Cost Calibration Jcost Comp Exp Eq Jlog

Calibration in log-coordinates

The cost function Jcost measures the price of a recognition event as a ratio x, with x = 1 meaning no change and zero cost. The identity Jcost(exp t) = Jlog t says that if you write the ratio as an exponential, the cost becomes the function Jlog(t) = cosh t - 1. This is not a deep theorem; it is true by definition of Jlog. Its importance is that it turns the multiplicative ratio x into the additive coordinate t, where the familiar tools of calculus apply directly.

In these log-coordinates, the calibration axiom takes a simple form. The second derivative of Jlog at t = 0 equals 1, which means the cost curve has unit curvature at the identity. This fixes the scale of the cost function uniquely, completing the characterization of Jcost. The machine-checked library of formal theorems proves this in a short chain: Jlog equals cosh t - 1, its derivatives are sinh and cosh, and cosh 0 = 1.

What the identity does not claim is equally important. It does not say that Jcost is the only possible cost function; the uniqueness of Jcost is a separate theorem resting on five axioms. It does not say that the second derivative being 1 is a derived fact; it is the calibration axiom itself, assumed rather than proved. And it does not claim anything about physical constants or the fine-structure constant, which remain open targets.

/-- Identity: (Jcost ∘ exp) equals Jlog pointwise. -/
lemma Jcost_comp_exp_eq_Jlog : (fun t : ℝ => Jcost (Real.exp t)) = Jlog := rfl

/-- The calibration theorem: second derivative at zero equals 1 -/
theorem Jlog_second_deriv_at_zero : deriv (deriv Jlog) 0 = 1 := by
  rw [deriv2_Jlog]
  exact cosh_zero

/-- Calibration for Jcost in log-coordinates: second derivative at zero is 1. -/
theorem Jcost_comp_exp_second_deriv_at_zero :
    deriv (deriv (fun t : ℝ => Jcost (Real.exp t))) 0 = 1 := by
  -- Jcost ∘ exp = Jlog by definition
  have h : (fun t : ℝ => Jcost (Real.exp t)) = Jlog := rfl
  rw [h]
  exact Jlog_second_deriv_at_zero

What this page does not claim

The identity does not prove that Jcost is the only possible cost function. The unit curvature is an axiom, not a derived consequence. No claim is made about the fine-structure constant or any physical constant.

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/Cost/Calibration.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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: