A Derivative Complexity Criterion for Elementary Integrability

v2 (25 April 2026): Revised and strengthened. (1) Open Problem P2. 5 (Differential Galois interpretation) updated from "Conjecture" to "now proved in companion paper" — references Theorem 5. 3 of Paper 1 and the Picard–Vessiot theory of van der Put no elementary formula for the first antiderivative is claimed. ──────────────────────────────────────── The Liouville–Risch theorem characterises whether a smooth function has an elementary antiderivative, but provides little information about the complexity structure of the derivative sequence. This paper develops a derivative-sequence complexity theory — Layer III of a three-layer integration framework — that fills this gap. We introduce six named conditions (E1, E2, E2-vanishing, E2-cycling, E2*, E3, N1, N2) on the derivative sequence of a smooth function y. The central results are: (i) the Liouville–Risch decomposition implies E1 and E2 (proved) ; (ii) E2* (constant-coefficient recurrence) implies elementary, and E3 (algebraic extension closure) implies elementary (both proved) ; (iii) y = R (x) e^ (Q (x) ) satisfies E2* if and only if R is a polynomial and Q (x) = λx (proved) ; and (iv) E2-cycling without E2* implies non-elementarity, proved for the Liouvillian class F via the Risch equation. A 14-function validation table classifies all examples with every entry justified by a proved theorem.