Discrete distributions and statistical mechanics of small systems

What the math gives to ML

The paper provides a constructive renormalization mechanism for nonnegative integer-valued distributions: thinning independent copies and summing them produces a semigroup whose fixed points are discrete-stable laws. Its transferable asset is a scale-consistency condition on count statistics, with the probability generating function predicting how aggregation changes the distribution, mean, variance, and tail behavior. A concrete neural-network use is to regularize stochastic routing or adaptive-computation counts so that their statistics remain predictable when batch size, microbatching, or stream duration changes. This is most promising for sparse mixture-of-experts routing, where uncontrolled count fluctuations cause capacity overflow and unstable expert utilization.