Random transpositions on Dyck paths mix in order n log n

Unrefereed, AI-assisted analytic proof candidate for AIM-PROBABILITY-0039. For length-2n Dyck words, the original independent ordered-coordinate transposition proposal/rejection walk has proposed worst-start mixing time Theta(n log n), with lower bound (n/2) log n - O(n). The written proof uses two-cut matroid entropy, a uniform classical height-channel log-Sobolev bound, discrete hypercontractivity, and a constant-factor return to the original clock. Seven finite producer replay stages and negative controls corroborate bounded identities, not the universal theorem. Five producer-coordinated internal AI editorial reports and responses are supplied. No external peer review, independent reproduction, formal verification, priority clearance, cutoff, sharp upper constant, computational-runtime guarantee or fastest-sampler claim. Original prose and data CC0-1.0; original code MIT; third-party materials retain upstream terms and are not relicensed.