Limits of Stochastic Semigroups and Block-Triangular Majorisation

What the math gives to ML

The paper gives a constructive description of stochastic maps that preserve a Gibbs distribution when inverse temperature tends to infinity. Its key transferable asset is that this limit is not obtained by simply substituting the limiting distribution into the finite-temperature constraint: it produces block-upper-triangular stochastic operators with substochastic cross-block flow and a finite extremal decomposition. This suggests an energy- or hierarchy-aware neural mixing layer whose transition matrix preserves a prescribed Gibbs-like distribution while annealing toward one-way information flow between learned groups. The most practical first test is a constrained attention or MoE router, compared with ordinary softmax routing at equal compute.