Foundation Primitive Recognition Calculus Prccalibration Independence Cost Lambd

The gauge family

A recognition cost (a function that assigns a penalty to each possible recognition event) in the Recognition Science framework is not a free choice. The framework's central theorem proves that any cost function meeting five plain conditions must equal J(x) = (x + 1/x)/2 − 1. But before that theorem can bite, one must ask which of its conditions actually does the work. The declaration costLambda_satisfiesCompositionLaw answers a specific part of that question: it shows that the composition law, one of the five conditions, is satisfied not just by the canonical cost J but by an entire family of functions.

The family is defined in multiplicative coordinates as costLambda(c, x) = (x^c + x^(−c))/2 − 1, where c is any positive real number and x is positive. The c = 1 member is exactly J. The declaration proves that every member of this family satisfies the composition law, which in log coordinates takes the form of the d'Alembert identity: G(t+u) + G(t−u) = 2G(t)G(u) + 2, where G(t) = cosh(c·t) − 1. This identity is the engine of the upstream classification theorem, so showing that the whole family obeys it is nontrivial: the composition law alone does not pin down J.

The declaration is part of a larger independence result. The full non-calibration hypothesis set of the uniqueness theorem, namely reciprocity, normalization, the composition law, and continuity on the positive reals, is satisfied by every member of the family for every positive c. The only hypothesis that distinguishes c = 1 from all other values is calibration, the condition that G''(0) = 1, which holds if and only if c = 1. The family is faithful: distinct values of c give genuinely distinct cost functions. The conclusion is that the unit of scale is the one irreducible gauge choice in the framework; nothing in the cost laws forces it.

What the declaration does not claim is equally important. It does not prove that the composition law is satisfied by all possible cost functions, only by this specific family. It does not establish that the composition law is irrelevant; rather, it shows that the law is necessary but not sufficient. It does not claim that calibration is arbitrary or unjustified, only that it is a separate hypothesis not derivable from the other four. The declaration is a precision tool: it isolates exactly where the uniqueness theorem's power comes from, which is the calibration condition, and it does so by exhibiting a family that satisfies everything else.

theorem costLambda_satisfiesCompositionLaw (c : ℝ) :
    Cost.FunctionalEquation.SatisfiesCompositionLaw (fun x => costLambda c x) :=
  (Cost.FunctionalEquation.composition_law_equiv_coshAdd _).mpr (costLambda_coshAddIdentity c)

/-- The `c = 1` member is exactly `Cost.Jcost` on the positives. -/
theorem costLambda_one_eq_Jcost {x : ℝ} (hx : 0 < x) :
    costLambda 1 x = Cost.Jcost x := by
  rw [costLambda_eq_cosh 1 hx, one_mul]
  show Real.cosh (Real.log x) - 1 = (x + x⁻¹) / 2 - 1
  exact costLambda_one_eq_J x hx

/-- The remaining hypothesis, calibration (`G''(0) = 1`), holds iff `c = 1`, because
`G(costLambda c)'' (0) = c²`. -/
theorem costLambda_isCalibrated_iff {c : ℝ} (hc : 0 < c) :
    Cost.FunctionalEquation.IsCalibrated (fun x => costLambda c x) ↔ c = 1 := by
  have hG : Cost.FunctionalEquation.G (fun x => costLambda c x)
      = fun t => Real.cosh (c * t) - 1 := by
    funext t; exact G_costLambda c t
  unfold Cost.FunctionalEquation.IsCalibrated
  rw [hG]
  exact curvature_one_iff_J hc

/-- **Item 2, airtight: calibration is the ONLY hypothesis of the uniqueness theorem
that pins `J`.** Every gauge member satisfies the full non-calibration hypothesis set of
`law_of_logic_forces_jcost` (reciprocity, normalization, the RCL composition law, and
continuity on the positives); calibration holds iff `c = 1`; and the family contains
genuinely distinct members. Hence the four non-calibration hypotheses are satisfied by
more than one function and cannot determine `J`: the unit of scale is logically
independent of the entire law set except for the calibration choice. -/
theorem calibration_is_the_only_hypothesis_pinning_J :
    (∀ c : ℝ,
        Cost.FunctionalEquation.IsReciprocalCost (fun x => costLambda c x)
          ∧ Cost.FunctionalEquation.IsNormalized (fun x => costLambda c x)
          ∧ Cost.FunctionalEquation.SatisfiesCompositionLaw (fun x => costLambda c x)
          ∧ ContinuousOn (fun x => costLambda c x) (Set.Ioi 0))
      ∧ (∀ c : ℝ, 0 < c →
          (Cost.FunctionalEquation.IsCalibrated (fun x => costLambda c x) ↔ c = 1))
      ∧ (∃ c d : ℝ, 0 < c ∧ 0 < d ∧ c ≠ d
            ∧ (fun x => costLambda c x) ≠ (fun x => costLambda d x)) := by
  refine ⟨fun c => ⟨costLambda_isReciprocalCost c, costLambda_isNormalized c,
            costLambda_satisfiesCompositionLaw c, costLambda_continuousOn c⟩,
          fun c hc => costLambda_isCalibrated_iff hc, ?_⟩
  refine ⟨1, 2, one_pos, two_pos, by norm_num, ?_⟩
  intro h
  have h12 : (1 : ℝ) = 2 := costLambda_inj one_pos two_pos h
  norm_num at h12

What this page does not claim

The composition law alone does not force the canonical cost J. The declaration does not prove that all possible recognition costs satisfy the composition law. The calibration condition is not claimed to be derivable from the other four hypotheses.

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/PrimitiveRecognitionCalculus/PRCCalibrationIndependence.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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