Weighted Switchings and Linear Corner Growth in Uniform Contingency Tables
Alper Ferudun, Mercury Software GmbH

For a uniformly chosen nonnegative integer n-by-n matrix with both margin
vectors (3n,n,...,n), the corner expectation has order n. An explicit
finite weighted-switching bound gives liminf E[X_11]/n >= 2-sqrt(2).
The same corner law has total variation distance at least 1/4096 from
every geometric distribution, including n-dependent choices of parameter.
A general bound covers margins (L,t,...,t). The proof does not use an
independence approximation or a Markov-chain mixing theorem.

This answers the linear-order question and gives a counterexample to
unrestricted geometric entry approximation in OWR-5152-009. It does not
assert the exact limiting mean, a Gaussian limit, or the full phase
diagram. The fixed-base cloning results are not contradicted.

English unrefereed preprint. Author affiliation: Mercury Software GmbH.
OpenAI Codex assisted with derivation, literature review, computational
checks and preparation. No independent review, formal verification,
novelty certification or absolute priority claim. CC BY 4.0.
