Transport Topology for Fixed-Support Sinkhorn Layers

Fixed-support Sinkhorn scaling exposes an exact quotient Markov operator. For finite active scores and compatible positive marginals, the homogeneous column-potential derivative of a finite row-column cycle is $M_t=C_t^{\top }{R}_t$, formed from the two actual half-step plans. At a balanced fixed point it becomes $M=\mathrm{diag}(b{)}^{-1}{P}^{\top }\mathrm{diag}(a{)}^{-1}P$. These row-stochastic operators descend to potentials modulo constants, and their transposes transport zero-mass quotient cotangents. We give a self-contained calculus including score and marginal source terms and derive Dobrushin bounds for homogeneous and projected-source tails. The main structural result is a face-lattice dichotomy: a fixed support and compatible marginals admit a score-uniform one-step global quotient-mixing certificate if and only if every nonempty feasible face of the transportation polytope has pairwise two-hop column overlap. Otherwise finite scores can make the coefficient arbitrarily close to one. A windowed characterization extends this criterion to ordered products. Partition heat-bath layers and product-coordinate sweeps provide exact sparse examples in which every one-step coefficient can equal one while the full-window coefficient is zero. Forced bus mass, plan-distortion bounds, and register examples give additional sufficient certificates. The results concern fixed-support quotient transport; they do not establish complete neural-gradient bounds, downstream accuracy, or system speedups.