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. The largest possible dimension of the resulting family is \(\lceil 2^{d+1}/3\rceil-d-1\) for \(d\ge2\), even with uniform margins. For \(d\ge3\), equality holds precisely on classical extremal square-free layer sets, up to cube automorphisms. These extremizers have no surviving ratios. We also 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 vertex theorem. Two exact implementations verify all 65,805 nonempty supports through dimension four. This does not classify arbitrary collapsed marginal ratios or resolve the broad AIM specification question; priority is undetermined.