Foundation Jcost Hessian C7 Jcost Hessian Coefficient

The quadratic coefficient

The Recognition Science cost function J(x) = (x + 1/x)/2 - 1 measures the price of a recognition event. Near its equilibrium point x = 1, the cost behaves like a simple quadratic: if you move a small amount eps away from 1, the cost is approximately eps²/2. The exact formula, proved for every eps except -1, is J(1 + eps) = eps² / (2(1 + eps)). The factor 1/2 is the Taylor quadratic coefficient, and in the standard convention the Hessian coefficient is twice that, so it equals 1.

The declaration jcostHessianCoefficient is a definition in the framework's machine-checked library of formal theorems. It sets the Hessian coefficient to 2 times the quadratic coefficient, and a theorem proves the result is 1. This is not a numerical approximation; it is an exact algebraic identity, checked by the kernel with no axioms beyond the standard three. The same module packages the local kernel, the coefficient half, and the Hessian one into a single certificate structure.

What the declaration does not claim is broader. It says nothing about the global shape of J away from the equilibrium, nothing about the forcing chain that derives J itself, and nothing about the physical interpretation of the Hessian as a curvature in a recognition space. The coefficient 1 is a local property at x = 1, not a statement about the cost's behavior at other points. The theorem holds for all real eps except -1, where the formula's denominator vanishes; that excluded point is a pole, not a counterexample.

For a reader, the practical content is plain: at equilibrium, the cost function has a fixed quadratic curvature, and the constant that measures it is exactly 1. This is the kind of exact local fact that anchors later global arguments, even though this declaration alone does not carry them.

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

What this page does not claim

The declaration does not prove the global form of J or the forcing chain that derives it. It does not claim the Hessian coefficient is a curvature in any geometric sense beyond the algebraic identity. It does not apply at eps = -1, where the formula's denominator is zero.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

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