# Verification report

The written proof is the mathematical evidence. It combines the established Caffarelli contraction theorem with a Gaussian-mixture Hessian identity, Brenier change of variables, an explicit weak Gaussian integration-by-parts lemma, a scalar spectral inequality, conditional Gibbs variational inequality, and conditional truncation. The hypotheses and limiting steps are audited in `reproducibility/proof_audit.md`. No gap was identified in that originating-researcher audit; this is not independent or formal verification.

Scope: complete solution of the first numbered question on page 695 of Oberwolfach Report 14/2022 with sharp constant one. The second numbered question, generalized dual Sudakov minoration, remains unaddressed. The entire frozen dataset row is not declared solved.

Supplementary numerical evidence:

- 1,415 scalar finite laws with stable log-sum-exp Gauss-Hermite quadrature; no violation exceeding 1e-7 in the sampled cases.
- Twelve quantile-transport calculations (six laws on two meshes), with omitted quantile tails of 1e-9 per side; tested entropy and integration-by-parts residuals are below 1e-6.
- These are floating-point checks without certified quadrature or tail enclosures, not a proof of the general theorem. The derivative comparison in the exploration is not claimed as a result.

The package replay runs both programs in normal and optimized Python, checks explicit tolerances, and requires matching numerical reports. The publication package records the executed replay receipt, build diagnostics, SHA-256 file hashes, and five-page direct visual inspection. The manuscript was also checked with the desktop LaTeX compiler. The source ZIP is self-contained and excludes third-party PDFs.

Related works by Liu and by Chu-Raginsky are credited. The Caffarelli theorem and transport measure-theory inputs are attributed to their sources. The literature search is bounded and does not certify absolute novelty. AI assistance is disclosed in ordinary manuscript text. No peer review, independent review, or formal verification is claimed.