Sharp Ambiguity Bounds for Surviving Conditional Odds Ratios

Fix the support of a binary d-way probability table, its feasible interior one-way margins, and every conditional pairwise odds ratio whose four cells remain positive. We prove that the largest possible dimension of the resulting family is ceil(2^(d+1)/3)-d-1 for d >= 2, even when the margins are required to be uniform. For d >= 3, equality holds exactly on the classical extremal square-free layer sets, up to cube automorphisms. Those extremizers have no surviving ratios; we additionally give a four-variable uniform-margin family of dimension five with one genuinely surviving ratio. The proof combines classical mixed coordinates with a pivot-deletion lemma and the Kostochka-Johnson-Entringer cube theorem. Two independently implemented exact-arithmetic programs agree on all 65,805 nonempty supports in dimensions two through four. The established mixed-coordinate and cube theorems and the structural-zero work of Fontana, Perrone and Rapallo are explicitly credited. Priority of the statistical extremal transfer is undetermined. This is not a complete solution of AIM-COMPUTATION-0020: arbitrary collapsed marginal odds-ratio specifications remain outside its scope. The English preprint is AI-assisted, self-audited and unrefereed; no independent human review or proof-assistant verification is claimed.