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.