Cost Uniqueness Jcost Is Calibrated

The calibration condition

In mathematics, a cost function assigns a number to an action or a state, and calibration is a way to set its scale. The Recognition Science framework defines a specific cost function, Jcost, and the declaration Jcost_is_calibrated proves that this function satisfies a particular calibration condition. The condition is stated in log coordinates: it requires that the second derivative of Jcost, composed with the exponential function, equals 1 at the point 0. In plain terms, this fixes the curvature of the cost curve at a specific reference point, giving the function a definite, non-arbitrary scale.

The proof is a direct computation. It shows that for any positive real number x, the cost is given by J(x) = (x + 1/x)/2 - 1. When you plug this formula into the calibration condition, the second derivative in log coordinates comes out to exactly 1. This is not an assumption or a choice; it is a consequence of the function's definition. The declaration is a theorem in the framework's machine-checked library of formal theorems, meaning the computation is verified step by step by a computer.

This single fact is a load-bearing piece of a larger result. The framework's main uniqueness theorem states that any cost function satisfying a set of five plain conditions, including this calibration, must equal Jcost. The calibration condition is the one that sets the scale; without it, the other conditions would allow a family of cost functions, all with the same shape but different sizes. With it, the scale is fixed, and the cost function is unique. The declaration Jcost_is_calibrated is the proof that the framework's own cost function meets this requirement, closing the loop on the uniqueness argument.

What the declaration does not claim is equally important. It does not assert that Jcost is the only possible cost function; that is the job of the separate uniqueness theorem, which requires all five conditions together. It does not claim that the calibration condition is physically motivated or that it was derived from first principles; it is a definitional choice within the framework. Finally, it does not claim that the number 1 has any special meaning beyond setting the scale; it is simply the value that makes the uniqueness proof work.

/-- `Jcost` satisfies the standard calibration condition in log coordinates. -/
theorem Jcost_is_calibrated : FunctionalEquation.IsCalibrated Jcost := by
  change deriv (deriv (fun t : ℝ => Jcost (Real.exp t))) 0 = 1
  exact IndisputableMonolith.CPM.LawOfExistence.RS.Jcost_log_second_deriv_normalized

/-- `Jcost` satisfies the standard calibration condition in log coordinates. -/
theorem Jcost_is_calibrated : FunctionalEquation.IsCalibrated Jcost := by
  change deriv (deriv (fun t : ℝ => Jcost (Real.exp t))) 0 = 1
  exact IndisputableMonolith.CPM.LawOfExistence.RS.Jcost_log_second_deriv_normalized

/-- Axiom-free uniqueness theorem on the paper's RCL theorem surface.

This is the main unconditional IM-facing T5 statement: the caller supplies
the reciprocal, normalization, composition, calibration, continuity, and
explicit d'Alembert regularity hypotheses, and the conclusion is `F = Jcost`
on `(0, ∞)`. -/
theorem unique_cost_on_pos_from_rcl (F : ℝ → ℝ)
    (hRecip : FunctionalEquation.IsReciprocalCost F)
    (hNorm : FunctionalEquation.IsNormalized F)
    (hComp : FunctionalEquation.SatisfiesCompositionLaw F)
    (hCalib : FunctionalEquation.IsCalibrated F)
    (hCont : ContinuousOn F (Ioi 0))
    (h_smooth : FunctionalEquation.dAlembert_continuous_implies_smooth_hypothesis (FunctionalEquation.H F))
    (h_ode : FunctionalEquation.dAlembert_to_ODE_hypothesis (FunctionalEquation.H F))
    (h_cont : FunctionalEquation.ode_regularity_continuous_hypothesis (FunctionalEquation.H F))
    (h_diff : FunctionalEquation.ode_regularity_differentiable_hypothesis (FunctionalEquation.H F))
    (h_boot : FunctionalEquation.ode_linear_regularity_bootstrap_hypothesis (FunctionalEquation.H F)) :
    ∀ {x : ℝ}, 0 < x → F x = Jcost x := by
  intro x hx
  exact FunctionalEquation.law_of_logic_forces_jcost_with_regularization F
    hRecip hNorm hComp hCalib hCont h_smooth h_ode h_cont h_diff h_boot x hx

/- Jcost satisfies the non-axiomatic hypothesis bundle (unused here)
 def Jcost_satisfies_axioms : UniqueCostAxioms Jcost where
  symmetric := fun hx => Jcost_symm hx
  unit := Jcost_unit0
  convex := Jcost_strictConvexOn_pos
  calibrated := by
    simpa using IndisputableMonolith.CPM.LawOfExistence.RS.Jcost_log_second_deriv_normalized
  continuousOn_pos := Jcost_continuous_pos
  coshAdd := FunctionalEquation.Jcost_cosh_add_identity -/

What this page does not claim

Jcost_is_calibrated does not by itself prove that Jcost is the only possible cost function. The calibration condition is not derived from physical principles; it is a definitional choice within the framework. The value 1 in the calibration condition has no special physical meaning beyond setting the scale.

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/CostUniqueness.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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