# Verification of the Gauss factorization manuscript

The result is a complete written proof of the explicitly stated theorems, accepted after the originating researcher's detailed self-audit. It is not an independently reviewed proof, a proof-assistant formalization, or a certified novel solution of the entire source problem.

## General arguments checked in the written proof

Relative Frobenius descent uses the degree-zero Cartier kernel over rings with nilpotents. A vector-bundle section vanishes on the whole relative curve precisely when a finite matrix on the base vanishes. This represents the closed factorization subfunctor and allows iteration.

For the plane case, the point classification is Homma's theorem. A tangent-space calculation supplies the scheme equality. The coordinate forms are a regular sequence; the degree inequality qm−2 greater than 3m−3 implies the Serre-dual multiplication map is surjective. This accounts for possible first-order line-bundle deformations instead of silently excluding them. Frobenius kills the dual-number deformation of the source, and Euler's identity identifies the tangent space with the coefficient subspace. Regularity excludes hidden thickenings.

Characteristic two at the first height is handled separately. Relative and absolute Frobenius coefficients are not conflated. The plane Galois-component statement is not extrapolated to arbitrary space curves.

## Exact regression tests

The private research suite passed 144 checks: 136 finite mathematical regressions and eight source-file integrity checks. The latter require the retained private source archive and are not included in the portable run.

The public checker runs the 136 mathematical checks using exact SymPy polynomial arithmetic and exact modular elimination. It covers:

- Odd-characteristic gradient, Euler, Hessian and singular-parameter resultant identities.
- Characteristic-two Frobenius pullback, residual differential and smoothness identities.
- Coefficient monomial counts, Hasse derivative patterns and Hilbert-series bounds.
- Exact Macaulay multiplication ranks for selected nonmonomial regular sequences.
- Dual-number examples and the characteristic-two cubic exception.

Finite tests do not prove the general representability, cohomology or scheme assertions. Those arguments are in main.tex and the retained detailed proof. No numerical tolerance or randomized acceptance criterion is used.

## Publication scope

Only the relative factorization and smooth-plane theorem scopes are marked complete. No vertical space-curve Hilbert component is constructed and the whole AIM source record remains unresolved. The manuscript is AI-assisted, self-audited and unrefereed; novelty remains undetermined.