# Verification report

Result: a complete, self-audited counterexample to unrestricted stability of positive definite Hessian metrics under small affine deformations. The written proof is in `main.tex`, with a detailed proof and fourteen-point audit in `reproducibility/`. This is not independent review or machine-checked formal verification.

On the fixed torus, the sole Christoffel coefficient Gamma^1_22=t defines a complete, flat, torsion-free, volume-preserving connection converging smoothly to the Euclidean connection. For every hypothetical positive Hessian metric with arbitrary periodic coefficients, Codazzi gives b_v-t a=c_u. Integration forces t times the positive integral of a to vanish. Products extend the counterexample to all dimensions at least two. Averaging and simultaneous diagonalization also give the stated criterion for translation-invariant connections.

The normal and optimized supplementary checker receipts agree: 53 symbolic identities, 350 exact rational tests. Neither finite tests nor symbolic simplification substitutes for the positivity/integration argument or the general averaging proof. The indefinite metric sanity check is retained.

Source scope was checked against all three pages of the original problem collection. Two complete Baues–Goldman pages identify the exact classical shear family; two Koszul pages identify the hyperbolic hypothesis; the two-page public Yagi preview supplies averaging background. Yagi's remaining pages were not read. Literature coverage is bounded, novelty uncertified, and no first-solution or priority claim is made.

The built-in LaTeX compilation and local Tectonic compilation succeeded. All four PDF pages were visually inspected: equations, bibliography, name-only byline and affiliation footnote are legible, with no clipping or overlapping text. One nonfatal underfull-box diagnostic was observed, not an overflow. AI assistance is disclosed in the body text. Portable rebuild and archive checks are recorded separately in the package-readiness receipt.