Coarse-grained kinetic scale tightens thermodynamic spectral bounds of Markov cycles

What the math gives to ML

The paper provides a constructive spectral comparison principle for continuous-time Markov cycles: affinity alone does not determine the sharpest oscillation bound, because the geometric means of the forward and backward rates provide an additional coarse-grained kinetic scale. A nonuniform cycle is compared with a uniform cycle having the same products of forward and backward rates and the same affinity; the uniform comparison cycle bounds the oscillation frequency in every winding sector, with equality only for uniform rates. This can transfer to recurrent and state-space neural layers by parameterizing a cyclic continuous-time hidden-state generator and using the comparison spectrum as a cheap, differentiable stability and frequency controller.