Cost Functional Equation Aczel

The Aczél Closure

A functional equation is a rule that a function must obey, and the question is whether only one function can obey it. In Recognition Science, the cost function J(x) = (x + 1/x)/2 - 1 is singled out by exactly such a rule. The equation says that the cost of recognizing a composite event equals a fixed combination of the costs of its parts. Written in symbols, the composition law is F(xy) + F(x/y) = 2F(x)F(y) + 2F(x) + 2F(y). The Aczél closure theorem states that J is the unique function satisfying this law together with four other plain conditions: reciprocal symmetry, normalization to zero at unity, calibration, and continuity on positive inputs.

The classical background here is the theory of functional equations, where a rule like this often admits many wild solutions unless a regularity condition is added. Without continuity, one can construct pathological functions that satisfy the composition law but behave erratically. The theorem's five conditions, taken together, eliminate those alternatives and leave exactly one well-behaved solution. This is the same pattern that appears in other uniqueness results: a simple rule plus a smoothness requirement selects a single nice answer from a large family of possible ones.

In Recognition Science, the framework models reality as maintaining a ledger, a discrete record of recognition events, and the cost of each recognition is forced by the equation rather than chosen freely. The Aczél closure theorem is the bridge that lets the framework use this classical uniqueness result without carrying extra assumptions. The machine-checked library of formal theorems proves that the five conditions imply J directly, with no regularity parameters left for the caller to supply.

What this establishes in plain language is that the cost function is not an arbitrary choice. If you accept the five conditions as describing what recognition costs must look like, then J is the only possible answer. That uniqueness is what lets the rest of the framework build on a single fixed foundation, not on a menu of alternatives.

/-- **Law of Logic cost theorem, Aczél closure**: The J-cost function is the unique
    reciprocal cost satisfying the RCL, normalization, calibration, and continuity.

    This version uses the global Aczél axiom internally and requires NO regularity
    hypothesis parameters from the caller. -/
theorem law_of_logic_forces_jcost_aczel (F : ℝ → ℝ)
    (hRecip : IsReciprocalCost F)
    (hNorm : IsNormalized F)
    (hComp : SatisfiesCompositionLaw F)
    (hCalib : IsCalibrated F)
    (hCont : ContinuousOn F (Set.Ioi 0)) :
    ∀ x : ℝ, 0 < x → F x = Cost.Jcost x := by
  exact law_of_logic_forces_jcost F hRecip hNorm hComp hCalib hCont

/-- **Law of Logic cost theorem, Aczél closure**: The J-cost function is the unique
    reciprocal cost satisfying the RCL, normalization, calibration, and continuity.

    This version uses the global Aczél axiom internally and requires NO regularity
    hypothesis parameters from the caller. -/
theorem law_of_logic_forces_jcost_aczel (F : ℝ → ℝ)
    (hRecip : IsReciprocalCost F)
    (hNorm : IsNormalized F)
    (hComp : SatisfiesCompositionLaw F)
    (hCalib : IsCalibrated F)
    (hCont : ContinuousOn F (Set.Ioi 0)) :
    ∀ x : ℝ, 0 < x → F x = Cost.Jcost x := by
  exact law_of_logic_forces_jcost F hRecip hNorm hComp hCalib hCont

What this page does not claim

This module does not prove the five conditions are necessary; it proves they are sufficient. The theorem does not derive the value of any physical constant such as the fine-structure constant. The Aczél closure is a legacy compatibility surface, not the axiom-free core of the framework.

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/FunctionalEquationAczel.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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