Beyond Fixed Points: Ruelle-Conley Ergodic-Homological Equilibrium in Non-Convergent Game Dynamics

This work formulates the Algorithmic Meta-Game of Learning Dynamics and introduces a new dynamical solution concept—Ruelle-Conley Ergodic-Homological Equilibrium (RCE)—for multi-agent systems whose learning trajectories fail to converge to static Nash equilibria and instead settle onto limit cycles, quasi-periodic tori, or general hyperbolic attractors. By integrating Conley Index Theory with Ruelle's Linear Response Theory on Sinai-Ruelle-Bowen (SRB) physical measures, we establish the Cohomological Adjoint Equation for polynomial-time spectral auditing of strategic stationarity, the Branch-Hysteresis Rent and Ergodic Price of Algorithmic Autonomy (EPoAA), the Koszul-Amari Cohomological Invariance and Carnot-KL Berry-Phase Theorem on the probability simplex, and the Toroidal Arnold-Tongue Homological Collapse and Semicircle Collusion Law, together with a Pigouvian cross-adjoint tax mechanism. Authorship & Citation Note:The theoretical framework, solution concepts, and analytical results in this work are the original contribution of Xuanyang Zhang (ORCID: 0009-0009-7646-6169). This preprint is archived as a timestamped record of version 1.0; please cite this record and its DOI when referencing or building upon these results.