optimal_h
The definition supplies the step size h that minimizes the composite error bound δ(h) = ε/h² + (1+B) K h /3 arising in quantitative stability analysis for near-solutions of the d'Alembert equation. Researchers studying stability of functional equations or transfer of bounds from the cost functional to hyperbolic functions would cite the resulting O(ε^{1/3}) rate. The expression is obtained by setting the derivative of δ with respect to h to zero and extracting the positive real root of the resulting cubic.
claimLet ε, B, K > 0. The minimizing step size is given by $h = (6ε/((1+B)K))^{1/3}$.
background
The d'Alembert module studies quantitative closeness of C³ even functions H with H(0)=1 to the hyperbolic cosine when the defect Δ_H(t,u) = H(t+u) + H(t-u) - 2 H(t) H(u) is small. The composite error δ(h) balances the contribution of the defect bound ε scaled by 1/h² against the third-derivative remainder scaled by h, with B the uniform bound on |H| and K the uniform bound on |H'''|.
proof idea
The definition is the explicit positive root obtained by setting dδ/dh = 0, which produces the cubic equation h³ = 6ε/((1+B)K).
why it matters in Recognition Science
The definition supplies the concrete h that realizes the O(ε^{1/3}) error rate inside the main stability estimate (Theorem 7.1) and its corollary transferring bounds to the cost functional F. It therefore supports the quantitative passage from small defect in the shifted cost H = J + 1 to proximity with cosh, consistent with the J-uniqueness step (T5) of the forcing chain.
scope and limits
- Does not prove the stability theorem or the bound on δ(h_opt).
- Does not treat the degenerate cases K = 0 or ε = 0.
- Does not incorporate dependence on the second-derivative value a or the interval length T.
- Does not extend to higher-order Taylor remainders or non-C³ functions.
formal statement (Lean)
171noncomputable def optimal_h (ε B K : ℝ) : ℝ :=
proof body
Definition body.
172 (6 * ε / ((1 + B) * K)) ^ (1/3 : ℝ)
173
174/-! ## Theorem 7.1: Main Stability Estimate -/
175
176/-- **Theorem 7.1 (d'Alembert Stability)**
177
178Let H ∈ C³([-T,T]) be even with H(0) = 1, and set a := H''(0) > 0.
179
180Define:
181- ε := sup_{|t|,|u| ≤ T} |Δ_H(t,u)| (defect bound)
182- B := sup_{|t| ≤ T} |H(t)| (function bound)
183- K := sup_{|t| ≤ T} |H'''(t)| (third derivative bound)
184- δ(h) := ε/h² + (1+B)·K·h/3 (error function)
185
186Then for every h with 0 < h ≤ T and every t with |t| ≤ T - h:
187
188 |H(t) - cosh(√a·t)| ≤ (δ(h)/a) · (cosh(√a·|t|) - 1)
189
190When a = 1 and δ(h) is small, this shows H ≈ cosh on compact intervals. -/