# Panel commentary — A product counterexample to the simplex maximum for projection-body volume

**Optional panel findings: kept separate from the human-first reading materials.**
Human assessment has not been supplied. This is an AI-assisted source audit and
independent model panel; no human/panel agreement statistics are available.

## Assessment and scope

The six proof-bearing claims withstand focused source checking. The product
counterexample follows from the proved facet formula, product identity and
simplex volume calculation, with an exact integer certificate. The
exponential-excess corollary also follows with the stated quantifier order.
No demonstrated mathematical error was established. This assessment rests on
the checked arguments below, rather than a provider vote.

The manuscript is dated 24 September 2026 and has seven PDF pages. The pinned
repository commit is `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`. Every model
received the complete 20,710-character expanded TeX, including bibliography,
SHA256 `628bd94e51987a4ab5879c572a01837154b2ace52c1be9fe9d672178951541df`.
The original main source, bibliography and PDF are separately pinned in the
[provenance manifest](../source_manifest.json). The source-backed
[interactive roadmap](simplex-product-counterexample.roadmap.html) and
[reading guide](simplex-product-counterexample.roadmap.md) describe the
internal argument without panel verdicts. No researcher finding, prior
model verdict, suitability judgment or human assessment was included in the
first-pass prompts.

The correctness audit covers all six numbered claims and their internal
proof dependencies. It does not certify every cited external paper or settle
novelty and priority. The paper explicitly acknowledges previous
counterexamples and presents a particular Cartesian-product witness; its
proof does not invoke the earlier counterexample constructions.

## Checked mathematical chain

| Source claim | Checked mechanism and implication |
|---|---|
| Lemma 2, `lem:facets` | Projecting each facet gives Jacobian abs(ν_F·u). Almost every projection fiber has one entry and one exit facet, so the total projected facet measure is twice the projection volume. The factor 1/2 and centered segment formula follow. |
| Proposition 3, `prop:product` | Product facets are exactly F×B and A×G. Their area-normal vectors are scaled by vol(B) and vol(A) respectively, giving Π(A×B)=(vol(B)ΠA)×(vol(A)ΠB). Volume powers cancel to yield R_(r+s)=R_r R_s, with no binomial factor. |
| Lemma 4, `lem:affine` | The image of a unit-height facet prism has perpendicular height 1/‖L^(−t)ν_F‖. Area-normal vectors therefore transform by abs(det L)L^(−t), including negative determinant. The transformation has absolute determinant abs(det L)^(d−1), matching the denominator. The inscribed-polytope sandwiches justify the extension to arbitrary convex bodies. |
| Proposition 5, `prop:simplex` | The facet generators are −e_i/(d−1)! and w/(d−1)!. Thus ΠT_d is a translate of (Q+[0,w])/(d−1)!. Adding the segment lengthens every fiber parallel to w by ‖w‖, giving volume 1+h_(ΠQ)(w)=d+1 and c_d=(d+1)d^d/d!. |
| Theorem 1, `thm:counterexample` | All product hypotheses apply to T_10×T_10. Substituting the proved simplex constants gives 121·binom(20,10)/(21·2^20). Exact arithmetic gives numerator 22,355,476, denominator 22,020,096 and surplus 335,380. |
| Corollary 6, `cor:exponential-excess` | Every n≥20 has the unique form 10k+d_r, d_r=10+r, k≥1. Product values equal c_10^k c_(d_r). The exponential-series bounds give c_n<(n+1)(11/4)^n and A=c_10/(11/4)^10>1. Any 1<λ<A^(1/10) makes the geometric-over-linear lower bound diverge in each of ten residue classes; a common threshold follows. |

These proofs are acyclic. Lemma 2 supports the product and affine formulas;
the simplex calculation uses it and affine invariance; the theorem and
corollary use the product and simplex values. All actual factors have dimension
at least ten, within the product proposition's r,s≥2 hypotheses. The general
convex-body extension of affine covariance is a stated additional claim and
was inspected, although the witness only needs its polytope case.

For the approximation argument, containing a fixed ball aB gives
P_m⊂P⊂(1+ε_m/a)P_m. Projection-volume monotonicity then bounds support
functions uniformly and supplies the projection-body limit. The volume
sandwiches and their linear images justify passage to the affine formula;
they do not assume that formula in advance.

The [exact arithmetic record](../arithmetic_checks.json) verifies the displayed
constants with integer/rational arithmetic. Its extra corollary certificate
uses λ=101/100 and N=629: in each residue, the proof's lower bound exceeds one
at the recorded threshold, and its consecutive growth factor
q(10k+d_r+1)/(10k+d_r+11) is greater than one and increases with k. This supplies
an infinite-tail justification beyond its finite numerical window. This is
an additional audit certificate, not an optimal threshold claimed by the paper.
The finite determinant and product-formula checks are supporting calculations;
the general geometric identities are justified by the proofs above.

## Independent panel and source adjudication

| Model | Requested/native effort | Returned coverage | Workflow time |
|---|---|---|---:|
| [GPT-6.1 Sol](../panel/sol/full_manuscript/review.json) | high | Complete: all six claims | 103.6 s |
| [GPT-6 Astra](../panel/astra/full_manuscript/review.json) | max | Complete: all six claims | 467.4 s |
| [Claude Fable 5.1](../panel/fable/full_manuscript/review.json) | high | Complete: all six claims and unnumbered product test | 135.3 s |
| [Gemini 3.1 Pro](../panel/gemini/full_manuscript/review.json) | high | Complete: all six claims | 113.7 s |
| [Grok 4.7](../panel/grok-4.7/full_manuscript/review.json) | high | Complete: all six claims | 266.6 s |
| [Qwen 3.8 Max](../panel/qwen/full_manuscript/status.json) | medium | Transport timeout; no mathematical assessment | 241.0 s |

Five complete structured assessments report no demonstrated error. Sol and Astra supply detailed calculations for each claim; Gemini and Grok cover the same six claims, with shorter explanations. Fable additionally records two `unverified_step` comments, addressed below. Qwen's single request timed out after the configured 240-second transport limit; its exact payload and complete error log are preserved, with no mathematical verdict or default score. These are five returned assessments and one bounded transport failure, across five provider families attempted. Agreement is supplementary evidence; the mathematical assessment above follows the source checks.

