universalResponseCert
The universal equilibrium response certificate asserts that three equilibrium domains share a response coefficient of 1 under the common J-cost kernel, supplying the formal basis for equating Nash, market, and health equilibria at r=1. A researcher modeling unified equilibria in Recognition Science would cite this to invoke the shared quadratic coefficient of 1/2 and Hessian coefficient of 1. The definition is a direct structure constructor that assembles the certificate from the domain cardinality result, the universal response theorem, and J-
claimLet $UniversalResponseCert$ be the structure with fields: the cardinality of the set of equilibrium domains equals 3, the response coefficient equals 1 for every equilibrium domain, and a $JCostHessianCert$. The definition constructs an instance of this structure by supplying the domain count of 3, the constant response coefficient of 1, and the J-cost Hessian certificate.
background
In the C7 module on Universal Equilibrium Response, any recognition science equilibrium modeled by the same local J-cost kernel inherits the quadratic coefficient 1/2 and Hessian coefficient 1. This supplies the common core for the claim that Nash, market, and health equilibria coincide at r=1. The module proves only the shared J-kernel; empirical sensitivity comparisons across fields remain outside its scope.
proof idea
The definition is a one-line structure constructor. It supplies the three_domains field directly from the equilibrium domain count theorem, the universal_response field from the response coefficient universal theorem, and the kernel field from the J-cost Hessian certificate.
why it matters in Recognition Science
This definition supplies the shared J-kernel that underpins the C7 claim of universal equilibrium response across domains. It feeds the formal common core for equating distinct equilibrium types at r=1 in the Recognition Science framework, relying on the J-cost Hessian properties established in the Foundation module. The construction closes the interface for any downstream model that assumes the same local kernel, though it leaves open the empirical validation of cross-field sensitivities.
scope and limits
- Does not establish empirical equivalence of sensitivities across Nash, market, and health domains.
- Does not derive the specific form of the J-cost kernel from more primitive axioms.
- Does not address equilibria outside the three enumerated domains.
- Does not compute numerical values for response coefficients in specific physical systems.
formal statement (Lean)
45noncomputable def universalResponseCert : UniversalResponseCert where
46 three_domains := equilibriumDomain_count
proof body
Definition body.
47 universal_response := responseCoefficient_universal
48 kernel := jcostHessianCert
49
50end UniversalEquilibriumResponseC7
51end Applied
52end IndisputableMonolith