On Competing Restoring Modes and Intermediate-Entropy Stability in Evolutionary Game Dynamics

Evolutionary stability can be limited by different recovery processes on opposite sides of a parameter interval. We characterise such a competition in a noisy five-strategy Iterated Prisoner’s Dilemma with additive symmetric mutation. Continuation of its high-cooperation stationary branch finds two Hopf-type stability crossings at Generous Tit for Tat forgiveness q = 0.06858452 and 0.32707445. Between them, the weakest tangent-Jacobian restoring rate reaches 0.03200788 at q = 0.2895306043. An oscillatory eigenmode improves with forgiveness while a compositional mode deteriorates; their transverse crossing determines the non-smooth maximin optimum. Implicit eigenvalue sensitivity and direct optimization agree to a relative discrepancy of 3.23 × 10⁻⁹. A Walsh-basis calculation gives the exact ALT–GTFT payoff asymmetry 2.45q and links the weakening compositional mode to the change in population-weighted interaction entropy rate. At the stability optimum, that rate is strictly bracketed by its values at the two dynamical boundaries (0.107352 < 0.112838 < 0.151597); strategy and stationary-outcome entropies fail the same test. Architecture controls distinguish phase formation from entropy bracketing: 4/36 balanced deterministic mappings form two-sided regimes, 11/24 matched mildly stochastic architectures satisfy the full criterion versus 0/24 state-permuted controls, and 1/96 prospective continuous-policy systems is a strict pass. The result establishes a conditional, reproducible intermediate-entropy stability regime and an explicit mechanism for its local restoring optimum.