Foundation Dalembert Proof

The D'Alembert Equation

The d'Alembert functional equation, named after Jean le Rond d'Alembert and studied in depth by János Aczél in the 1960s, asks for functions H that satisfy H(t+u) + H(t-u) = 2H(t)H(u). The two familiar solutions are H(t) = cos(αt) and H(t) = cosh(αt), with the constant function H(t) = 1 as a third possibility. This equation is central to the Recognition Science framework because it describes how a measure of similarity between two items combines when the items are themselves combined.

In Recognition Science, the framework models a ledger, a discrete record of recognition events, and assigns a cost to each event. The framework proves that any cost function satisfying basic conditions of symmetry and normalization must obey a specific composition law. In logarithmic coordinates, this law becomes the d'Alembert equation. The machine-checked library of formal theorems establishes that the only continuous solutions to this equation, under a calibration condition, are the hyperbolic cosine functions.

The result is a uniqueness theorem. The theorem dAlembert_classification proves that if H is a continuous solution to the d'Alembert equation with H(0) = 1 and H''(0) = 1, then H(t) = cosh(t) for all t. This classification is what forces the specific form of the cost function in the framework, leading to the golden ratio and other derived constants. The proof relies on the classical theory of functional equations, particularly the work of Aczél.

The practical consequence is that the framework's core equation is not an arbitrary choice. It is the unique polynomial form compatible with the basic requirements of symmetry, normalization, and multiplicative consistency. This uniqueness is what gives the framework its predictive power, as it rules out alternative formulations of the cost function.

/-- The classification theorem for d'Alembert equation (Aczél).

Continuous solutions to H(t+u) + H(t-u) = 2·H(t)·H(u) with H(0) = 1 are:
1. H(t) = 1 (constant)
2. H(t) = cos(αt) for some α ∈ ℂ
3. H(t) = cosh(αt) for some α ∈ ℝ

With the calibration H''(0) = 1, only H = cosh survives. -/
theorem dAlembert_classification (H : ℝ → ℝ)
    (h : IsDAlembertSolution H)
    (hCont : Continuous H)
    (hCalib : deriv (deriv H) 0 = 1)
    -- Regularity hypotheses (from Aczél theory)
    (hSmooth : dAlembert_continuous_implies_smooth_hypothesis H)
    (hODE : dAlembert_to_ODE_hypothesis H)
    (hODECont : ode_regularity_continuous_hypothesis H)
    (hODEDiff : ode_regularity_differentiable_hypothesis H)
    (hBoot : ode_linear_regularity_bootstrap_hypothesis H) :
    ∀ t, H t = cosh t :=
  dAlembert_cosh_solution H h.1 hCont h.2 hCalib hSmooth hODE hODECont hODEDiff hBoot

/-- The classification theorem for d'Alembert equation (Aczél).

Continuous solutions to H(t+u) + H(t-u) = 2·H(t)·H(u) with H(0) = 1 are:
1. H(t) = 1 (constant)
2. H(t) = cos(αt) for some α ∈ ℂ
3. H(t) = cosh(αt) for some α ∈ ℝ

With the calibration H''(0) = 1, only H = cosh survives. -/
theorem dAlembert_classification (H : ℝ → ℝ)
    (h : IsDAlembertSolution H)
    (hCont : Continuous H)
    (hCalib : deriv (deriv H) 0 = 1)
    -- Regularity hypotheses (from Aczél theory)
    (hSmooth : dAlembert_continuous_implies_smooth_hypothesis H)
    (hODE : dAlembert_to_ODE_hypothesis H)
    (hODECont : ode_regularity_continuous_hypothesis H)
    (hODEDiff : ode_regularity_differentiable_hypothesis H)
    (hBoot : ode_linear_regularity_bootstrap_hypothesis H) :
    ∀ t, H t = cosh t :=
  dAlembert_cosh_solution H h.1 hCont h.2 hCalib hSmooth hODE hODECont hODEDiff hBoot

What this page does not claim

This page does not claim that the d'Alembert equation itself is a new discovery. It does not claim that the classification theorem is proved without regularity assumptions. It does not claim that the framework derives the fine-structure constant.

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$ lake env lean IndisputableMonolith/Foundation/DAlembert/Proof.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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