From Fisher--Rao Simplex Flows to Canonical Jump Generators: A $\Gamma$-Convergence Theory of Discrete Flow Matching

We prove that Fisher–Rao simplex flows with profile-generated endpoint dequantization admit a discrete boundary theory. For a reduced well-prepared mesoscopic decomposition, the projected action $\Gamma$-converges to a finite-state graph action with universal mobility $m_\star = 1/\pi^2$. The limit pair $(\pi, J)$ satisfies the discrete continuity equation, and under a generator-compatible closure law the limiting action equals the minimal quadratic traffic action among all CTMC realizations of the same flux, attained by a canonical one-sided generator. On product simplices the limit tensorizes to Hamming graphs, so only single-coordinate transitions survive. We further show that any nonzero limiting edge flux forces blow-up of microscopic normal current density, and hence velocity blow-up under a uniform density bound. Deterministic experiments confirm the exact occupied-cell formula, the critical-scaling trichotomy, the generator identification, the tensorized Hamming structure, and the predicted singularity exponents.