IndisputableMonolith.Foundation.JCostHessianC7
Module that pins the exact quadratic numerator in the local expansion of the recognition cost J about the identity, and shows the Hessian coefficient equals one. Curvature-cost and alpha-genesis work cite it when comparing geometric quadratics (Regge, Dirichlet, cube curvature) to genuine J-cost. The argument is direct Taylor/Hessian calculus on J together with algebraic identities from the Cost library, closed by a small certificate bundle.
claimFor the recognition cost $J(x)=(x+x^{-1})/2-1$, the local expansion about the identity is $J(1+\varepsilon)=\tfrac12\varepsilon^2+O(\varepsilon^3)$; equivalently the Hessian coefficient of $J$ at $1$ equals $1$. The module records the exact quadratic numerator of that expansion and packages the identity as a reusable certificate.
background
Recognition Science takes the unique cost $J$ forced by the Recognition Composition Law and T5 uniqueness: $J(x)=(x+x^{-1})/2-1$, also $J(x)=\cosh(\log x)-1$. Near the fixed point $x=1$, only the quadratic jet matters for comparing infinitesimal defects to geometric energy forms.
This foundation module sits on IndisputableMonolith.Cost and isolates that jet: the exact quadratic numerator in $J(1+\varepsilon)$, the Taylor quadratic coefficient, and the Hessian coefficient at unity. Sibling lemmas name the one-sided shift $J(1+\cdot)$, the local quadratic kernel, and the equality of the Hessian coefficient to one.
Downstream curvature work needs a clean, named fact that the recognition quadratic is normalized with leading coefficient $1/2$ (Hessian $1$), so that any competing bulk or seed quadratic can be judged against a single RS-native scale rather than an ad hoc prefactor.
proof idea
Definition-and-lemma module, not a single deep theorem. It introduces the local quadratic kernel and the named Taylor/Hessian coefficients, then proves the coefficient identities by direct differentiation (or series) of $J$ at $1$, using Cost-library simplifications such as the evaluation and derivative rules for $J$. The equality of the Hessian coefficient to one is the main computational lemma; a small certificate structure bundles the normalized expansion for import by later modules.
why it matters in Recognition Science
Parent consumers are the Alpha Genesis curvature verdict and the Curvature Cost Form bridge. The verdict module records that the seed $4\pi\cdot 11$ is a category error relative to genuine J-cost; the cost-form module is the M2B bridge from the Regge-to-J-curvature plan and states that the bulk Regge/Dirichlet quadratic cannot be the source of the one-cell recognition cost. Both comparisons require a fixed, proved local quadratic for $J$ rather than a floating normalization. In the forcing chain this is the analytic face of T5 J-uniqueness: once $J$ is fixed, its Hessian at the identity is forced, and curvature or alpha arguments that invoke a "recognition quadratic" must match this coefficient. The module therefore closes a foundation gap between abstract J-uniqueness and concrete geometric cost comparisons.
scope and limits
- Does not derive J-uniqueness or the RCL; those are assumed from Cost and the forcing chain.
- Does not identify J-cost with Regge, Dirichlet, or Einstein-Hilbert quadratics.
- Does not compute global or finite-defect costs away from the identity.
- Does not fix dimension, eight-tick structure, or the value of alpha.
- Does not by itself reject the $4\pi\cdot 11$ seed; that verdict lives downstream.