Foundation Logic As Functional Equation Rcl Is Unique Functional Form Of Logic

The forced form of comparison

A comparison operator takes two positive quantities and returns a real-valued cost of comparing them. The framework Recognition Science asks what structural laws such an operator must satisfy to be a well-posed logic. The answer, proved in its machine-checked library of formal theorems, is that any operator obeying six plain laws must have a derived cost function that satisfies a specific bilinear composition law. The theorem RCL_is_unique_functional_form_of_logic states that there exists a polynomial P and a real constant c such that for all positive u and v, P u v = 2*u + 2*v + c*u*v, and this P is multiplicatively consistent with the derived cost.

The six laws are identity, non-contradiction, excluded middle, scale invariance, route independence, and non-triviality. Identity says comparing a quantity with itself costs zero. Non-contradiction says the cost is symmetric in its two arguments. Excluded middle is a continuity condition. Scale invariance says the cost does not change if both inputs are scaled by the same positive factor. Route independence requires a polynomial combiner of degree at most two that satisfies a d'Alembert composition law relating the cost of products and quotients to the costs of the factors. Non-triviality rules out the degenerate operator that is identically zero.

The proof proceeds by showing these laws imply the hypotheses of a classical functional equation result. The derived cost function is normalized, symmetric, continuous on positive reals, and admits a symmetric polynomial combiner satisfying the d'Alembert law. The theorem bilinear_family_forced then forces the combiner to have the form P u v = 2*u + 2*v + c*u*v. This is the unique functional form: no other polynomial of degree at most two can satisfy the composition law under these hypotheses.

In Recognition Science, this result is the first step in a chain that forces the golden ratio as a self-similar scaling, an eight-tick recognition cycle, and three spatial dimensions. The theorem itself does not fix the value of the constant c; that requires an additional calibration condition. It also does not establish that any particular comparison operator exists satisfying all six laws, only that if one does, its derived cost must have this form.

The declaration does not claim that the constant c is determined by the six laws alone. It does not claim that the polynomial P is unique as a function, only that its form is forced up to the constant c. It does not claim that the derived cost function itself equals the specific function J(x) = (x + 1/x)/2 - 1; that is a separate theorem requiring a smoothness package and a calibration condition.

/-- A comparison operator on positive reals takes two positive quantities and
returns a real-valued cost of comparing them. The four Aristotelian
constraints below are the structural content of comparison being a
well-posed operation. -/
abbrev ComparisonOperator := ℝ → ℝ → ℝ

/-- **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⟩

/-- A comparison operator **satisfies the laws of logic** if all four
Aristotelian constraints hold, together with scale invariance (the bridge
from two-argument to one-argument form) and non-triviality (so that the
derived cost is not vacuously zero). -/
structure SatisfiesLawsOfLogic (C : ComparisonOperator) : Prop where
  identity            : Identity C
  non_contradiction   : NonContradiction C
  excluded_middle     : ExcludedMiddle C
  scale_invariant     : ScaleInvariant C
  route_independence  : RouteIndependence C
  non_trivial         : NonTrivial C

/-- **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⟩

What this page does not claim

The constant c is not determined by the six laws alone. The derived cost function is not claimed to equal J(x) = (x + 1/x)/2 - 1 without additional smoothness and calibration assumptions. The theorem does not assert the existence of any comparison operator satisfying the six laws.

Verify this page

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

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