State convertibility and fluctuation theorems from a dynamical reference: majorization meets martingales

What the math gives to ML

The paper provides a constructive criterion for state convertibility under a changing reference distribution: a transition matrix must preserve the reference while mapping the target state, and this is equivalent to a martingale coupling between the relative-population distributions. This gives an implementable constraint for stochastic neural layers whose outputs are probability distributions, ensuring that transformations do not create relative-population variability unavailable in the input. The most promising transfer is a martingale-constrained probabilistic layer or regularizer, with convex-order violations serving as a quantitative certificate of an impossible or overly expressive transition.