cost_stability_transfer
plain-language theorem explainer
The cost stability transfer theorem shows that a quantitative stability bound for near-solutions H of the d'Alembert equation implies a corresponding bound on the deviation of the associated cost function from the J-cost. Researchers deriving error estimates for the reciprocal cost in Recognition Science would cite this when moving from the functional equation to explicit uniqueness bounds. The proof is a direct invocation of the CostStabilityTransferHypothesis after assuming the d'Alembert stability estimate.
Claim. Let $H:ℝ→ℝ$ be $C^3$, even, with $H(0)=1$ and curvature $a=H''(0)>0$, and let $ε, B, K$ be the uniform defect, function, and third-derivative bounds on $[-T,T]$. Suppose the d'Alembert stability estimate holds: for $0<h≤T$ and $|t|≤T-h$, $|H(t)-cosh(√a t)|≤(δ_error(ε,B,K,h)/a)(cosh(√a |t|)-1)$. Then for $0<h≤T$ and $e^{-(T-h)}<x<e^{T-h}$, $|H(log x)-1-J(x)|≤(δ_error(ε,B,K,h)/a)(cosh(√a |log x|)-1)$, where $J(x)=(x+x^{-1})/2-1$.
background
The module formalizes stability estimates for near-solutions of the d'Alembert equation $H(t+u)+H(t-u)=2H(t)H(u)$, with defect $Δ_H(t,u)=H(t+u)+H(t-u)-2H(t)H(u)$. StabilityHypotheses requires $H$ to be $C^3$ and even with $H(0)=1$ and positive curvature $a=H''(0)$. StabilityBounds collects the uniform defect bound $ε$, the sup-norm $B$ on $H$, and the bound $K$ on $|H'''|$. StabilityEstimate is the quantitative bound showing $H$ is close to $cosh(√a t)$ with error controlled by $δ_error(ε,B,K,h)/a$ times $(cosh(√a |t|)-1)$ for $|t|≤T-h$ (equation 7.3 in the paper).
proof idea
The proof is a one-line wrapper that applies the CostStabilityTransferHypothesis. It invokes the hypothesis with the assumed StabilityEstimate h_stab together with the step size $h$ and the conditions $0<h≤T$ to obtain the stated bound on $|H(log x)-1-Jcost x|$.
why it matters
This result transfers the d'Alembert stability theorem to the cost functional, feeding directly into the cost stability corollary cost_stability_calibrated (when curvature equals 1). It fills the Cost Stability Corollary (Corollary 7.1) in the paper on uniqueness of the canonical reciprocal cost. In the Recognition Science framework it supports quantitative uniqueness statements for the J-cost consistent with T5 J-uniqueness and the Recognition Composition Law.
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