IndisputableMonolith.Decision.NashEquilibriumFromJCost
This module derives Nash equilibrium from J-cost minimization in the Recognition Science decision framework. It shows that joint homeostasis at r=1 forces J-cost to zero and certifies stability against unilateral deviation. Decision theorists cite the zero-cost and stability lemmas to connect the J function to equilibrium concepts. The module organizes its argument as a chain of cost lemmas imported from the Cost module.
claimAt joint homeostasis where every agent satisfies $r=1$, the J-cost vanishes ($J=0$), which certifies a Nash equilibrium: no agent can lower its cost by deviating while others remain fixed.
background
The module imports the J-cost definition from IndisputableMonolith.Cost, where $J(x)=(x+x^{-1})/2-1$. Joint homeostasis is the state in which all agents sit at the fixed point $r=1$ on the phi-ladder. The Recognition Composition Law is presupposed to ensure that total cost is additive across agents. The module then introduces the Nash equilibrium certificate as the object that records the zero-cost condition together with the deviation penalty.
proof idea
This is a definition-and-lemma module. It first records the algebraic identity that J(1)=0 at homeostasis, then proves that any deviation r≠1 raises J-cost, and finally assembles these facts into the Nash stability statement. All steps are one-line wrappers that apply the algebraic properties of J already established in the upstream Cost module.
why it matters in Recognition Science
The module supplies the decision-theoretic bridge that lets the Recognition Science forcing chain (T5–T8) reach equilibrium concepts. It feeds the parent claim that stable configurations arise precisely when total J-cost is minimized at the phi-fixed point. By grounding Nash equilibrium in the zero-cost property at r=1, the module closes the loop between the Recognition Composition Law and multi-agent stability.
scope and limits
- Does not address equilibria outside the J-cost model.
- Does not treat states away from joint homeostasis.
- Does not compute explicit numerical equilibria for finite agent sets.
- Does not invoke the full eight-tick octave or spatial dimension D=3.