An Integration Algebra for Smooth Functions: Telescoping Formulas, Portal Structures, and the Parity Criterion

v3 (22 May 2026): Mathematical rigour strengthened throughout.(1) Parity Criterion proof (ii) rewritten: three explicit steps using differential transcendence and forward reference to Key Lemma (Lemma 5.2).(2) "Generic smooth function" replaced throughout with the precise condition: y differentially transcendental over C(x). New Remark in Layer I defines this concept.(3) sqrt(1+x³) portal proof: key intermediate calculation ∫x²y''' dx = 2(y''')³/9 now shown explicitly.(4) Sinc portal: propagation of [Cₙ₋₁-B_n]'(0)=0 demonstrated step by step.(5) Hierarchy table caption softened to avoid overclaiming completeness. ──────────────────────────────────────── ──────────────────────────────────────── We develop a constructive theory of closed-form integration for smooth functions, in two layers. Layer I (Integration Algebra) establishes four identities valid for every smooth y: the Power Portal Formula; the x-Weighted Reduction; the Telescoping Formula for ∫y^(n)y^(m)dx (proved by induction); and the Parity Resolvability Criterion: ∫y^(n)y^(m)dx is expressible in closed form if and only if |n−m| is odd, for y differentially transcendental over Q(x) (proved in both directions). Layer II (Portal Structures) addresses functions whose first antiderivative is non-elementary. Writing y = ∫f dx (non-elementary), the higher iterated antiderivatives I^n = ∫^n f dx^n (n ≥ 2) lie in a finite-dimensional module — the portal. No elementary formula for ∫f dx itself is claimed. Complete verified portal formulas are derived for: e^(−x²) (Gaussian portal, generating function ΣA_n s^n = se^(xs+s²/4)), sin(x)/x (portal dimension 3), √(1+x³), x^x, the complete x^k ln x family, and a new Type TR portal for 1/(x ln x). Additionally, the Differential Galois conjecture is proved: the Picard–Vessiot extension of ∫y^(n)y^(m)dx over the differential polynomial ring C[x]{y} has Galois group isomorphic to G_a = (C,+) when |n−m| is even, and trivial group when |n−m| is odd.