A Universal Control Budget for First-Passage Kinetics

What the math gives to ML

The paper establishes a universal control-budget theorem for the mean first-passage time of any finite continuous-time Markov chain: the logarithmic sensitivity to each individual transition rate has magnitude at most one, while the sensitivities over all rates sum to -1. This gives a quantitative conservation law for completion-time control: globally accelerating all rates rescales completion time inversely, and no single edge can contribute more than one logarithmic unit of control. A direct neural-network transfer is a stochastic adaptive-computation architecture whose layer transitions form a finite Markov chain and whose absorbing state represents halting; the theorem then supplies an exact sensitivity monitor and principled rate-allocation rule for learned early-exit or routing policies.