gameTheoryCert
plain-language theorem explainer
The declaration populates the master GameTheoryCert structure with the sigma charges and conservation properties for the four 2x2 outcomes. Game theorists working inside Recognition Science cite it to confirm that mutual cooperation carries positive sigma while defection is the sole violator and that the coordination dividend sits at 1/phi. The construction is a direct record literal that assembles the pre-proved sibling lemmas for each field.
Claim. The structure GameTheoryCert is defined by the record with fields CC_sigma : jointSigma(cooperate, cooperate) = 1, DD_sigma : jointSigma(defect, defect) = -1, DD_violates : not isSigmaConservative(defect, defect), nonDD_conservative : every non-defect-defect outcome is sigma-conservative, dividend_band : 0.617 < cooperationDividend < 0.622, and dividend_strict_unit : 0 < cooperationDividend < 1.
background
The module Cooperation From sigma-Conservation treats game-theoretic equilibria as J-cost minima on the multi-agent ledger. Recognition Science predicts that any Nash equilibrium is a stationary point of the joint J-cost functional and that a two-player game is cooperative precisely when joint sigma is conserved across moves. Defect-defect is the unique sigma-non-conservative outcome in the prisoner's-dilemma normal form, while the cooperation dividend equals 1/phi in (0.617, 0.622). Upstream, the Economics.GameTheoryFromRS module supplies a parallel certificate structure whose fields include five game types and the Jcost relations Jcost(1) = 0 together with the strict positivity of Jcost(r) for r not equal to 1.
proof idea
The definition is a one-line record construction that directly assigns each field of GameTheoryCert. It pulls CC_sigma from the sibling theorem CC_sigma_pos, DD_sigma from DD_sigma_neg, the violation from DD_not_sigma_conservative, the conservation property from nonDD_sigma_conservative, the numerical band from cooperationDividend_band, and the strict bounds from the pair cooperationDividend_pos together with cooperationDividend_lt_one.
why it matters
This definition supplies the concrete inhabitant of GameTheoryCert that is referenced by the higher-level certificate in Economics.GameTheoryFromRS. It realizes element 84 of the framework, confirming the sigma-based account of why reciprocal-altruism strategies dominate iterated play. The construction ties the game-theoretic dividend directly to the phi-ladder and to the sigma-conservation law that follows from the Recognition Composition Law.
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