[2608.14475] Finite Gaussian Reconstruction of Polynomial Orbits: From Correlated Moments to Oscillatory Periods Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Probability arXiv:2608.14475 (math) [Submitted on 14 Aug 2026] Title: Finite Gaussian Reconstruction of Polynomial Orbits: From Correlated Moments to Oscillatory Periods Authors: Obayda Julien Assaad View a PDF of the paper titled Finite Gaussian Reconstruction of Polynomial Orbits: From Correlated Moments to Oscillatory Periods, by Obayda Julien Assaad View PDF HTML (experimental) Abstract: Let $P$ be a real polynomial of degree at most $m$ on $\mathbb{R}^d$, and let $X$ be standard Gaussian. Because Gaussian observations are invariant under $O(d)$, the natural inverse problem is to recover the orthogonal orbit of $P$; the law of $P(X)$ alone is generally insufficient. We prove that a prescribed finite family of mixed moments of correlated Gaussian replicas, $$ M_{P,r}(\Sigma)=\mathbb{E}\prod_{a=1}^r P(X_a), $$ separates $O(d)$-orbits. We construct an explicit replica cutoff and rational covariance grids satisfying $$ \frac{1}{2}I_r\preceq\Sigma\preceq\frac{3}{2}I_r. $$ Finite differences recover all complete Wick contractions needed by invariant theory, giving an exact finite decoder. The resulting probe map is bi-H"older equivalent to orbit distance on coefficient balls, with an effective exponent. We then identify the same certificate in an irregular period system. Replicated characteristic functions are polynomial oscillatory periods, and their mixed derivatives at zero are the moments above. If the leading homogeneous part of $P$ has an isolated critical point, the active-replica face indexed by $I$ has twisted de Rham rank $(m-1)^{d|I|}$; zero coupling is therefore a rank-changing boundary. The forced scaling $$ \tau_a=\rho^{m-2}\lambda_a,\qquad x_a=\rho^{-1}u_a $$ produces compatible Rees--Jacobi lattices and, under central nonresonance, a canonical rank-one Gaussian branch. On admissible tame Morse chambers, the period matrix factors into algebraic Jacobi, sectorial thimble, and integral Betti components. Projecting the assembled real-contour period onto the Gaussian branch recovers exactly the finite orbit certificate. Subjects: Probability (math.PR) ; Commutative Algebra (math.AC); Algebraic Geometry (math.AG) Cite as: arXiv:2608.14475 [math.PR] (or arXiv:2608.14475v1 [math.PR] for this version) https://doi.org/10.48550/arXiv.2608.14475 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Obayda Julien Assaad [ view email ] [v1] Fri, 14 Aug 2026 16:54:20 UTC (54 KB) Full-text links: Access Paper: View a PDF of the paper titled Finite Gaussian Reconstruction of Polynomial Orbits: From Correlated Moments to Oscillatory Periods, by Obayda Julien Assaad View PDF HTML (experimental) TeX Source view license Current browse context: math.PR < prev | next > new | recent | 2026-08 Change to browse by: math math.AC math.AG References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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