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| Main Author: | |
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| Format: | Recurso digital |
| Language: | English |
| Published: |
Zenodo
2026
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| Subjects: | |
| Online Access: | https://doi.org/10.5281/zenodo.20344519 |
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- <p>v4 (22 May 2026): Minor cleanup.<br>(1) HAL identifiers removed from companion paper bibliography entry<br> — the HAL deposits were rejected due to affiliation policy and<br> the identifiers are no longer valid. Zenodo DOI is the sole<br> public identifier.<br>(2) Gel'fand–Dikii (1975) and Mikhailov–Novikov–Wang (2007) added<br> to bibliography. Originality Remark updated to acknowledge the<br> classical background of the parity criterion and precisely<br> enumerate new contributions.</p> <p>────────────────────────────────────────</p> <p>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.</p>