From Axioms to Attractors: A Common Lyapunov Law for Equilibria Across Sectors

Across physics, core equilibrium relations—Born’s rule, entropy increase, and the curvature–matter balance—are usually postulates, not outcomes. We introduce a single restoration mechanism, the Deterministic Statistical Feedback Law (DSFL), which tracks a quadratic mismatch between a statistical blueprint and a physical response and proves it is a Lyapunov functional. Under standard reversibility or coercivity assumptions, this residual decays exponentially at an explicit, sector-dependent rate set by a spectral gap or a coercivity constant. As a flagship illustration, we restate the classical equivalence between exponential variance decay and a noncommutative Poincaré (spectral-gap) inequality for reversible quantum Markov semigroups, making the rate bookkeeping explicit. We also present two sharp demonstrators: finite-dimensional pure-dephasing Lindbladians, where the envelope is fixed by the slowest dephasing pair, and a coercive PDE template with an exact residual energy identity and a clean exponential envelope. The framework places classical, quantum, and PDE or field examples on the same Lyapunov footing and makes their rates directly comparable across sectors.