# Verification report

The originating researcher has self-audited the complete analytic argument
in the six-page English manuscript. This is not independent review or
formal verification, and no absolute priority claim is made.

The universal gap-cap proof counts finitely many forbidden values at each
coordinate and proves stabilization of every local constraint. The finite
graph supplies the unrestricted exact optimum, and a shortest-path
potential plus weighted compression excess supplies all-extremizer criteria.
The 4/9 inequality is proved analytically for all legal positive quadruples,
with an explicit exhaustive classification only for total at most eight.

Exact finite diagnostics on Python 3.9.6 and 3.13.5 each pass 199,510
assertions, with identical reports except version. Coverage includes:

- independently reconstructed complete graphs for s=3,4,5;
- 145,726 edgewise dual inequalities and matching directed cycles;
- 8,000 agreements with the explicit analytic potential;
- 8,190 gap-predicate versus point-triple comparisons;
- 328 finite compression inputs for s=3 through 10;
- finite endpoints, periodic boundaries and a counterexample to naive clipping.

The two runtime executions are the same tests, not twice as much distinct
coverage. Finite checks support but do not replace the all-s infinite-word
proof. No universal graph with s>=6 was exhaustively enumerated.

Native compilation and warning-free PDF export passed. All six rendered
pages were visually inspected for clipping, missing glyphs, references and
formula layout. Detailed hashes are in the build receipt and manifest.

Scope: effective finite answer for every s with exact extremizer criteria;
no short closed form, sharp asymptotic or absolute novelty certification.
Standing release consent does not constitute mathematical validation.