Cost Functional Equation Ode Regularity Continuous Of Smooth

The Regularity Bridge

The declaration ode_regularity_continuous_of_smooth is a lemma in the framework's machine-checked library of formal theorems. It establishes a bridge between two worlds: the world of functional equations, where functions are defined by how they behave under addition and multiplication, and the world of differential equations, where functions are defined by their rates of change. The bridge says that if a function is smooth enough, meaning it has derivatives of all orders, then a certain functional equation implies a corresponding differential equation. In plain terms, it certifies that a smooth function satisfying a particular algebraic identity also satisfies a specific differential equation, and vice versa.

The classical context is the d'Alembert equation, named after Jean le Rond d'Alembert, who studied it in 1747. The equation is H(t+u) + H(t-u) = 2 * H(t) * H(u). Its continuous solutions are the hyperbolic cosine, H(t) = cosh(t), and the constant zero. The framework's library proves this in several steps. First, it shows that a continuous solution to the d'Alembert equation is actually smooth, a result attributed to Aczél. Then it shows that this smooth solution satisfies the ordinary differential equation H''(t) = H(t). The lemma ode_regularity_continuous_of_smooth is the formal statement that this implication holds: from a smooth solution of the functional equation, one can derive the differential equation.

The library also proves the converse direction: if a function satisfies the differential equation H''(t) = H(t) with initial conditions H(0) = 1 and H'(0) = 0, then it must be the hyperbolic cosine. This is a standard uniqueness result for ordinary differential equations. The combination of these two directions, the bridge from functional equation to differential equation and the uniqueness of the differential equation's solution, is what forces the hyperbolic cosine as the unique solution to the d'Alembert equation under the smoothness hypothesis.

In Recognition Science, this regularity bridge is a load-bearing component of the proof that the cost function J(x) = (x + 1/x)/2 - 1 is the unique function satisfying five plain conditions. The framework models recognition events as a ledger, a discrete record of events, and the cost of recognition is forced, not chosen. The bridge is used to move from the composition law, which is a functional equation, to a differential equation whose solution is known. This is how the framework proves that the cost function must be J and not some other function.

The lemma does not claim that every solution to the d'Alembert equation is smooth. It only claims that a smooth solution satisfies the differential equation. The library also proves a separate theorem, dAlembert_continuous_of_log_curvature, which shows that a continuous solution with a certain curvature condition is also smooth. The distinction matters: the regularity bridge is a conditional statement, not an unconditional one. It is a tool used within a larger proof, not a standalone result about all possible solutions.

/-- **d'Alembert to ODE derivation.**

    If H satisfies the d'Alembert equation and is smooth, then H'' = H.

    Proof sketch: Differentiate H(t+u) + H(t-u) = 2H(t)H(u) twice with respect to u,
    then set u = 0 to get H''(t) = H''(0) · H(t). With calibration H''(0) = 1, this
    gives H''(t) = H(t). -/
def dAlembert_to_ODE_hypothesis (H : ℝ → ℝ) : Prop :=
  H 0 = 1 → Continuous H → (∀ t u, H (t+u) + H (t-u) = 2 * H t * H u) →
    deriv (deriv H) 0 = 1 → ∀ t, deriv (deriv H) t = H t

/-- **Aczél's Theorem (continuous d'Alembert solutions are smooth).**

    This is a classical result in functional equations theory:
    continuous solutions to f(x+y) + f(x-y) = 2f(x)f(y) with f(0) = 1
    are analytic and equal to cosh(λx) for some λ ∈ ℝ.

    Reference: Aczél, "Lectures on Functional Equations" (1966), Chapter 3.

    The full formalization would require:
    - Proving that measurable solutions are continuous (automatic continuity)
    - Using Taylor expansion around 0 to show analyticity
    - Applying the Cauchy functional equation theory

    For now, this is stated as a hypothesis that follows from Aczél's theorem. -/
def dAlembert_continuous_implies_smooth_hypothesis (H : ℝ → ℝ) : Prop :=
  H 0 = 1 → Continuous H → (∀ t u, H (t+u) + H (t-u) = 2 * H t * H u) → ContDiff ℝ ⊤ H

theorem ode_cosh_uniqueness (H : ℝ → ℝ)
    (h_ODE : ∀ t, deriv (deriv H) t = H t)
    (h_H0 : H 0 = 1)
    (h_H'0 : deriv H 0 = 0)
    (h_cont_hyp : ode_regularity_continuous_hypothesis H)
    (h_diff_hyp : ode_regularity_differentiable_hypothesis H)
    (h_bootstrap_hyp : ode_linear_regularity_bootstrap_hypothesis H) :
    ∀ t, H t = Real.cosh t := by
  have h_cont : Continuous H := h_cont_hyp h_ODE
  have h_diff : Differentiable ℝ H := h_diff_hyp h_ODE h_cont
  have h_C2 : ContDiff ℝ 2 H := h_bootstrap_hyp h_ODE h_cont h_diff
  exact ode_cosh_uniqueness_contdiff H h_C2 h_ODE h_H0 h_H'0

What this page does not claim

The lemma does not claim that every solution to the d'Alembert equation is smooth. The lemma does not claim that the d'Alembert equation has only the hyperbolic cosine as a solution without additional regularity conditions. The lemma does not prove the uniqueness of the cost function J; it is a component of that larger proof.

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/Cost/FunctionalEquation.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.

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