listed vertices. The converse follows by linearity. Thus this is a complete half-space description. The supporting-face rank checks establish that all its inequalities define facets. The analogous single-block argument applies to T20.

## Geometric premises

These identities are mathematical inputs to the computation, not theorems established by running it:

1. A k-simplex with edge-column matrix E has k-volume \(\sqrt{\det(E^TE)}/k!\). Measures multiply for Cartesian products in orthogonal subspaces (Fubini).
2. For a d-polytope with facet areas \(s_F\), outward unit normals \(\nu_F\), and an interior point c, its volume is the sum of the facet-pyramid volumes \(s_F h_F/d\). In the code, if the facet is \(n_F\cdot z=b_F\) and \(s_F\nu_F=\alpha_F n_F\), its contribution is \(\alpha_F(b_F-n_F\cdot c)/d\).
3. Cauchy's polytope projection formula:
   \[
   h_{\Pi P}(u)=\tfrac12\sum_F s_F|\nu_F\cdot u|,\qquad
   \Pi P=\sum_F[-s_F\nu_F/2,s_F\nu_F/2].
   \]
   The factor one-half counts projected entry and exit facets. This identifies the computed zonotope with the projection body.
4. A zonotope \(\sum_j[0,g_j]\) has d-volume \(\sum_{|I|=d}|\det(g_i:i\in I)|\). Translation does not affect volume. Thus the full generator is \(g_F=s_F\nu_F\), not \(s_F\nu_F/2\); there is no extra factor \(2^{-d}\).

These are standard finite-dimensional volume identities. The checker does **not** assume the paper's multiplicativity formula \(R(A\times B)=R(A)R(B)\) or its closed form for simplex projection-body volume. Product measure is used only to evaluate the explicit facets. Extending the benchmark from T20 to every 20-simplex additionally uses affine invariance of R; the strict violation of the stated numerical bound already follows from the computed values.

## Exact results

Write \(D=9!\,10!=1\,316\,818\,944\,000\). All 22 area-normal vectors for K are the primitive outward normals divided by D. The primitive normals are the 20 negative coordinate vectors and the two positive block-sum vectors. For T20, the 21 corresponding vectors have denominator \(19!\).

| Quantity | K = T10 × T10 | T20 |
|---|---:|---:|
| Dimension | 20 | 20 |
| Vertices | 121 | 21 |
| Facets | 22 | 21 |
| Maximal minors evaluated | 231 | 21 |
| Zero primitive minors | 110 | 0 |
| Primitive minors with absolute value 1 | 121 | 21 |
| Body volume | \(1/(10!)^2\) | \(1/20!\) |
| Projection-body volume | \(121/(9!\,10!)^{20}\) | \(21/(19!)^{20}\) |
| Normalized volume R | \(295410156250000/321489\) | \(640000000000000000/707107401\) |

The ratio obtained from the reconstructed geometries is
\[
\frac{R_{20}(K)}{R_{20}(T_{20})}
=\frac{5588869}{5505024}
=\frac{22355476}{22020096}>1.
\]
The exact relative excess is \(83845/5505024\). The paper's unreduced cross-multiplied integer gap is \(335380>0\), recovered as a final comparison rather than supplied as the computational evidence.

The projection-volume formulas in the table summarize the computed determinant sums. Full rational values, signed determinants, facet indices, scales, incidences, Gram data, and pyramid contributions are retained in `exact-results.json`.

## Reproduction and scope

Unzip `projection-body-exact-check.zip` and run:

```sh
python3 exact_check.py reproduced
```

No external packages or network access are needed. Compare `reproduced/exact-results.json` and `reproduced/vertices.json` with the archived outputs. The bundle includes the exact executed checker, both original uploaded ZIPs (including original license and source provenance), source-hash checks, a calculation transcript, and a checksum manifest.

This validates the explicit dimension-20 construction and its reported strict inequality under the stated geometric premises. It does not formally verify those identities, the entire manuscript, its exponential-excess corollary, historical priority, or all other repository papers. The source itself acknowledges earlier counterexamples in dimensions at least nine; this check makes no novelty claim.
