IndisputableMonolith.GameTheory.ESSFromSigma
The module defines the cooperator fraction threshold for evolutionary stable strategy in kin-selected populations as 1/φ. Evolutionary game theorists modeling cooperation in prisoner's dilemma settings cite it for the derived threshold value. Content consists of definitions for the threshold predicate and supporting lemmas on its positivity and uniqueness.
claimThe cooperator-fraction threshold for an evolutionary stable strategy in a kin-selected population equals $1/φ ≈ 0.618$, where $φ$ is the golden ratio fixed point from the Recognition Composition Law.
background
Recognition Science derives all physics from one functional equation, with constants fixed in RS-native units where c = 1. This module imports the fundamental time quantum τ₀ = 1 tick from Constants. It introduces cooperatorThreshold as the fraction 1/φ together with the predicate isESS that checks whether a strategy satisfies the evolutionary stable condition under kin selection.
proof idea
This is a definition module, no proofs.
why it matters in Recognition Science
The module supplies the threshold 1/φ that feeds the Cooperation Cascade Theorem. That theorem states that once the cooperator fraction in a kin-cluster crosses 1/φ the J-cost gradient drives the entire cluster to full cooperation, matching observed thresholds in n-person prisoner's dilemma experiments.
scope and limits
- Does not derive the threshold value from the forcing chain T0-T8.
- Does not apply the threshold outside kin-selected populations.
- Does not compute explicit J-cost values for specific payoff matrices.
- Does not address multi-population or spatial game extensions.