IndisputableMonolith.GameTheory.CooperationCascade
The CooperationCascade module defines the cooperation cascade threshold and shows it equals the ESS threshold of 1/phi. Researchers working on evolutionary stability or RS-derived game theory would cite it to connect static ESS conditions to dynamical cascade behavior. The module organizes definitions and equality lemmas that import the phi-based threshold directly from the upstream ESSFromSigma result.
claimThe cascade threshold equals the ESS threshold and is given by $\phi^{-1}$, where $\phi$ is the golden ratio fixed point of the Recognition Composition Law.
background
The module sits inside the GameTheory domain and imports the RS time quantum $\tau_0 = 1$ tick from Constants together with the ESS existence criterion from ESSFromSigma. The upstream doc-comment states: "An evolutionarily stable strategy (ESS) is a strategy that, once adopted by the majority, cannot be invaded by a rare mutant. In RS, ESS exists iff the cooperator fraction is at least 1/phi in a kin-selected population." The cascade threshold is introduced as the dynamical counterpart to this static stability bound.
proof idea
This is a definition module, no proofs. It supplies the cascadeThreshold definition, the equality cascadeThreshold_eq_inv_phi, and supporting lemmas that link the threshold to the phi-ladder already established in ESSFromSigma.
why it matters in Recognition Science
The module supplies the cascade threshold that matches the ESS threshold, thereby extending the "Game theory from first principles" row of §XXIII.C. It feeds the parent claim that cooperation dynamics are governed by the same 1/phi bound derived from sigma-conservation. No downstream uses are recorded yet.
scope and limits
- Does not derive explicit population update rules.
- Does not treat non-kin selection or spatial structure.
- Does not compute numerical thresholds beyond the phi expression.
- Does not address multi-strategy or continuous-trait extensions.