# Verification of the half ball counterexample package

The mathematical proof and its self-audit are included in reproducibility/complete_counterexample.md and reproducibility/proof_audit.md. The proof concerns exactly the stated nested half-ball/cone bodies. It uses ordinary unnormalized Euclidean volume and independent uniform samples.

The complete C++ computation includes 1048576 cells and uses only outward-rounded integer intervals. Ordinary and undefined-behavior-instrumented runs produce identical 20767-byte certificates. The instrumented replay exited successfully without diagnostics. The resulting gap lower bound is 49211553366/281474976710656, greater than 1/6000.

The separately coded Python implementation replays 8448 cells, tests arithmetic against exact fractions and recomputes the scalar certificate and explicit cone inequalities. Normal and optimized executions agree. Non-certifying controls include a full-ball normalization identity, 30 independent section-integral comparisons and a two-million-sample Monte Carlo experiment. They are not proof inputs.

Both LaTeX compilers succeeded. All seven full-page PNGs were inspected for equations, line breaks, author footnote, bibliography and page transitions. No overflow, missing glyphs or unresolved references were found. Package hashes and archive contents are checked in the accompanying manifests.

This is an originating-researcher self-audit, not an independent expert review or a formal proof-assistant certificate. Rademacher's construction and the previously established general nonmonotonicity result are explicitly credited. Priority is not certified.