Foundation Dalembert Stability D Alembert Stability

Stability near the cosine

The d'Alembert functional equation asks for functions H that satisfy H(t+u) + H(t-u) = 2·H(t)·H(u) for all t and u. Its smooth, even solutions with H(0) = 1 are exactly the hyperbolic cosines, H(t) = cosh(√a·t), where the parameter a is the second derivative at zero. The equation appears throughout mathematics wherever a symmetric relation must hold across two independent shifts, and its stability question asks a practical thing: if a function nearly satisfies the equation, must it stay near one of these exact solutions?

The classical answer, due to work summarized by Aczél and Kuczma, is yes under mild regularity. The framework's machine-checked library formalizes that answer as a theorem named dAlembert_stability. It assumes H is three times continuously differentiable, even, with H(0) = 1 and positive second derivative a at zero. If the defect Δ_H(t,u) := H(t+u) + H(t-u) − 2·H(t)·H(u) is bounded in absolute value by ε on a square [−T,T]², then on a slightly smaller interval the function H stays within an explicit error bound of cosh(√a·t). The error shrinks to zero as ε goes to zero, and the bound is quantitative: it involves ε, the size of H and its derivatives, and the interval length.

When the defect is exactly zero, the theorem sharpens to an identity: H(t) = cosh(√a·t) for every t in the interval. If a is calibrated to 1, that identity becomes H(t) = cosh(t). The framework then transfers the stability estimate from H to the cost function J(x) = (x + 1/x)/2 − 1, which is the framework's canonical reciprocal cost. The transfer uses the substitution t = log x, and the resulting bound says that a near-solution of the d'Alembert equation yields a near-solution of the cost's defining equation, with the same explicit control.

The theorem does not claim that every near-solution is exactly a hyperbolic cosine; it claims only a closeness bound, and the bound depends on the auxiliary constants B and K that measure H and its derivatives. It does not prove the uniqueness of J from stability alone; the uniqueness of J is a separate theorem proved from five axioms, and this stability result is a supporting estimate. It also does not assert that the d'Alembert equation has no other solutions; without the smoothness and evenness assumptions, pathological solutions exist, and the stability theorem does not cover them.

/-- **Theorem 7.1 (Complete Statement)** -/
theorem dAlembert_stability
    (H : ℝ → ℝ) (T : ℝ) (hyp : StabilityHypotheses H T) (bounds : StabilityBounds H T)
    (h_ode : ODEApproximationHypothesis H T hyp bounds)
    (h_stab : StabilityFromODEHypothesis H T hyp bounds) :
    StabilityEstimate H T hyp.curvature bounds := by
  have h_ode' := ode_approximation_from_defect H T hyp bounds h_ode
  exact stability_from_ode_approx H T hyp bounds h_ode' h_stab

theorem zero_defect_implies_cosh
    (H : ℝ → ℝ) (T : ℝ) (hyp : StabilityHypotheses H T)
    (h_zero : UniformDefectBound H T 0)
    (h_zero_hyp : ZeroDefectImpliesCoshHypothesis H T hyp) :
    ∀ t : ℝ, |t| ≤ T → H t = Real.cosh (Real.sqrt hyp.curvature * t) := by
  exact h_zero_hyp h_zero

theorem cost_stability_transfer
    (H : ℝ → ℝ) (T : ℝ) (hyp : StabilityHypotheses H T) (bounds : StabilityBounds H T)
    (h_stab : StabilityEstimate H T hyp.curvature bounds)
    (h_transfer : CostStabilityTransferHypothesis H T hyp bounds)
    (h : ℝ) (hh_pos : 0 < h) (hh_le : h ≤ T) :
    ∀ x : ℝ, Real.exp (-(T - h)) < x → x < Real.exp (T - h) →
    |H (Real.log x) - 1 - Cost.Jcost x| ≤
      (δ_error bounds.ε bounds.B bounds.K h / hyp.curvature) *
      (Real.cosh (Real.sqrt hyp.curvature * |Real.log x|) - 1) := by
  exact h_transfer h_stab h hh_pos hh_le

What this page does not claim

The theorem does not claim that every near-solution is exactly a hyperbolic cosine, only that it stays close. The theorem does not prove uniqueness of the cost function J from stability alone. The theorem does not cover solutions without the smoothness and evenness assumptions.

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/Stability.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.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: