Cost Aczel Proof Exists Integral Ne Zero

The bootstrap lemma

In mathematics, a functional equation is an equation where the unknown is a function rather than a number. The d'Alembert equation, H(t+u) + H(t-u) = 2·H(t)·H(u), is one of the oldest and most studied. It asks for functions H that behave like cosine or hyperbolic cosine under addition and subtraction of inputs. A classical theorem, due to János Aczél, states that any continuous solution of this equation with H(0) = 1 is not merely continuous but infinitely differentiable, in fact real analytic.

The declaration exists_integral_ne_zero is a lemma inside the machine-checked proof of that theorem. It establishes a small but essential fact: if H is continuous and H(0) = 1, then the integral of H from 0 to some small positive number δ is not zero. The proof is short. Since H(0) = 1 and H is continuous, H stays close to 1 near 0, so its integral over a sufficiently small interval cannot vanish. This is a standard consequence of continuity, but it is the load-bearing step that makes the rest of the bootstrap argument work.

Why does this matter? The integration bootstrap is a technique for showing that a continuous function is smooth. One defines a new function Phi(t) as the integral of H from 0 to t. If Phi has a nonzero value at some point δ, then one can solve for H(t) in terms of Phi(t+δ) and Phi(t-δ). Since Phi is one degree smoother than H, this representation shows H is smoother than it appeared. Repeating the argument gives infinite differentiability. The lemma exists_integral_ne_zero guarantees that the needed nonzero value of Phi is always available under the hypotheses.

In the Recognition Science framework, this lemma is part of a larger chain that forces the cost function J(x) = (x + 1/x)/2 - 1. The framework models physical structure as a ledger of recognition events with a forced cost. The d'Alembert equation arises naturally in that setting, and the smoothness theorem is a step toward classifying all possible cost functions. But the lemma itself is purely classical mathematics; it makes no claim about physics, recognition, or the framework's axioms. It is a fact about continuous functions on the real line.

The lemma does not claim that the integral is nonzero for every δ, only that there exists at least one positive δ with that property. It does not assert anything about the value of the integral, only that it is not zero. And it does not, by itself, prove the Aczél classification theorem; it is one ingredient in a longer proof. The full classification, which identifies all continuous solutions as the constant 1, cosh(αt), or cos(αt), is a separate theorem that builds on this lemma and several others.

private lemma exists_integral_ne_zero (H : ℝ → ℝ) (h_one : H 0 = 1) (h_cont : Continuous H) :
    ∃ δ : ℝ, 0 < δ ∧ Phi H δ ≠ 0 := by
  have h_pos : (0 : ℝ) < H 0 := by rw [h_one]; exact one_pos
  have h_ev : ∀ᶠ x in nhds (0 : ℝ), (0 : ℝ) < H x :=
    h_cont.continuousAt.eventually (Ioi_mem_nhds h_pos)
  obtain ⟨ε, hε_pos, hε⟩ := Metric.eventually_nhds_iff.mp h_ev
  refine ⟨ε / 2, by positivity, ?_⟩
  intro h_eq
  have hδ_pos : (0 : ℝ) < ε / 2 := by positivity
  obtain ⟨c, hc_mem, hc_eq⟩ := exists_hasDerivAt_eq_slope (Phi H) H hδ_pos
    ((phi_differentiable H h_cont).continuous.continuousOn)
    (fun x _ => phi_hasDerivAt H h_cont x)
  rw [phi_zero, h_eq, sub_zero, zero_div] at hc_eq
  linarith [hε (show dist c 0 < ε by
    simp only [Real.dist_eq, sub_zero, abs_of_pos hc_mem.1]; linarith [hc_mem.2])]

private theorem dAlembert_contDiff_smooth (H : ℝ → ℝ) (h_one : H 0 = 1) (h_cont : Continuous H)
    (h_dAl : ∀ t u, H (t + u) + H (t - u) = 2 * H t * H u) :
    ContDiff ℝ smooth H :=
  contDiff_infty.mpr (dAlembert_contDiff_nat H h_one h_cont h_dAl)

What this page does not claim

The lemma does not claim the integral is nonzero for every δ, only for at least one positive δ. The lemma does not by itself prove the full Aczél classification theorem. The lemma makes no claim about physics or the Recognition Science framework's axioms.

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

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