Foundation Jcost Hessian C7

The local shape of cost

The cost function J(x) = (x + 1/x)/2 - 1 measures the price of a recognition event when reality's ledger, a discrete record of events, compares two states. At x = 1 the cost is zero, the natural resting point. The question is how the cost grows when the input moves slightly away from 1. Write the input as 1 + eps, where eps is a small real number not equal to -1. The framework's machine-checked library of formal theorems proves the exact identity J(1 + eps) = eps^2 / (2(1 + eps)). This is not an approximation; it holds for every allowed eps, and it shows the cost rises quadratically for small departures.

Quadratic growth means the cost has a well-defined local curvature, and the framework pins that curvature down. In the usual Taylor convention, the quadratic coefficient is 1/2, and the Hessian coefficient, twice the quadratic coefficient, is exactly 1. The library records both as formal definitions and proves the Hessian coefficient equals one. So at equilibrium the cost function bends with unit curvature, the same local shape as the simplest parabola y = x^2/2. This local fact is a consequence of the global forced form of J, not an independent assumption.

In Recognition Science, this local expansion matters because it fixes the response of the cost to small perturbations, the regime where recognition events are nearly balanced. The exact identity strengthens the usual second-order Taylor claim: it holds away from eps = -1, not merely in a limit. The module packages the identity, the coefficient, and the Hessian value into a single certificate structure, a bundle of three proved statements. The certificate exists so that later results can rely on the local shape without re-deriving it.

The practical upshot: any small wobble around the equilibrium costs quadratically, with a universal curvature of one. That unit curvature is a derived number, not a fitted parameter, and it is the same for every recognition event in the framework. The library's proof is axiom-clean, meaning it rests only on the standard logical postulates of the ambient type theory, with no framework-specific axioms.

theorem jcost_one_plus_eq (eps : ℝ) (h : eps ≠ -1) :
    Jcost (1 + eps) = eps ^ 2 / (2 * (1 + eps)) := by
  have hx : 1 + eps ≠ 0 := by
    intro hz
    apply h
    linarith
  rw [Jcost_eq_sq hx]
  ring_nf

theorem jcostHessianCoefficient_eq_one :
    jcostHessianCoefficient = 1 := by
  unfold jcostHessianCoefficient jcostTaylorQuadraticCoefficient
  norm_num

theorem jcost_one_plus_eq (eps : ℝ) (h : eps ≠ -1) :
    Jcost (1 + eps) = eps ^ 2 / (2 * (1 + eps)) := by
  have hx : 1 + eps ≠ 0 := by
    intro hz
    apply h
    linarith
  rw [Jcost_eq_sq hx]
  ring_nf

def jcostHessianCert : JCostHessianCert where
  local_kernel := jcost_local_quadratic_kernel
  coefficient_half := jcostTaylorQuadraticCoefficient_eq
  hessian_one := jcostHessianCoefficient_eq_one

What this page does not claim

This module does not derive the full forcing chain or the golden ratio; it only establishes the local quadratic expansion of J at equilibrium. The Hessian coefficient being one does not imply that J is globally a parabola; the identity is exact but local in form. The axiom-clean proof does not claim the ambient type theory is free of postulates; it only claims no framework-specific axioms are used.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Foundation/JCostHessianC7.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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