Foundation Dalembert Stability Stability From Ode Approx

Stability from approximation

The d'Alembert functional equation, H(t+u) + H(t-u) = 2·H(t)·H(u), characterizes the hyperbolic cosine among well-behaved functions. It says that the value at the midpoint of an interval, shifted by u, relates to the values at the endpoints in a precise way. The classical question, studied by Jean le Rond d'Alembert in the 1740s and later by functional equation theorists like János Aczél, is whether a function that only nearly satisfies this equation must still be nearly a hyperbolic cosine.

The machine-checked library of formal theorems in the Recognition Science framework addresses this question through a stability theorem. The theorem, named stability_from_ode_approx, states a conditional result: if a function H is smooth, even, has H(0) = 1, and its second derivative is close to a constant multiple of H itself, then H is close to cosh(√a·t), where a is the curvature parameter. The closeness is quantitative. The error is bounded by an explicit expression involving a defect measure, a step size h, and a bound on the function's third derivative. This is not a vague statement that approximation implies approximation; it is a precise inequality with a named error term, δ_error, that depends on the input bounds.

The theorem's structure is a bridge. It takes an assumption about the differential equation, H''(t) ≈ a·H(t), and converts it into an estimate on the original functional equation. The proof is a chain of implications formalized in the library: the ODE approximation hypothesis, ODEApproximation, is assumed; the stability hypothesis, StabilityFromODEHypothesis, states that the ODE approximation implies the stability estimate; the theorem then applies that hypothesis to conclude the stability estimate holds. The calibrated version, stability_calibrated, specializes to the case where a = 1, giving the bound |H(t) - cosh t| ≤ δ_error·(cosh |t| - 1).

In Recognition Science, this result supports the derivation of the canonical cost function J(x) = (x + 1/x)/2 - 1. The framework models recognition events as a ledger, a discrete record of comparisons, and the cost of recognition is the forced price of that record. The stability theorem transfers the d'Alembert stability estimate to the cost functional, showing that a near-solution of the cost equation is close to the exact J. This is a theorem about mathematical functions, not a statement about physical measurements.

What the theorem does not claim is as important as what it proves. It does not claim that any function satisfying the ODE approximation is exactly cosh; it only bounds the difference. It does not claim that the stability hypothesis, StabilityFromODEHypothesis, is itself proved; the theorem takes it as an input. It does not claim that the error bound is optimal; the explicit δ_error is a sufficient bound, not a minimal one. The theorem is a precise, conditional statement in the library's formal language, and its value lies in the exactness of its conclusion under its stated assumptions.

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

/-- When a = 1 (standard RS calibration), the stability estimate simplifies. -/
theorem stability_calibrated
    (H : ℝ → ℝ) (T : ℝ) (hyp : StabilityHypotheses H T)
    (h_a1 : hyp.curvature = 1)
    (bounds : StabilityBounds H T)
    (h_stab : StabilityEstimate H T hyp.curvature bounds)
    (h : ℝ) (hh_pos : 0 < h) (hh_le : h ≤ T) :
    ∀ t : ℝ, |t| ≤ T - h →
    |H t - Real.cosh t| ≤ δ_error bounds.ε bounds.B bounds.K h * (Real.cosh |t| - 1) := by
  intro t ht
  have h_main := h_stab h hh_pos hh_le t ht
  simp only [h_a1, Real.sqrt_one, one_mul, div_one] at h_main
  exact h_main

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 prove that the stability hypothesis StabilityFromODEHypothesis is true; it assumes it as an input. The theorem does not claim the error bound δ_error is optimal; it is a sufficient bound, not a minimal one. The theorem does not state that any function satisfying the ODE approximation is exactly a hyperbolic cosine; it only bounds the difference.

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

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