IndisputableMonolith.GameTheory.CooperationCascade
The CooperationCascade module defines the threshold at which cooperation becomes self-reinforcing in a Recognition Science population and equates it to the ESS threshold. Game theorists deriving stability from sigma-conservation would cite it to link cascade onset with the golden-ratio fixed point. The module organizes its content as a chain of definitions and equivalences that import the 1/phi condition from ESSFromSigma and verify its consequences for subcritical and full-cooperation regimes.
claimThe cascade threshold equals $1/phi$, identical to the minimum cooperator fraction required for an evolutionarily stable strategy (ESS) in a kin-selected population under sigma-conservation.
background
Recognition Science obtains game-theoretic results from the J-cost functional and the phi self-similar fixed point forced in the UnifiedForcingChain. The Constants module supplies the base time quantum tau_0 = 1 tick. The imported ESSFromSigma module states that an evolutionarily stable strategy exists if and only if the cooperator fraction is at least 1/phi in a kin-selected population. This module introduces the cooperation cascade as the dynamical counterpart of that threshold, where adoption by a sufficient fraction triggers runaway cooperation.
proof idea
This is a definition module whose argument proceeds by successive definitions and equivalences. It first introduces cascadeThreshold, proves equality to inv phi by reduction to the phi fixed-point property imported from ESSFromSigma, then derives the implications cascade_implies_ESS and subcritical_does_not_cascade as direct consequences of the threshold comparison.
why it matters in Recognition Science
The module supplies the cascade side of the game-theory development in section XXIII.C, feeding the parent results on ESS certification and full-cooperation cascades. It closes the link between the phi-ladder and strategy stability, confirming the 1/phi threshold as the critical point where cooperation dominates under the Recognition Composition Law.
scope and limits
- Does not derive explicit population dynamics or time evolution of the cascade.
- Does not treat non-kin-selected or multi-strategy games beyond cooperator-defector.
- Does not incorporate spatial structure or the D=3 dimension.
- Does not address corrections from quantum or relativistic regimes.