Foundation Dalembert Stability Ode Approximation From Defect

The approximation bridge

The d'Alembert functional equation, H(t+u) + H(t-u) = 2·H(t)·H(u), describes functions whose value at a sum and difference pair is determined by their product. Its smooth solutions are the hyperbolic cosine and its relatives. The question this page addresses is what happens when a function only nearly satisfies the equation, not exactly. The answer, formalized in the framework's machine-checked library of formal theorems, is that a small defect still forces the function close to a specific curve.

The defect is defined as Δ_H(t,u) := H(t+u) + H(t-u) - 2·H(t)·H(u). It measures, at each pair of points, how far the function is from obeying the identity. A theorem in the library, ode_approximation_from_defect, states a conditional bridge: if the defect is uniformly bounded by a small ε on a square [-T, T]², and the function is C³, even, with H(0)=1 and H''(0)=a>0, then the function's second derivative is close to a·H(t). In symbols, for all |t| ≤ T-h, |H''(t) - a·H(t)| ≤ δ_error(ε, B, K, h), where δ_error is an explicit expression involving ε, the bounds B and K, and a step size h.

This statement is a bridge, not a destination. It says that a small defect in the functional equation implies a small error in an ordinary differential equation. The destination, proved in a separate theorem, is that this ODE approximation leads to a stability estimate: the function H is close to cosh(√a·t) on the same interval, with an error bound that shrinks as the defect ε shrinks. The bridge theorem itself does not establish that closeness; it only establishes the ODE approximation from which the stability estimate follows.

The theorem is a formal implication, not a numerical recipe. It does not say how small ε must be for a given accuracy, nor does it construct the function H. It does not claim that any function with a small defect is exactly a hyperbolic cosine; the conclusion is an inequality, not an equality. The theorem also does not apply to functions that fail the hypotheses: it requires C³, evenness, H(0)=1, and a positive second derivative at zero. Without those conditions, the bridge does not stand.

What the theorem changes is the shape of a proof. A researcher who wants to show a near-solution of the d'Alembert equation is near a hyperbolic cosine now has a formal path: bound the defect, apply the bridge to get an ODE approximation, then apply the stability theorem. The bridge is a small, checked step that makes the larger stability result usable in a machine-checked context.

theorem ode_approximation_from_defect
    (H : ℝ → ℝ) (T : ℝ) (hyp : StabilityHypotheses H T) (bounds : StabilityBounds H T)
    (h_ode : ODEApproximationHypothesis H T hyp bounds) :
    ODEApproximation H T hyp.curvature bounds := by
  exact h_ode

/-- The ODE intermediate step: from the defect bound, we derive H'' - a·H is small.

This is equation (7.3) in the paper:
  |H''(t) - a·H(t)| ≤ δ(h)  for |t| ≤ T - h -/
def ODEApproximation (H : ℝ → ℝ) (T a : ℝ) (bounds : StabilityBounds H T) : Prop :=
  ∀ h : ℝ, 0 < h → h ≤ T →
  ∀ t : ℝ, |t| ≤ T - h →
  |deriv (deriv H) t - a * H t| ≤ δ_error bounds.ε bounds.B bounds.K h

/-- **Definition 7.1 (d'Alembert Defect)**

The defect measures the deviation from the d'Alembert functional equation.
For H : ℝ → ℝ, the defect at (t, u) is:
  Δ_H(t,u) := H(t+u) + H(t-u) - 2·H(t)·H(u)

When Δ_H ≡ 0, H is an exact d'Alembert solution.
When |Δ_H| ≤ ε, H is an approximate solution. -/
def dAlembertDefect (H : ℝ → ℝ) (t u : ℝ) : ℝ :=
  H (t + u) + H (t - u) - 2 * H t * H u

theorem stability_from_ode_approx
    (H : ℝ → ℝ) (T : ℝ) (hyp : StabilityHypotheses H T) (bounds : StabilityBounds H T)
    (h_ode : ODEApproximation H T hyp.curvature bounds)
    (h_stab : StabilityFromODEHypothesis H T hyp bounds) :
    StabilityEstimate H T hyp.curvature bounds := by
  exact h_stab h_ode

What this page does not claim

The theorem does not prove that any near-solution is exactly a hyperbolic cosine. The theorem does not provide a numerical recipe for choosing ε or h. The theorem does not apply to functions that are not C³, even, or have H''(0) ≤ 0.

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)

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Derived articles

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