Foundation Jcost Hessian C7 Jcost One Plus Eq

The local expansion

The recognition cost function J(x) = (x + 1/x)/2 - 1 measures how far a recognition event ratio x sits from its natural equilibrium at x = 1, where the cost is zero. The declaration jcost_one_plus_eq establishes a precise local fact: when x is written as 1 + eps, with eps not equal to -1, the cost is exactly eps^2 / (2(1 + eps)). This is not a Taylor approximation; it is an algebraic identity that holds for every real eps except the single excluded point.

The formula shows that near equilibrium, the cost grows quadratically with the deviation eps. The leading term is eps^2 / 2, and the framework records this as a Taylor coefficient of 1/2. In the standard convention where the Hessian is twice the quadratic coefficient, the Hessian coefficient at equilibrium is exactly 1. The framework's machine-checked library of formal theorems proves these statements with no gaps and no additional axioms.

In Recognition Science, this local expansion is the seed of a broader structural claim. The framework models recognition events as entries in a discrete ledger, a record of comparisons, and the cost of each recognition is forced by five plain conditions. The exact quadratic form at equilibrium is the first step toward understanding how small perturbations behave: a recognition ratio slightly off from 1 incurs a cost that rises with the square of the deviation, not linearly.

The declaration does not claim that the full cost function is globally quadratic, nor that the Hessian coefficient 1 applies anywhere except at the single equilibrium point x = 1. It also does not assert that the Taylor expansion converges or that higher-order terms are absent; the exact formula shows the denominator (1 + eps), so the behavior far from equilibrium is not quadratic. The theorem is a local statement, precise and limited to the neighborhood of x = 1.

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 cost function is globally quadratic; it is only locally quadratic near x = 1. The Taylor expansion converges for all eps; the exact formula includes a denominator that diverges at eps = -1. The Hessian coefficient 1 applies at any point other than the single equilibrium point x = 1.

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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Derived articles

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