# Verification report

Problem: AIM-ALGEBRAIC_GEOMETRY-0125. Report date: 6 September 2026.

## Exact versions

The analytic proof reviewed independently is retained unchanged as reproducibility/proof_candidate_v1.md, SHA256 582d69d3ee975598e626e0578f92e7c562c8eff47b88c5c6304a4d8a7601b16f. Its historical candidate-stage heading is preserved, not silently rewritten.

The separate analytic review is reproducibility/all_degree_exact_version_review.md, SHA256 3d1941a15f7a4d31c150fea4decc438bb0bb01a811d50fcb87f3569e128d455d. It reports PASS with no required mathematical correction.

The final English manuscript has main.tex SHA256 670c2ee5d4077f4458f2e641899d31dfebd684285024dd18c3c5f33bb0a35f07 and references.bib SHA256 4ea2ad879f49da52ec4e911b2eaf5876dbd39267cb36905c3d042cda0758c298. The separate manuscript-transfer report in reproducibility/manuscript_review/ binds its conclusion to these exact versions, including the two final grammatical corrections.

These reviews were performed in a separate model context, not by a human referee. There is no Lean, Coq or other formal proof-kernel certificate.

## Lemma-by-lemma gap audit

1. Prime-field moment: parameters remain indeterminates while coordinates are summed. Every pure parameter power of degree p-1 vanishes because its monomial misses a coordinate. The constant coefficient is exactly 1, including p=2. Individual degrees below p-1 make evaluation on the nonzero coefficient grid injective.
2. Count conversion: the affine zero indicator, the origin and the factor p-1 are all retained. The sign is N_projective = 1 + (-1)^m C modulo p.
3. Odd p not dividing d: the final parameter has degree at most p-3. There are at least two count-good scalar choices and at most one singular choice. The argument treats p dividing t separately and checks all coordinate boundary strata. Geometric diagonal normalization is not assumed to take place over F_p.
4. Dividing characteristic: all homogeneous partials AND F=0 are required. Prefix conditions rule out a torus critical point; the B=0 condition instead proves a torus critical point lies off F=0. Every zero-coordinate stratum is checked, including the exceptional m=3 single-zero case.
5. Positive compositions: explicit bounds give a proper prefix equal to p when d>p. For d=p>m, the weighted composition interval attains the needed residue, with m=p-1 separated.
6. Equal-prime boundary d=p=m: Wilson's theorem gives the coordinate relation. The recursion first proves coordinates lie in F_p; only then is Fermat's little theorem applied. The coefficient count excludes the other potential exponent solution.
7. Characteristic two, odd degree: parity of t, coordinate boundary strata, and the geometric torus are handled separately. Algebraic-closure coordinates are never silently treated as rational.
8. Exhaustion: the degree/characteristic cases are disjoint and cover every p, m>=3, d>=m. Equations have a noncancelling pure-power term. Smooth positive-dimensional projective hypersurfaces are geometrically integral because distinct positive-degree components meet and repeated components are singular.

The n=1 remark uses the classical existence of an irreducible finite-field polynomial. The numerical AIM alternative is answered without substituting separable rational connectedness for rational connectedness. Non-uniruledness is only a credited application of Riedl-Woolf's Corollary 3.10.

## Exact finite checks

| Check | Retained outcome | What it does not prove |
| --- | --- | --- |
| Root arithmetic implementation | 76 coefficient examples, 75 exact projective enumerations, 894 compositions | Rational point scans do not prove geometric smoothness. |
| Separately authored arithmetic implementation | 10,293 prefix compositions, 240 residue-DP compositions, 144 formal coefficient families | Finite ranges do not prove the unbounded theorem. |
| Exact SymPy 1.14.0 chart computation | Eight selected equations, all 26 projective charts yield the unit ideal | This shares the root form builder and is not an independent universal proof kernel. |
| Portable package replay | Exact output comparisons recorded in build evidence | Re-execution is not another independently authored implementation. |

The chart ideals include the defining equation and all homogeneous partials before dehomogenization, which is essential when p divides d. The first chart run failed solely because SymPy was unavailable. That environment failure and the successful retry are both retained; it is not a counterexample.

The unbounded theorem rests on the analytic proof. Computation is supplementary and is not used to infer smoothness from one nonzero entry of a higher-dimensional Frobenius matrix.

## Source and novelty gate

The separate family-level source comparison finds a plausibly original uniform combination of established ingredients. It positively identifies the Koblitz/McCarthy families, Kloosterman's diagonal smoothness criterion and Gabber's undeformed chain. It does not certify absolute priority. See source_novelty_review.md and the exact report in reproducibility.

## Build and visual gate

The final PDF is nine pages. Every full page was rendered and visually inspected for formulas, citations, margins, footnote placement and text flow. Tectonic reports no undefined references/citations, missing glyphs, overfull/underfull boxes or LaTeX errors. The package manifest records the final PDF and source hashes and isolated archive-build receipt. Earlier build attempts remain in the private dossier.

No review disagreement or unresolved degree/characteristic gap is being carried into the final paper. This status is an internal research acceptance, not journal acceptance or human peer review.

## Public release presentation

On 6 September 2026 the author requested Zenodo and EulerSolve publication
under the previously approved Creative Commons Attribution 4.0 International
License (CC BY 4.0). The public edition adds that license and an unrefereed
preprint notice to the author footnote, and normalizes the title's joining
hyphen to the canonical metadata. The body from the abstract through the
references is byte-for-byte unchanged from the reviewed manuscript. The
reviewed source hashes above identify that preserved pre-release version;
the release manifest records the new presentation-only source and PDF hashes.
No mathematical claim, proof, bibliography entry, review or AI disclosure was
changed. Public availability is not peer review or a priority guarantee.