The Replicator-Optimization Mechanism — A Scale-Relative Formalism for Persistence-Conditioned Dynamics

Abstract

Abstract Four independent fields—physics, biology, economics, and cultural evolution—have converged on the same mathematical machinery for describing persistence-conditioned dynamics. The convergence is not metaphorical but literal: the same fitness landscapes, selection operators, and transmission kernels appear independently. We synthesize these into the Replicator-Optimization Mechanism (ROM): a unified apparatus instantiable at any scale. Key Contributions Cross-field synthesis: Physics, biology, economics, and cultural evolution share identical formal structure Political application: ROM instantiated with friction from stake-voice mismatch as primitive, legitimacy as survival probability Machine-checked proofs: Core algebraic results verified in Lean 4 with Mathlib (28 theorems, zero sorry placeholders) Key results: Simplex preservation, survival monotonicity, moving equilibrium existence, impossibility of static equilibrium under varying friction Links arXiv: arXiv:2601.06363 Lean 4 proofs: github.com/studiofarzulla/lean-formalizations ASCRI: systems.ac/4/DAI-2503 Research Lab: Dissensus AI