Foundation Generalized Dalembert Rcl Is Unique Functional Form Of Logic Continuo

The continuous uniqueness theorem

The d'Alembert functional equation is a classical object from 18th-century mechanics. It asks for functions H that satisfy H(x+y) + H(x-y) = 2 H(x) H(y) for all real numbers x and y. The equation first appeared in Jean le Rond d'Alembert's 1747 work on vibrating strings, where it described how a wave's shape evolves. Its continuous solutions were classified by János Aczél and Pl. Kannappan in the 20th century: any continuous H with H(0) = 1 must be one of just three forms, the constant 1, a hyperbolic cosine cosh(αx), or a trigonometric cosine cos(αx). The classification is a standard result in functional equations, and it is the backbone of the Recognition Science theorem under discussion.

Within Recognition Science, a framework that derives physical structure from the cost of recognition events, the theorem RCL_is_unique_functional_form_of_logic_continuous uses this classical classification to prove a stronger uniqueness statement. The framework models a ledger, a discrete record of events, and assigns a cost to each recognition. The theorem states that if a comparison operator satisfies five laws, identity, non-contradiction, excluded middle, scale invariance, and a continuous version of route independence, plus a technical smoothness input, then its derived cost function must satisfy a specific bilinear identity: F(xy) + F(x/y) = P(F(x), F(y)) where P(u, v) = 2u + 2v + c·u·v for some constant c. This is a theorem proved in the framework's machine-checked library of formal theorems, not a hypothesis.

The proof works by transforming the cost function into log coordinates, where the composition law becomes the d'Alembert equation. The framework's library first proves the Aczél–Kannappan classification internally, then shows that the continuous solutions are exactly the constant zero, a parabola αt², a hyperbolic cosine minus one, or one minus a trigonometric cosine. Each of these four forms satisfies the bilinear identity, and the classification ensures no other continuous solution exists. The theorem thus extends the earlier polynomial-degree-restricted version to a purely continuous setting, replacing a stronger but less natural hypothesis with plain continuity.

The theorem does not claim that the constant c is forced to a particular value, nor does it identify which of the four solution branches is physically realized. It also does not claim that the technical smoothness inputs follow from continuity alone; in fact, a counterexample in the library shows that the quartic log-cost blocks the proposed second-derivative identity, so the additional inputs are genuinely needed. The result is a structural classification: it narrows the possible cost functions to a small, explicit family, but it does not by itself select the unique J(x) = (x + 1/x)/2 − 1 that the framework's broader forcing chain derives.

/-- **Generalized Translation Theorem (named-hypothesis form)**.
Under the continuous-combiner hypothesis plus the explicit analysis
package, the Law of Logic forces the bilinear RCL family. -/
theorem RCL_is_unique_functional_form_of_logic_continuous
    (C : ComparisonOperator)
    (h : SatisfiesLawsOfLogicContinuous C)
    (hInputs : ContinuousCombinerAnalysisInputs C h) :
    ∃ (P : ℝ → ℝ → ℝ) (c : ℝ),
      (∀ x y : ℝ, 0 < x → 0 < y →
        derivedCost C (x * y) + derivedCost C (x / y)
          = P (derivedCost C x) (derivedCost C y)) ∧
      (∀ u v, P u v = 2*u + 2*v + c*u*v) :=
  continuous_combiner_bilinear C h hInputs

/-- **Aczél–Kannappan classification** (proved theorem, not axiom):
every continuous solution of the d'Alembert functional equation
`H(x+y) + H(x-y) = 2 H(x) H(y)` with `H(0) = 1` is either the
constant 1, a hyperbolic cosine, or a trigonometric cosine.

The proof reduces to
`IndisputableMonolith.Cost.FunctionalEquation.dAlembert_classification`,
which assembles the integration bootstrap, universal-coefficient ODE
derivation, and ODE uniqueness lemmas into the disjunction. -/
theorem aczel_kannappan_continuous_dAlembert
    (H : ℝ → ℝ) (hCont : Continuous H) (h_one : H 0 = 1)
    (hEq : ∀ x y : ℝ, H (x + y) + H (x - y) = 2 * H x * H y) :
    (∀ x, H x = 1) ∨
    (∃ α : ℝ, ∀ x, H x = Real.cosh (α * x)) ∨
    (∃ α : ℝ, ∀ x, H x = Real.cos (α * x)) :=
  IndisputableMonolith.Cost.FunctionalEquation.dAlembert_classification
    H h_one hCont hEq

/-- A classified log-coordinate cost: parabolic, hyperbolic, trigonometric,
or zero. This is the algebraic target left after the smoothness/affine-forcing
analysis has been done. -/
def ClassifiedLogCost (G : ℝ → ℝ) : Prop :=
  (∀ t, G t = 0) ∨
  (∃ α : ℝ, ∀ t, G t = α * t^2) ∨
  (∃ α : ℝ, ∀ t, G t = Real.cosh (α * t) - 1) ∨
  (∃ α : ℝ, ∀ t, G t = 1 - Real.cos (α * t))

/-- Explicit package of the extra analysis needed to force bilinearity from
an arbitrary continuous combiner. This is deliberately a hypothesis package,
not an axiom. The quartic-log obstruction shows the package is not automatic
from `SatisfiesLawsOfLogicContinuous`. -/
structure ContinuousCombinerAnalysisInputs
    (C : ComparisonOperator)
    (h : SatisfiesLawsOfLogicContinuous C) : Prop where
  finite_smoothness : ContinuousCombinerMollifierFiniteSmoothness C h
  second_derivative :
    ContinuousCombinerSecondDerivativeInput C h
      (continuous_combiner_log_smoothness_bootstrap C h finite_smoothness)
  psi_affine :
    ContinuousCombinerPsiAffineCompletion C h
      (continuous_combiner_log_smoothness_bootstrap C h finite_smoothness)
      second_derivative

What this page does not claim

The theorem does not force the constant c to a specific value. The theorem does not identify which solution branch is physically realized. The theorem does not claim the technical smoothness inputs follow from continuity alone.

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

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