quartic_not_aczel_second_derivative_identity
plain-language theorem explainer
The declaration shows that G(t) = t^4 fails the second-derivative identity extracted from the smooth Aczél equation. Researchers examining continuous-combiner routes in the Recognition Science foundation would cite it to block that path. The proof assumes the witnessing ψ exists, specializes at t=1, rewrites the second derivative via the companion lemma, and obtains an immediate numerical contradiction.
Claim. There is no function $ψ : ℝ → ℝ$ such that $2 G''(t) = ψ(G(t)) · G''(0)$ holds for all $t ∈ ℝ$, where $G(t) = t^4$.
background
AczelSecondDerivativeIdentity asserts the existence of ψ such that 2 G''(t) = ψ(G(t)) G''(0) for every t, with ψ understood as the partial ∂₂P(u,0) from the smooth Aczél equation. The module treats the second-derivative obstruction that arises when one attempts to realize a continuous combiner. The quartic G(t) = t^4 is introduced as the finite-polynomial-closure counterexample because its second derivative vanishes at 0 yet equals 12 at t=1.
proof idea
The term proof assumes the identity, extracts the pair (ψ, hψ), specializes hψ at t=1, rewrites the left-hand side with the lemma quartic_second_derivative that supplies the explicit form 12 t², and applies norm_num to reach the contradiction 24 = 0.
why it matters
The result supplies a concrete obstruction that rules out the continuous-combiner route for the quartic log-cost already used in the paper. It thereby supports the shift toward the discrete eight-tick octave and the Recognition Composition Law in the T0–T8 forcing chain. No downstream declarations depend on it, so it functions as a local closure of a potential loophole in the generalized d'Alembert foundation.
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