cascadeThreshold_eq_inv_phi
plain-language theorem explainer
The cascade threshold equals the reciprocal of the golden ratio. Game theorists deriving cooperation thresholds in n-person prisoner's dilemmas from the J-cost gradient would cite this equality to connect the cascade condition to the ESS threshold. The proof is a one-line reflexivity that follows at once from the definition of cascadeThreshold as cooperatorThreshold.
Claim. Let $τ$ denote the cascade threshold. Then $τ = φ^{-1}$, where $φ$ is the golden ratio.
background
The Cooperation Cascade module treats the cascade threshold as the critical cooperator fraction that triggers full cooperation via the J-cost gradient in a kin-cluster. This threshold is introduced as cascadeThreshold and set equal to cooperatorThreshold. The module sits inside the game-theory row of the Recognition framework and imports the ESS threshold construction from ESSFromSigma together with the phi-based constants.
proof idea
The proof is a one-line reflexivity that applies the definition cascadeThreshold := cooperatorThreshold and the prior identification of that quantity with 1/phi.
why it matters
This equality confirms that the cascade threshold coincides with the ESS threshold, thereby linking the cascade side of the Cooperation Cascade Theorem to the stable-strategy results in ESSFromSigma. It supplies the concrete numerical anchor (1/φ) required by the module's statement that crossing the threshold drives the cluster to full cooperation. The result closes one step in the game-theory derivation from the J-functional and the phi fixed point.
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