IndisputableMonolith.Foundation.GeneralizedDAlembert.SecondDerivative
Module housing second-derivative identities for the quartic counterexample to unrestricted route-independence combiners. It records that a degree-4 polynomial fails the second-derivative form of Aczél's equation that degree-≤2 combiners satisfy. Foundation workers cite it when relaxing the Translation Theorem's polynomial-degree hypothesis to a continuity (or C²) regularity condition. The argument is direct differentiation of the quartic-log combiner.
claimFor the quartic combiner $P_4$, the second mixed partials fail the Aczél-type identity $\partial_{xx}P + \partial_{yy}P = F(\partial_x P, \partial_y P)$ that holds for every total-degree-$\le 2$ polynomial combiner; explicitly, $\partial_{xx}P_4$ is computed and shown not to match the required functional form.
background
The parent module GeneralizedDAlembert weakens the Translation Theorem's demand that the route-independence combiner $P$ be a polynomial of total degree at most two. A quartic-log counterexample shows some regularity is necessary, yet degree $\le 2$ is stronger than needed; the program is to replace the degree bound by continuity (or $C^2$) hypotheses.
Aczél's classical d'Alembert / cosine functional equation and its polynomial solutions motivate the second-derivative test used here. Differentiating a candidate combiner twice yields an algebraic identity that every admissible $P$ must satisfy. The quartic fails that identity, so it is excluded once second derivatives exist, without forcing global degree $\le 2$.
Sibling declarations name the explicit second derivative of the quartic and the negation of the Aczél second-derivative identity for that quartic.
proof idea
The module is a short calculation layer on top of GeneralizedDAlembert. It defines the second partials of the quartic combiner by direct term-by-term differentiation, then compares the resulting expression against the algebraic shape required by the differentiated Aczél / d'Alembert identity. The mismatch is recorded as a negative lemma (quartic does not satisfy the second-derivative identity). No heavy tactic machinery; the content is elementary multivariable calculus plus the already-imported combiner algebra.
why it matters in Recognition Science
Inside the Recognition Science foundation, the Translation Theorem currently forces $P$ to be degree $\le 2$. This module supplies the concrete quartic obstruction that justifies keeping a regularity hypothesis while dropping the rigid degree bound (Move 3 of GeneralizedDAlembert: discharge polynomial regularity using continuity). Downstream, a continuity-only or $C^2$-only version of route independence can cite the negative second-derivative lemma to show the quartic is already ruled out. The module does not yet feed a named parent theorem (used_by is empty); it is scaffolding for the regularity relaxation that protects the forcing chain's early steps without over-restricting combiners.
scope and limits
- Does not prove the full continuity-only Translation Theorem.
- Does not classify all $C^2$ solutions of the Aczél second-derivative identity.
- Does not treat non-polynomial or non-smooth combiners beyond the quartic example.
- Does not connect second derivatives to the J-cost or T5–T8 forcing steps directly.