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

Abstract: Thermodynamic bounds on the spectrum of a Markov cycle depend on the cycle affinity and maximum escape rate, but they ignore how the local transition rates are distributed around the cycle. In this work, we show that a single additional cycle-wide kinetic quantity, the geometric means of forward and backward rates, yields a strictly stronger bound for every nonuniform cycle. By comparing each winding sector of the eigenmode with a product-matched uniform cycle, we bound oscillation frequencies, tighten the admissible spectral region, and show that only uniform cycles can saturate the Uhl-Seifert boundary. Our results show that retaining a coarse-grained kinetic scale can sharpen thermodynamic spectral bounds without requiring knowledge of the full generator. Our method also provides a complex-analytic framework which unifies generator-specific kinetic information with established affinity-winding bounds.