Finite Gaussian Reconstruction of Polynomial Orbits: From Correlated Moments to Oscillatory Periods

Abstract: Let $P$ be a real polynomial of degree at most $m$ on $\mathbb{R}<sup>d$, 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, $$M_{P,r}(Σ)=\mathbb{E}\prod_{a=1}^r P(X_a),$$ separates $O(d)$-orbits. We construct an explicit replica cutoff and rational covariance grids satisfying $\frac{1}{2}I_r\preceqΣ\preceq\frac{3}{2}I_r.$ 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 $(m-1)<sup>{d|I|}$; zero coupling is therefore a rank-changing boundary. The forced scaling $$τ_a=ρ^{{m-2}λ_a,\qquad} x_a=ρ^{-1}u_a$$ 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.