responseCoefficient_universal
The theorem shows that the response coefficient equals 1 for every equilibrium domain. Researchers comparing equilibria across game theory, markets, and health states cite it to justify a uniform Hessian response under the shared J-cost kernel. The proof is a direct term reduction that unfolds the coefficient definition and invokes the base equality for the J-cost Hessian.
claimFor any equilibrium domain $D$ (one of game theory, financial market, or health state), the response coefficient equals 1.
background
The C7 module shows that any RS equilibrium modeled by the same local J-cost kernel inherits the quadratic coefficient 1/2 and Hessian coefficient 1. EquilibriumDomain is the inductive type with constructors gameTheory, financialMarket, and healthState. The responseCoefficient definition maps every such domain to the identical jcostHessianCoefficient value. This rests on the upstream theorem jcostHessianCoefficient_eq_one, which states that the coefficient equals 1 by unfolding to the Taylor quadratic term and applying numerical normalization.
proof idea
The proof is a one-line term wrapper. It unfolds responseCoefficient to jcostHessianCoefficient and then applies the theorem jcostHessianCoefficient_eq_one.
why it matters in Recognition Science
This theorem supplies the universal_response field inside the universalResponseCert definition, which certifies the common kernel across the three domains. It formalizes the shared J-cost structure behind the C7 claim that Nash, market, and health equilibria coincide at r=1. The module explicitly notes that empirical cross-field sensitivity comparisons lie outside the present formal core.
scope and limits
- Does not prove empirical equivalence of sensitivities across domains.
- Does not derive the J-cost kernel from more primitive axioms.
- Applies only to the three enumerated equilibrium domains.
- Ignores higher-order terms beyond the Hessian coefficient.
Lean usage
example (D : EquilibriumDomain) : responseCoefficient D = 1 := responseCoefficient_universal Dformal statement (Lean)
35theorem responseCoefficient_universal (D : EquilibriumDomain) :
36 responseCoefficient D = 1 := by
proof body
Term-mode proof.
37 unfold responseCoefficient
38 exact jcostHessianCoefficient_eq_one
39