Ordered-Box Moment Geometry and Sharp Finite-Information Stability for Symmetric Real-Rooted Canonical Products
Year: 2026
Type: preprint
Abstract: We study two linked finite-information problems. The first is an obstacle-constrained equal-weight moment problem with coordinate-dependent bounds and fixed first $J$ power sums. We prove that every nonempty finite ordered-box fiber has unique upper and lower $p_{J+1}$-principal profiles, and that the same two profiles simultaneously extremize every separable objective with nonnegative $(J+1)$-st derivative. The theorem extends to countably infinite nonnegative ordered boxes dominated by a summable upper sequence. The second problem concerns normalized symmetric real-rooted canonical products $$F(z) = \prod_{n \ge 1} \left(1 - \frac{z^2}{\gamma_n^2}\right)$$ with simple positive zeros satisfying a counting envelope. Observing the first $N$ positive zeros and the first $J$ reciprocal even moments reduces the hidden tail to an ordered-capacity moment class. A general-$J$ strictification theorem realizes this geometry inside the original simple-zero class. This yields an exact arbitrary-envelope minimax capacity $\Lambda_J(q)$, a strict simple-zero lower realization, and a quantitative complex-disk upper bound. For noisy observations we introduce a directional logarithmic width $\Gamma_J(q;\eta)$ and a weighted higher stop-loss modulus $\Xi_J(q;\eta)$, with the exact zero-noise collapse $$\Gamma_J(q;0) = \Xi_J(q;0) = \Lambda_J(q).$$ The infinite-dimensional capacity is finitely certifiable through monotone finite truncations and an explicit analytic tail bound. For regularly varying quantiles $\rho_n = n^{1/\nu} \ell(n)$, $0 < \nu < 2$, the resulting dimensionless minimax law is sharp: $$\omega^\#_{N,J,R}(\delta;\Psi) \asymp \delta + N \rho_N^{-2J-2}.$$ The paper is restricted to the paired symmetric model and makes no claim for arbitrary nonsymmetric real-rooted canonical products.
Author: Lin, Tao
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7207690085
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