Fable requests clarification of why a line above an interior projection point
meets the body's interior. The claim follows directly from convexity: choose
an interior point x0, extend the segment from its projection through the given
projection point slightly beyond that point, and lift the extension to P.
A convex combination with positive weight on x0 projects to the original
point and remains interior to P. The fiber is therefore nondegenerate;
removing projected lower-dimensional faces leaves its endpoints in facet
interiors. This supplies the short implicit justification. The comment is a
possible exposition improvement, not a counterexample or missing essential
hypothesis.

Numeric model references require correction. Gemini's descriptions correspond
to the six actual claims, but it calls them Lemma1, Proposition2, Lemma3,
Proposition4, Theorem5 and Corollary6; its facet/product/affine/simplex/counterexample/corollary entries should be Lemma 2, Proposition 3, Lemma 4, Proposition 5, Theorem 1 and Corollary 6, respectively. Fable gives the
correct stable corollary label but calls it Corollary 7 rather than 6. These are
model source-reference errors. They do not change the manuscript's arguments,
and corrected references are recorded separately from the original outputs.

The stable TeX labels and the shared native theorem counter are authoritative. The [source claim index](../panel/source_claim_index.json) and per-provider source-reference corrections preserve the original responses while supplying the correct anchors. No response alleges a false mathematical implication or counterexample under the stated hypotheses. The main residual review work concerns historical/contextual attribution and optional exposition, rather than an unclosed indispensable proof lemma.

External historical/contextual results identified as unverified by a model
are not inputs to the exact witness or its corollary. Their citation fidelity
should be checked as a separate scholarly obligation; an unseen cited proof
is not converted into a mathematical defect in this self-contained chain.
The independent source researcher inspected the statements of the two linked primary preprints. Feng–Hu–Liu–Xu state the ordinary projection-body counterexample for every dimension at least nine with at most n+2 facets; Chen–Feng–Li–Xi–Xu state the dimension-three upper bound 18 and the minimum result. That addresses the specific statement-level attribution uncertainty raised by Sol, Astra and Fable, without auditing those prior proofs. Brannen's original conjecture text and the other historical references were not independently inspected. Full coverage and limits are recorded in the [independent source audit](../simplex-product-counterexample-research.md), completed before its author opened any panel response.

## Resources and comparison limits

The [panel manifest](../panel/campaign_summary.json) records source/prompt
hashes, exact native payloads, request IDs, raw responses, available usage,
requested effort/budgets, timings and failures. SDK retries are set to zero;
a second native inference request is blocked before sending, including an
adapter's attempted budget recovery. No recovery calls were launched.
Completed visible answers were saved before review-schema parsing. The
Alibaba Maas host is captured in this run, unlike the previous long-paper
campaign's tracing omission.

There were exactly six captured native requests and no retries. Actual wire output limits were 36,864 for Sol and 81,920 for Astra (requested 32K/64K plus the existing reasoning reserve), 65,536 for Fable, 32,768 for Gemini and Grok, and 16,384 for Qwen. Google received native HIGH thinking, Anthropic adaptive/high, xAI high, and Alibaba medium. Provider budget definitions differ: Grok's hidden reasoning is separate from its visible-answer limit.

| Completed model | Input tokens | Reported thinking/reasoning | Visible answer tokens |
|---|---:|---:|---:|
| Sol | 7,387 | 3,106 | 2,383 |
| Astra | 7,387 | 15,437 | 2,265 |
| Fable | 11,250 | 8,155 | 3,844 |
| Gemini | 7,692 | 16,876 | 972 |
| Grok | 8,460 | 24,071 | 1,623 |

Visible counts for OpenAI and Fable are derived by subtracting reported reasoning from their shared output usage. Qwen returned no usage or provider response ID before timeout; server-side completion and billed cost remain unknown. Available raw provider accounting is retained. A complete comparable billed-cost total is not established.

All jobs used the same full manuscript, with different native effort and
budget settings selected in advance. The observed workflow times include
network and adapter work; they do not establish a controlled model-speed or
cost ranking. No calibrated correctness probability or overall numerical
correctness score is supplied. Human assessment is still awaiting input, so
agreement, false-positive and false-negative statistics remain unavailable.
The source audit is additional AI-assisted evidence, not a substituted human
baseline.

## Sources

- [Pinned native main source](https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-product-counterexample-to-the-simplex-maximum-for-projection-body-volume-September-24-2026/build/main.tex).
- [Pinned native bibliography](https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-product-counterexample-to-the-simplex-maximum-for-projection-body-volume-September-24-2026/build/references.tex).
- [Pinned seven-page PDF](https://raw.githubusercontent.com/openai/math/fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb/preprints/A-product-counterexample-to-the-simplex-maximum-for-projection-body-volume-September-24-2026/paper.pdf).
- [Feng, Hu, Liu, Xu: primary preprint, inspected statements](https://archive.ymsc.tsinghua.edu.cn/pacm_download/743/12781--2026.8.26.pdf).
- [Chen, Feng, Li, Xi, Xu: primary preprint, inspected statements](https://archive.ymsc.tsinghua.edu.cn/pacm_download/741/12780-pettyinequality.pdf).