docs/claim5_source_and_falsification_audit.md

Claim 5 source and falsification audit

Public source and scope

• Primary source: https://ar5iv.labs.arxiv.org/html/2603.02429
• Retrieved: 2026-07-22 with a browser User-Agent
• Retrieved HTML SHA-256: 7031d50a4323d0de060fe8d4c7e65dea340e5cd576a2e9b5f42663d5c1576716
• Audited scope: Section 6, Lemma 6.1; Appendix E, Lemma E.1; Appendix I, Lemmas I.3-I.5.

Lemma 6.1 states, under -beta I <= Hessian(V) <= H <= beta I, that each of

1. E_mu[||grad V(x)||^2], and
2. E_mu[p^T H p]

is at most tr(H) + beta KL(mu || pi).

Valid-assumption counterexample family

For every integer d >= 1 and scalar s > 1, set

• V(x) = ||x||^2 / 2, H = I, and beta = 1;
• pi = N(0,I_d)_x x N(0,I_d)_p;
• mu = N(0,sI_d)_x x N(0,sI_d)_p.

All assumptions hold because Hessian(V)=I and -I <= I <= H=I <= I. The two left sides both equal d s. The joint Gaussian KL is

KL(mu || pi) = d (s - 1 - log(s)).

Therefore the claimed right side is d (s - log(s)), and each left side exceeds it by exactly d log(s), which is positive for every s > 1. This is a family of analytic counterexamples in every dimension, not a finite-sample or discretization effect.

Source proof mismatch

Appendix E substitutes U=quantity/(4 beta) into the Donsker-Varadhan bound and invokes the Appendix I moment-generating estimates. Its displayed rearrangement yields the weaker bounds

2 tr(H) + 4 beta KL(mu || pi),

not the tr(H) + beta KL(mu || pi) bounds in the lemma statement. The executable counterexamples violate the statement while satisfying this weaker Appendix E bound in every preregistered case.

Reproduction classification

Claim 5 is falsified as literally stated. This finding does not assert that the weaker Appendix E inequalities are false, nor does it determine which constants the downstream convergence theorems actually require.