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 C₍-₁-Bₙ' (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 Iⁿ = ∫ⁿ f dxⁿ (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 ΣAₙ sⁿ = se^ (xs+s²/4) ), sin (x) /x (portal dimension 3), √ (1+x³), xˣ, the complete xᵏ 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 Cxy has Galois group isomorphic to Gₐ = (C, +) when |n−m| is even, and trivial group when |n−m| is odd.