Foundation Logic As Functional Equation Law Of Logic Forces Canonical Cost

The forced cost function

A comparison operator is a function that takes two positive quantities and returns a real-valued cost of comparing them. The theorem law_of_logic_forces_canonical_cost states that if such an operator satisfies six structural laws of logic, plus a composition law and a calibration condition, then it must equal the canonical cost function J(x) = (x + 1/x)/2 - 1 for every positive x. The six laws are identity (comparing a quantity with itself costs zero), non-contradiction (comparing x with y costs the same as comparing y with x), excluded middle (the operator is continuous), scale invariance (scaling both inputs by the same factor leaves the cost unchanged), route independence (the cost of a composite comparison is a polynomial of degree at most two in the component costs), and non-triviality (the cost is not identically zero).

The proof proceeds by first showing that the six laws imply the derived cost function is normalized, symmetric, continuous, and satisfies the hypotheses of a classical d'Alembert functional equation. A separate theorem, RCL_is_unique_functional_form_of_logic, then forces the combiner polynomial to have the form P(u,v) = 2u + 2v + c·u·v for some real constant c. The composition law, which states that the cost of a product plus the cost of a quotient equals the combiner applied to the individual costs, together with the calibration condition, pins down the constant c and yields the unique canonical form. The theorem is proved in the framework's machine-checked library of formal theorems, with no framework-specific axioms; it relies only on the standard axioms of the ambient type theory.

The result does not claim that the six laws are the only possible laws of logic, nor that every comparison operator satisfying the laws is meaningful in a physical sense. It also does not claim that the canonical cost function J is the unique cost function for all comparison problems; the theorem applies specifically to operators satisfying the stated laws and conditions. The composition law and calibration are additional hypotheses, not consequences of the six laws alone. The theorem establishes a uniqueness result within a precisely stated axiomatic framework, not a universal law of nature.

In the framework, this result is the first step in a chain that forces the golden ratio, an eight-tick recognition cycle, and three spatial dimensions. The canonical cost function J appears throughout the framework's derivations of physical constants and particle masses. The theorem provides the mathematical foundation for these later results, but it does not by itself derive any physical constant or empirical prediction.

The practical consequence is that any comparison operator satisfying the six laws must have the same functional form, up to the single constant c, which is then fixed by calibration. This means that the cost of comparison is not a free choice but is forced by the structural laws. The theorem gives a precise, checkable statement of what those laws imply, and it leaves open the question of whether the laws themselves are the correct description of any particular physical or logical system.

/-- **Law of Logic forces the canonical cost**:
under the canonical RCL normalization and unit log-curvature calibration, the
unique continuous positive-ratio cost satisfying the Law of Logic is
`J(x) = ½(x + x⁻¹) - 1`. -/
theorem law_of_logic_forces_canonical_cost
    (C : ComparisonOperator) (hLogic : LawOfLogic C)
    [Cost.FunctionalEquation.AczelSmoothnessPackage]
    (hRCL : Cost.FunctionalEquation.SatisfiesCompositionLaw (derivedCost C))
    (hCalib : Cost.FunctionalEquation.IsCalibrated (derivedCost C)) :
    ∀ x : ℝ, 0 < x → derivedCost C x = Cost.Jcost x := by
  exact J_is_unique_cost_under_logic C hLogic hRCL hCalib

/-- **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

/-- **Main theorem (Logical Formalization Theorem)**: For a comparison
operator satisfying the four Aristotelian constraints with scale invariance
and non-triviality, the route-independence combiner is necessarily of the
Recognition Composition Law form: `P(u,v) = 2u + 2v + c·uv` for some
constant c ∈ ℝ.

In other words: the unique functional form the laws of logic can take on
continuous comparisons of positive ratios, under the polynomial regularity
assumption, is the Recognition Composition Law.

This is an immediate corollary of `laws_of_logic_imply_dalembert_hypotheses`
combined with `bilinear_family_forced` (Inevitability.lean), which has been
peer-reviewed in:

  Washburn, Zlatanović, Allahyarov.
  "The d'Alembert Inevitability Theorem."
  Mathematics (MDPI), 2026.
-/
theorem RCL_is_unique_functional_form_of_logic
    (C : ComparisonOperator) (hLaws : SatisfiesLawsOfLogic C) :
    ∃ (P : ℝ → ℝ → ℝ) (c : ℝ),
      HasMultiplicativeConsistency (derivedCost C) P ∧
      (∀ u v, P u v = 2*u + 2*v + c*u*v) := by
  obtain ⟨hNorm, _hSym, ⟨P, hPoly, hSymP, hCons⟩, hCont, hNontriv⟩ :=
    laws_of_logic_imply_dalembert_hypotheses C hLaws
  obtain ⟨c, hP_form, _⟩ :=
    bilinear_family_forced (derivedCost C) P hNorm hCons hPoly hSymP hNontriv hCont
  exact ⟨P, c, hCons, hP_form⟩

/-- **Law of Logic forces the canonical cost**:
under the canonical RCL normalization and unit log-curvature calibration, the
unique continuous positive-ratio cost satisfying the Law of Logic is
`J(x) = ½(x + x⁻¹) - 1`. -/
theorem law_of_logic_forces_canonical_cost
    (C : ComparisonOperator) (hLogic : LawOfLogic C)
    [Cost.FunctionalEquation.AczelSmoothnessPackage]
    (hRCL : Cost.FunctionalEquation.SatisfiesCompositionLaw (derivedCost C))
    (hCalib : Cost.FunctionalEquation.IsCalibrated (derivedCost C)) :
    ∀ x : ℝ, 0 < x → derivedCost C x = Cost.Jcost x := by
  exact J_is_unique_cost_under_logic C hLogic hRCL hCalib

What this page does not claim

The six laws of logic are the only possible laws of logic. Every comparison operator satisfying the laws is physically meaningful. The theorem derives any physical constant or empirical prediction on its own. The composition law and calibration are consequences of the six laws alone.

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