Weighted Switchings and Linear Corner Growth in Uniform Contingency Tables

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.