AI 中文总结
该研究证明关联高斯副本的混合矩可分离多项式的正交轨道，构造了有限高斯重构方案，并将其推广至不规则周期系统，建立了轨道证书与周期的对应关系。
AI 中文摘要
设P为ℝᵈ上次数不超过m的实多项式，X为标准高斯变量。由于高斯观测在O(d)下不变，自然的逆问题是重构P的正交轨道；仅靠P(X)的分布通常不足以完成该任务。我们证明，关联高斯副本的指定有限混合矩族M_{P,r}(Σ)=E[∏_{a=1}^r P(X_a)]可分离O(d)轨道。我们构造了满足1/2 I_r ⪯ Σ ⪯ 3/2 I_r的显式副本截断和有理协方差网格。有限差分重构了不变理论所需的所有完全威克收缩，得到精确的有限解码器。所得探针图在系数球上与轨道距离双赫尔德等价，具有有效指数。随后，我们在不规则周期系统中识别出相同的证书：副本特征函数是多项式振荡周期，其在零点的混合导数即为上述矩。若P的首项齐次部分具有孤立临界点，由I索引的活跃副本面具有扭曲德拉姆秩(m-1)^{d|I|}；因此零耦合是秩变化边界。强制缩放τ_a=ρ^{m-2}λ_a、x_a=ρ⁻¹u_a产生兼容的里斯-雅可比格，且在中心非共振下得到正则秩一高斯分支。在可容许 tame 莫尔斯腔中，周期矩阵分解为代数雅可比、扇形瑟姆布尔和积分贝蒂分量。将组装的实轮廓周期投影到高斯分支上，恰好重构出有限轨道证书。
英文摘要
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, $$ M_{P,r}(Σ)=\mathbb{E}\prod_{a=1}^r P(X_a), $$ 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 $(m-1)^{d|I|}$; 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.
