RCL_is_unique_functional_form_of_logic_continuous
Any comparison operator satisfying the continuous laws of logic together with the explicit analysis package for its combiner admits a bilinear route-independence function P of the form 2u + 2v + c u v. Researchers working on regularity conditions for the Law of Logic or extending the Translation Theorem beyond polynomials would cite this result. The proof is a one-line wrapper that invokes the continuous-combiner bilinear theorem.
claimLet $C$ be a comparison operator on positive reals. Suppose $C$ satisfies the continuous laws of logic (the Aristotelian conditions, scale invariance, and continuous route independence) and the continuous combiner analysis inputs hold. Then there exist a function $P : ℝ → ℝ → ℝ$ and a constant $c ∈ ℝ$ such that for all $x, y > 0$, derivedCost$_C(xy) +$ derivedCost$_C(x/y) = P($derivedCost$_C(x),$ derivedCost$_C(y))$ and $P(u,v) = 2u + 2v + c u v$.
background
The GeneralizedDAlembert module develops the continuous version of the Translation Theorem by replacing the polynomial route-independence hypothesis with continuity of the combiner. It first establishes the Aczél–Kannappan classification of continuous solutions to the d'Alembert equation, reducing them to the constant 1, hyperbolic cosines, or trigonometric cosines. A ComparisonOperator is a map ℝ⁺ × ℝ⁺ → ℝ that encodes the cost of comparing two positive quantities; derivedCost extracts the cost relative to the multiplicative identity 1. SatisfiesLawsOfLogicContinuous is the structure that retains the four Aristotelian constraints, scale invariance, and non-triviality while substituting ContinuousRouteIndependence for the polynomial version. ContinuousCombinerAnalysisInputs packages the finite smoothness, second-derivative identity, and psi-affine completion required to force bilinearity; the module doc notes that this package is not automatic from continuity alone, as shown by the quartic-log obstruction.
proof idea
The proof is a one-line wrapper that applies the theorem continuous_combiner_bilinear to the supplied hypotheses C, h, and hInputs. No additional tactics or reductions are performed.
why it matters in Recognition Science
This supplies the continuous analogue of the Translation Theorem, enabling the Law of Logic to be stated with continuous rather than polynomial route independence. It is invoked by laws_continuous_subsumes_polynomial to embed the polynomial case inside the continuous setting. In the Recognition Science framework it supports the claim that the Recognition Composition Law is the unique functional form of logic once continuity and the analysis package are assumed, linking to the forcing chain T5–T8 and the derivation of physical constants from the functional equation. The module doc flags that the analysis package remains necessary; the quartic-log counterexample shows that continuity alone does not discharge it.
scope and limits
- Does not prove that the analysis inputs follow from continuity of the combiner alone.
- Does not apply to operators that violate scale invariance or the Aristotelian constraints.
- Does not treat the non-continuous case, which still requires the polynomial hypothesis.
- Does not determine the numerical value of the constant c from the functional equation.
formal statement (Lean)
638theorem RCL_is_unique_functional_form_of_logic_continuous
639 (C : ComparisonOperator)
640 (h : SatisfiesLawsOfLogicContinuous C)
641 (hInputs : ContinuousCombinerAnalysisInputs C h) :
642 ∃ (P : ℝ → ℝ → ℝ) (c : ℝ),
643 (∀ x y : ℝ, 0 < x → 0 < y →
644 derivedCost C (x * y) + derivedCost C (x / y)
645 = P (derivedCost C x) (derivedCost C y)) ∧
646 (∀ u v, P u v = 2*u + 2*v + c*u*v) :=
proof body
Term-mode proof.
647 continuous_combiner_bilinear C h hInputs
648
649/-- Every polynomial-LoL operator is a continuous-LoL operator. The bilinear
650conclusion still requires the explicit analysis package at this level; the
651ordinary polynomial theorem in `LogicAsFunctionalEquation` remains the
652unconditional route. -/