ContinuousCombinerSecondDerivativeInput
This definition packages the second-derivative identity 2G''(t) = ψ(G(t)) G''(0) extracted from a smooth log-coordinate Aczél equation for the derived cost of a continuous Law of Logic. Analysts assembling continuous-combiner inputs in the Recognition framework cite it when preparing the hypotheses for the ψ-affine forcing step. It is realized as a direct wrapper around AczelSecondDerivativeIdentity applied to the exponentially rescaled derived-cost function.
claimLet $C$ be a comparison operator satisfying the continuous laws of logic and let $G(t)$ denote the derived cost of $C$ evaluated at $e^t$, assumed infinitely differentiable. Then there exists a function $ψ$ such that $2G''(t) = ψ(G(t)) · G''(0)$ for all real $t$.
background
The GeneralizedDAlembert module replaces the polynomial-degree restriction on route independence with continuity alone, invoking the Aczél–Kannappan classification of continuous solutions to the d'Alembert equation. SatisfiesLawsOfLogicContinuous is the structure that encodes identity, non-contradiction, excluded middle, scale invariance, continuous route independence, and non-triviality for such a combiner $C$. The upstream AczelSecondDerivativeIdentity states: for a smooth $G$, there exists $ψ$ such that $2 · (G'') (t) = ψ(G(t)) · (G''(0))$.
proof idea
This is a one-line wrapper that applies AczelSecondDerivativeIdentity to the function $t ↦$ derivedCost $C$ (Real.exp $t$).
why it matters in Recognition Science
It supplies the derivative-identity component of ContinuousCombinerAnalysisInputs, which is consumed by the theorem continuous_combiner_psi_affine_forcing to obtain the log-bilinear identity. This advances the module's goal of discharging the polynomial restriction in the Translation Theorem via continuity, as stated in the module doc, while the quartic-log obstruction shows the full package remains a hypothesis. It connects to the framework's use of the Aczél equation in the foundation layer.
scope and limits
- Does not prove the second-derivative identity holds automatically from continuity of the combiner.
- Does not discharge the separate ψ-affine completion step.
- Does not address the quartic log-cost obstruction to the identity.
formal statement (Lean)
544def ContinuousCombinerSecondDerivativeInput
545 (C : ComparisonOperator)
546 (_h : SatisfiesLawsOfLogicContinuous C)
547 (_hSmooth : ContDiff ℝ ((⊤ : ℕ∞) : WithTop ℕ∞)
548 (fun t : ℝ => derivedCost C (Real.exp t))) : Prop :=
proof body
Definition body.
549 AczelSecondDerivativeIdentity (fun t : ℝ => derivedCost C (Real.exp t))
550
551/-- **Named analysis input 2b: ψ-affine completion.**
552
553This is the remaining Stetkær/Aczél content: the derivative identity, symmetry
554of the combiner, and the Aczél equation force the actual log-bilinear identity.
555It is narrower than the former all-in ψ-affine forcing axiom because the
556second-derivative extraction is now named separately. -/