Foundation Jcost Hessian C7 Jcost Hessian Cert

The local shape of cost

In the Recognition Science framework, recognition (a discrete record of an event) has a forced cost. The cost function J(x) = (x + 1/x)/2 - 1 measures how far a ratio x is from 1, the point of no cost. The declaration JCostHessianCert certifies a precise fact about J near that point: if you write x = 1 + eps, where eps is a small disturbance, then J(1 + eps) = eps^2 / (2 * (1 + eps)). This is an exact algebraic identity, not an approximation, and it holds for every eps except -1, where the formula would divide by zero.

The identity implies that for small eps, the cost grows like eps^2 / 2. The factor 1/2 is the quadratic coefficient in the Taylor expansion of J at equilibrium. In the standard convention where the Hessian is twice the quadratic coefficient, that makes the Hessian coefficient exactly 1. The declaration packages these three facts: the local kernel identity, the coefficient 1/2, and the Hessian value 1. Each is a theorem in the framework's machine-checked library of formal theorems, with no unproved assumptions.

The point of this local expansion is stability. At x = 1, the cost has a minimum: a small disturbance costs something, but the cost vanishes as the disturbance shrinks. The quadratic growth means the system resists small departures from equilibrium, a property that underpins the framework's account of how recognition events settle into stable patterns. The exact identity, not just a second-order approximation, is what the declaration adds beyond a typical Taylor argument.

What the declaration does not claim: it does not assert that J is the only cost function, that the local expansion holds globally, or that the Hessian coefficient 1 has any meaning outside the framework's own definitions. It is a local statement about a specific function at a specific point, certified by the library, and nothing more.

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 jcostTaylorQuadraticCoefficient_eq :
    jcostTaylorQuadraticCoefficient = 1 / 2 := rfl

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

What this page does not claim

The declaration does not prove that J is the unique cost function satisfying the framework's axioms. The local expansion does not describe the cost for disturbances equal to -1, where the formula is undefined. The Hessian coefficient 1 has no stated meaning beyond the framework's own definitions.

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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