Foundation Dalembert Proof D Alembert Solution Deriv Zero

A symmetry at zero

The d'Alembert functional equation is a classical object in mathematics, studied since the 18th century. It asks for functions H that satisfy H(t+u) + H(t-u) = 2 H(t) H(u) for all real numbers t and u, with the normalization H(0) = 1. Its continuous solutions are exactly the familiar functions H(t) = cos(αt) and H(t) = cosh(αt), plus the constant function H(t) = 1. The equation appears across analysis, from trigonometry to the wave equation, where it encodes a symmetry between addition and subtraction.

One property follows directly from the equation's symmetry. If H is a solution and is differentiable at zero, then its derivative there must be zero: H'(0) = 0. The reason is that every solution is an even function, H(-t) = H(t), and any differentiable even function has zero slope at the origin. For the cosine and hyperbolic cosine solutions, this is visible: both have a flat tangent at t = 0. The constant solution is flat everywhere.

In Recognition Science, the framework models a ledger, a discrete record of recognition events, and the cost of recognition is forced by five plain conditions. The d'Alembert equation emerges as the unique form for multiplicative consistency in that derivation. The theorem dAlembert_solution_deriv_zero is a small but load-bearing step in that chain: it shows that any smooth solution must have zero derivative at the origin, a regularity fact that later steps rely on when they select the hyperbolic cosine solution and, ultimately, the cost function J(x) = (x + 1/x)/2 - 1.

What the theorem does not claim is just as important. It does not say that every solution is differentiable, nor that the derivative exists at all points. It assumes differentiability at zero as a hypothesis. It does not classify the solutions; the classification of continuous solutions as cosine, hyperbolic cosine, or constant is a separate theorem. And it does not, by itself, derive the cost function. The theorem is a lemma about a symmetry, not the full uniqueness argument.

For the reader, the payoff is a clean picture: the d'Alembert equation, a classical object, has a built-in flatness at the origin for any smooth solution. That flatness is what lets the framework's derivation proceed from symmetry to a unique cost. The theorem is a precise, machine-checked fact that supports a larger story without being the whole story.

/-- D'Alembert solutions satisfy H'(0) = 0 if differentiable. -/
theorem dAlembert_solution_deriv_zero (H : ℝ → ℝ) (h : IsDAlembertSolution H)
    (hDiff : DifferentiableAt ℝ H 0) :
    deriv H 0 = 0 := by
  have hEven := dAlembert_solution_even H h
  exact even_deriv_at_zero H hEven hDiff

/-- D'Alembert solutions are even. -/
theorem dAlembert_solution_even (H : ℝ → ℝ) (h : IsDAlembertSolution H) :
    Function.Even H := by
  have h0 := h.1
  have heq := h.2
  intro u
  have := heq 0 u
  simp only [zero_add, zero_sub, h0, two_mul] at this
  linarith

What this page does not claim

The theorem does not prove that all d'Alembert solutions are differentiable. The theorem does not classify the solutions as cosine, hyperbolic cosine, or constant. The theorem alone does not derive the cost function J(x) = (x + 1/x)/2 - 1.

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/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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