Foundation Logic As Functional Equation Route Independence Implies Multiplicativ

Route independence and the composition law

In the Recognition Science framework, a comparison operator is a function that assigns a real-valued cost to comparing any two positive quantities. The framework's library of machine-checked theorems asks what structural constraints such an operator must satisfy to be a well-posed operation. One of these constraints is route independence: the cost of comparing two quantities must not depend on the intermediate route taken through a third quantity. The theorem route_independence_implies_multiplicative_consistency proves that this single condition, taken alone, forces the derived cost function to obey a specific multiplicative composition law.

Formally, the theorem states that if a comparison operator satisfies route independence, then there exists a symmetric polynomial P of degree at most two such that for all positive x and y, the derived cost satisfies C(x·y) + C(x/y) = P(C(x), C(y)). This is a d'Alembert-style functional equation: it relates the cost of a product and a quotient to the costs of the factors. The theorem does not require any of the other Aristotelian constraints, such as identity, non-contradiction, or scale invariance. Route independence alone is sufficient to establish this multiplicative structure.

The theorem is a stepping stone, not the destination. It shows that route independence already imposes a strong algebraic shape on the cost function. In the full chain of proofs, this composition law combines with the other constraints to force the unique canonical cost function J(x) = (x + 1/x)/2 - 1. But the theorem itself says nothing about which polynomial P appears, nor does it determine the constant c in the eventual form P(u,v) = 2u + 2v + c·u·v. Those results require the additional assumptions of the full law of logic.

What the theorem does not claim is equally important. It does not assert that route independence alone determines the cost function; many different cost functions can satisfy the same composition law. It does not claim that the polynomial P is unique, only that one exists. And it does not establish that the derived cost is continuous, symmetric, or normalized; those properties come from other constraints in the framework. The theorem is a precise, narrow result: route independence implies multiplicative consistency, nothing more.

/-- **Translation lemma 4 (Route-independence ⇒ Multiplicative consistency
with a symmetric polynomial combiner)**: extracted directly from the
definition of `RouteIndependence`. -/
theorem route_independence_implies_multiplicative_consistency
    (C : ComparisonOperator)
    (hRI : RouteIndependence 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 := by
  obtain ⟨P, hPoly, hSym, hCons⟩ := hRI
  refine ⟨P, hPoly, hSym, ?_⟩
  intro x y hx hy
  exact hCons x y hx hy

/-- **Translation lemma 4 (Route-independence ⇒ Multiplicative consistency
with a symmetric polynomial combiner)**: extracted directly from the
definition of `RouteIndependence`. -/
theorem route_independence_implies_multiplicative_consistency
    (C : ComparisonOperator)
    (hRI : RouteIndependence 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 := by
  obtain ⟨P, hPoly, hSym, hCons⟩ := hRI
  refine ⟨P, hPoly, hSym, ?_⟩
  intro x y hx hy
  exact hCons x y hx hy

/-- **Translation lemma 4 (Route-independence ⇒ Multiplicative consistency
with a symmetric polynomial combiner)**: extracted directly from the
definition of `RouteIndependence`. -/
theorem route_independence_implies_multiplicative_consistency
    (C : ComparisonOperator)
    (hRI : RouteIndependence 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 := by
  obtain ⟨P, hPoly, hSym, hCons⟩ := hRI
  refine ⟨P, hPoly, hSym, ?_⟩
  intro x y hx hy
  exact hCons x y hx hy

What this page does not claim

Route independence alone determines the cost function. The polynomial P in the composition law is unique. Route independence implies continuity, symmetry, or normalization of the derived cost. The theorem establishes the full canonical cost function J(x) = (x + 1/x)/2 - 1.

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