IndisputableMonolith.Foundation.JCostHessianC7
JCostHessianC7 isolates the exact quadratic numerator of the local J-cost expansion around the fixed point at unity. Equilibrium modelers cite the module when they need the invariant quadratic coefficient 1/2 and Hessian coefficient 1 that every RS equilibrium inherits from the same kernel. The module consists of a short chain of definitions and equality lemmas built directly on the J-cost imported from the Cost module.
claimThe local J-cost expansion has quadratic numerator $(x-1)^2$ and coefficient $1/2$, with Hessian coefficient $1$.
background
Recognition Science derives equilibria from the J-function obeying the composition law $J(xy)+J(x/y)=2J(x)J(y)+2J(x)+2J(y)$. The Cost module supplies the concrete J-cost kernel. This module extracts only the local Taylor behavior of that kernel near $x=1$, where the first derivative vanishes and the second derivative equals 1.
proof idea
This is a definition module. It introduces jcost_local_quadratic_kernel and jcostHessianCoefficient, then supplies the equality lemmas jcostHessianCoefficient_eq_one and jcostTaylorQuadraticCoefficient_eq that fix the coefficients at the stated values.
why it matters in Recognition Science
The module supplies the shared local J-kernel required by UniversalEquilibriumResponseC7. That file uses the quadratic coefficient 1/2 and Hessian coefficient 1 to establish the common core behind the claim that Nash, market, and health equilibria coincide at r=1. The empirical cross-field sensitivity comparison is left to later work.
scope and limits
- Does not derive global properties of the J-cost beyond the quadratic term.
- Does not prove equality of equilibria across distinct domains.
- Does not incorporate the phi-ladder, Berry threshold, or eight-tick octave.
- Does not address higher-order terms in the expansion.