Foundation Logic As Functional Equation Laws Of Logic Imply Dalembert Hypotheses

From logic to a wave equation

The classical starting point is the d'Alembert equation, the wave equation that Jean le Rond d'Alembert wrote down in 1747 for a vibrating string. It describes how a disturbance spreads: the second derivative of a displacement with respect to time equals a constant times its second derivative with respect to position. Its solutions are traveling waves, and it became the template for field equations across physics. The equation itself is a statement about a function of two variables, and it has a distinctive property: the sum of the function at two related points equals a symmetric expression in the values at each point.

In Recognition Science, the framework asks what happens when comparison itself is the primitive. A comparison operator, a rule that assigns a real-valued cost to any pair of positive quantities, is constrained by six plain conditions: comparing a thing with itself costs zero, comparing x with y costs the same as comparing y with x, the cost varies continuously, scaling both inputs leaves the cost unchanged, the cost of a two-step comparison is independent of the route taken, and the cost is not identically zero. These are structural demands on what it means for comparison to be well-posed at all.

The theorem laws_of_logic_imply_dalembert_hypotheses proves that these six conditions force the derived cost function to have five specific properties. The derived cost, the cost of comparing any positive number with 1, is normalized at 1, symmetric under reciprocals, continuous, and nonzero somewhere. The route-independence condition forces the existence of a symmetric polynomial of degree at most two that satisfies a multiplicative consistency relation, which is exactly the d'Alembert composition law. In plain terms, the logic of comparison, stated as those six axioms, already contains the algebraic skeleton of the wave equation.

The framework's machine-checked library of formal theorems proves this in full. The theorem is a formal derivation, not a heuristic: from the six axioms alone, the d'Alembert structure follows. The library then extends the chain: the same axioms force a unique functional form for the cost, J(x) = (x + 1/x)/2 - 1, and from that cost the framework derives constants and dimensional structure. The d'Alembert hypotheses are the hinge: logic, formalized as constraints on comparison, turns into analysis.

What the declaration does not claim is equally precise. It does not claim that the six axioms are true of the actual universe; they are a definitional choice, a model of what comparison is. It does not claim that the wave equation itself is derived, only that the composition law that characterizes it is forced. And it does not claim that the framework's logic is the only route to the d'Alembert equation, which stands independently in classical physics. The theorem is about a formal implication, not about empirical fact.

/-- **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

/-- **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

/-- **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 axioms are not claimed to be true of the actual universe; they are a definitional choice. The wave equation itself is not derived, only the composition law that characterizes it. The framework's logic is not claimed to be the only route to the d'Alembert equation.

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)

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: