# Independent human assessment: simplex product counterexample

This worksheet contains no panel verdicts or AI source-audit conclusions. Fill it
before opening the panel commentary if you want an independent comparison.

## Manuscript and scope

**A product counterexample to the simplex maximum for projection-body volume**
(OpenAI, September 24, 2026). Seven pages including references; the main theorem
is stated on page 2 and its final calculation is on page 5.

- [Read the pinned PDF](paper.pdf).
- Repository revision: `fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb`.
- PDF SHA-256: `1d2211bb8d175f40fb55554586f02538fc9f4c04ecdc9793999a6f7ec514720a`.
- [Native source with bibliography expanded](manuscript.tex).

The paper studies the volume of the projection body itself. Read its definitions
before assessing the proposed comparison. Assess the main counterexample and the
additional exponential-growth corollary separately.

## Record your assessment

Reviewer: ____________________
Date: ____________________
Mathematical background relevant to this paper: ____________________
Resources or software used: ____________________
Did you see any panel commentary before finishing? ____________________

Use a status such as **checked**, **conditional on a stated input**, **specific
error**, or **not independently checked**. For an objection, record the exact
source location, hypotheses and mathematical reason. Distinguish a missing step
from a step whose verification you did not complete.

| Claim | PDF location | Your status | Derivation, objection, or remaining question |
|---|---|---|---|
| Theorem 1: proposed product counterexample | Statement p. 2; proof p. 5 | | |
| Lemma 2: facet formula | pp. 2–3 | | |
| Proposition 3: Cartesian-product identity | p. 3 | | |
| Lemma 4: affine covariance and general convex bodies | pp. 3–4 | | |
| Proposition 5: simplex calculation | pp. 4–5 | | |
| Corollary 6: exponential excess in large dimensions | p. 6 | | |

## Neutral checking prompts

1. Recompute the facet projection Jacobian and explain the multiplicity used in
   the facet formula, including exceptional directions and lower-dimensional faces.
2. Identify all facets of the Cartesian product and derive its projection body.
   Track dimensions and volume powers in the normalized functional.
3. Check affine covariance for polytopes and the limit argument for general bodies.
   Record which part is needed for the main counterexample.
4. Reconstruct the simplex projection body and calculate its volume from the
   stated generators; justify the extrusion calculation.
5. Recompute the final dimension-twenty comparison with exact arithmetic.
6. Check the corollary's exponential estimate and its quantifier over every
   sufficiently large dimension, including residue classes.
7. Separate proof inputs from background citations. Assess correctness and
   contribution relative to earlier work as distinct questions.

These are questions about the author's argument, not suggested findings.

## Overall assessment

Main counterexample: ____________________
Additional corollary: ____________________
Specific errors, if any: ____________________
Important steps not independently checked: ____________________
Exposition and scope: ____________________
What would change your assessment? ____________________

## Subsequent panel comparison

After recording your own assessment, open the separately labelled panel
commentary. Compare particular mathematical findings and source locations.
No human/panel agreement statistics have been calculated: the human assessment
has not yet been supplied.