Generalizing the multidimensional thermodynamic uncertainty relation to combinations of arbitrary counting variables

What the math gives to ML

The paper develops a multidimensional thermodynamic uncertainty relation for arbitrary combinations of counting observables, including currents, traffic, flows, coarse-grained observations, finite observation windows, and time-dependent driving. The transferable mechanism is a covariance-adjusted precision bound: correlated observables cannot all exhibit large mean rates and low fluctuations without a corresponding hidden dissipation or activity cost. In neural-network optimization, this can become an online diagnostic and learning-rate controller based on signed parameter motion, absolute update traffic, and layerwise update observables. The key falsifiable prediction is that a covariance-adjusted precision score crosses a reproducible threshold near optimizer divergence, earlier and more reliably than scalar gradient variance.