Foundation Logic As Functional Equation J Is Unique Cost Under Logic

The uniqueness theorem

A comparison operator is a rule that takes two positive quantities and returns a real number measuring the cost of comparing them. The framework asks what happens when such an operator is forced to behave like logic itself: comparing a thing with itself costs nothing, comparing x with y costs the same as comparing y with x, scaling both inputs by the same factor leaves the cost unchanged, and the cost of a two-step comparison does not depend on the route taken. These are the laws of non-contradiction, identity, excluded middle, scale invariance, and route independence, together with a requirement that the operator is not trivial. The theorem J_is_unique_cost_under_logic proves that any operator satisfying all six laws must have a derived cost function equal to J(x) = (x + 1/x)/2 - 1 for every positive x.

The proof works by showing that the six laws force the derived cost to satisfy a functional equation known as d'Alembert's equation, which has a unique continuous solution under a smoothness condition. The key step is that route independence implies the cost obeys a multiplicative consistency law: the cost of comparing a product equals a polynomial expression in the costs of comparing the factors. Solving that equation, with the normalization that comparing 1 with 1 costs zero, yields exactly the J function. The theorem is checked in the framework's machine-checked library of formal theorems, with no additional axioms beyond the standard logical ones.

The result matters because it turns a philosophical question into a mathematical one. If comparison is to behave like logic, there is no freedom in how the cost is measured: the function is forced. This is not a claim about how humans actually compare things, nor about what any particular physical system does. It is a statement about the structure any comparison operator must have if it satisfies those six laws. The theorem does not say that J is the only possible cost function in general, only that it is the unique one under these specific logical constraints.

The declaration also does not claim that the six laws are the only possible laws of logic, nor that they are the most natural ones. It takes them as given and derives the consequence. It does not claim that J applies to any actual measurement in the physical world; that would be a separate empirical question. What it establishes is a conditional: if comparison satisfies these laws, then the cost is J. The proof is complete and machine-checked, but the conditions are assumptions, not conclusions about reality.

In the broader framework, this theorem is the first link in a chain that derives the golden ratio, an eight-tick cycle, and three spatial dimensions from the same logical starting point. But those later results are not part of this declaration. The uniqueness theorem stands alone as a piece of pure mathematics about what logic forces when it is expressed as a comparison operator.

/-- **Cost Corollary (J is forced by the laws of logic + calibration)**:
Under the four Aristotelian constraints, the canonical c = 2 normalization,
and the unit log-curvature calibration, the unique continuous cost function
on positive ratios is the canonical reciprocal cost J.

This invokes the peer-reviewed and Lean-verified result:

  Washburn, Zlatanović. "Uniqueness of the Canonical Reciprocal Cost."
  Mathematics 14 (2026), 935.
-/
theorem J_is_unique_cost_under_logic
    (C : ComparisonOperator) (hLaws : SatisfiesLawsOfLogic C)
    [AczelSmoothnessPackage]
    -- The route-independence combiner is the canonical c = 2 RCL.
    (hRCL : SatisfiesCompositionLaw (derivedCost C))
    -- The derived cost satisfies the unit log-curvature calibration.
    (hCalib : IsCalibrated (derivedCost C)) :
    ∀ x : ℝ, 0 < x → derivedCost C x = Cost.Jcost x := by
  obtain ⟨hNorm, hSym, _, hCont, _⟩ :=
    laws_of_logic_imply_dalembert_hypotheses C hLaws
  -- Express IsSymmetric in the form needed by law_of_logic_forces_jcost.
  have hRecip : IsReciprocalCost (derivedCost C) := by
    intro x hx
    exact hSym x hx
  exact law_of_logic_forces_jcost (derivedCost C) hRecip hNorm hRCL hCalib hCont

/-- **Translation Theorem**: A comparison operator satisfying the four
Aristotelian constraints, together with scale invariance and non-triviality,
satisfies the hypotheses of the d'Alembert Inevitability Theorem on its
derived cost function.

This is the core technical content of the precursor paper. Once this is in
hand, the existing peer-reviewed and machine-verified theorems
(`bilinear_family_forced`, `law_of_logic_forces_jcost`) close the chain. -/
theorem laws_of_logic_imply_dalembert_hypotheses
    (C : ComparisonOperator) (hLaws : SatisfiesLawsOfLogic C) :
    IsNormalized (derivedCost C) ∧
    IsSymmetric (derivedCost C) ∧
    (∃ P : ℝ → ℝ → ℝ,
      (∃ a b c d e f : ℝ, ∀ u v, P u v = a + b*u + c*v + d*u*v + e*u^2 + f*v^2) ∧
      (∀ u v, P u v = P v u) ∧
      HasMultiplicativeConsistency (derivedCost C) P) ∧
    ContinuousOn (derivedCost C) (Set.Ioi 0) ∧
    (∃ x : ℝ, 0 < x ∧ derivedCost C x ≠ 0) := by
  refine ⟨?_, ?_, ?_, ?_, ?_⟩
  · exact identity_implies_normalized C hLaws.identity
  · exact non_contradiction_and_scale_imply_reciprocal C
      hLaws.non_contradiction hLaws.scale_invariant
  · exact route_independence_implies_multiplicative_consistency C
      hLaws.route_independence
  · exact excluded_middle_implies_continuous C hLaws.excluded_middle
  · exact hLaws.non_trivial

/-- **Cost Corollary (J is forced by the laws of logic + calibration)**:
Under the four Aristotelian constraints, the canonical c = 2 normalization,
and the unit log-curvature calibration, the unique continuous cost function
on positive ratios is the canonical reciprocal cost J.

This invokes the peer-reviewed and Lean-verified result:

  Washburn, Zlatanović. "Uniqueness of the Canonical Reciprocal Cost."
  Mathematics 14 (2026), 935.
-/
theorem J_is_unique_cost_under_logic
    (C : ComparisonOperator) (hLaws : SatisfiesLawsOfLogic C)
    [AczelSmoothnessPackage]
    -- The route-independence combiner is the canonical c = 2 RCL.
    (hRCL : SatisfiesCompositionLaw (derivedCost C))
    -- The derived cost satisfies the unit log-curvature calibration.
    (hCalib : IsCalibrated (derivedCost C)) :
    ∀ x : ℝ, 0 < x → derivedCost C x = Cost.Jcost x := by
  obtain ⟨hNorm, hSym, _, hCont, _⟩ :=
    laws_of_logic_imply_dalembert_hypotheses C hLaws
  -- Express IsSymmetric in the form needed by law_of_logic_forces_jcost.
  have hRecip : IsReciprocalCost (derivedCost C) := by
    intro x hx
    exact hSym x hx
  exact law_of_logic_forces_jcost (derivedCost C) hRecip hNorm hRCL hCalib hCont

What this page does not claim

The six laws are the only possible or most natural laws of logic. The cost function J applies to any actual physical measurement or comparison process. The theorem establishes anything about the golden ratio, eight-tick cycles, or three spatial dimensions, which are later results in 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/Foundation/LogicAsFunctionalEquation.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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