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  1. Spectral Crowding and Compression Limits for Arithmetic Covariance OperatorsScott SchubrEdition 2 · 2026-11-02 Public
  2. Diffusion budgets for contact-freezing lattice ellipsoidsScott SchubrEdition 2 · 2026-11-02 Public
  3. Stable star marginals and strict edge correlations for random forestsScott SchubrEdition 3 · 2026-11-02 Public
  4. Sharp output ranges and bias frontiers for stabilizer purity estimationScott SchubrEdition 2 · 2026-11-02 Public
  5. Covariance confidence bounds and parity certificates for stable count lawsScott SchubrEdition 3 · 2026-11-02 Public
  6. Log-concave polynomial prefix sums: algebraic certificates and a sharp zero sectorScott SchubrEdition 3 · 2026-11-02 Public
  7. Audibility of all odd blind modes in rapidly twisted polar-strip tubesScott SchubrEdition 3 · 2026-11-02 Public
  8. Sharp collision criteria for frozen color histogramsScott SchubrEdition 3 · 2026-11-02 Public
  9. Sharp noise transitions for an adaptive single-copy stabilizer-learning processScott SchubrEdition 3 · 2026-11-02 Public
  10. Leading bias and positive correction of heat-kernel Sinkhorn metricsScott SchubrEdition 3 · 2026-11-02 Public
  11. Chollet's inequality for positive semidefinite matrices of rank at most fourScott SchubrEdition 4 · 2026-11-02 Public
  12. Bose gases in weakly twisted annular-sector waveguidesScott SchubrEdition 3 · 2026-11-02 Public
  13. Optimal concavity powers for lattice counts on smooth toric Fano threefoldsScott SchubrEdition 3 · 2026-11-02 Public
  14. A square-integral threshold for subadditivity on shrinking corridorsScott SchubrEdition 5 · 2026-11-02 Public
  15. Small-ball estimates for Hölder centers of the Kloosterman processScott SchubrEdition 3 · 2026-11-02 Public
  16. Finite-projection holomorphic stabilization of contractible affine manifoldsScott SchubrEdition 4 · 2026-11-02 Public
  17. Power Concavity of Low-Dimensional Unimodular Fiber CountsScott SchubrEdition 5 · 2026-11-02 Public
  18. Positive Eigenfunctions Above the First Eigenvalue of the Affine 2-LaplacianScott SchubrEdition 4 · 2026-11-02 Public
  19. Finite Persistent Memory in Exact Sampling: Countable Rigidity and Algebraic Memory BoundsScott SchubrEdition 3 · 2026-11-02 Public
  20. Mixed-state escape bounds and proper agnostic learning of stabilizer states in odd-prime dimensionsScott SchubrEdition 4 · 2026-11-02 Public
  21. Resolving Goldreich's Open Problem 4.1: Linear Lower Bounds for Cubic Graph Isomorphism TestingScott SchubrEdition 7 · 2026-11-02 Public
  22. Eight-point fiber certificates and binary-input list recovery of folded Reed–Solomon codesScott SchubrEdition 3 · 2026-11-01 Public
  23. Single-time boundary heat-flux rigidity for smooth domainsScott SchubrEdition 4 · 2026-11-01 Public
  24. RH: Complete Signed Certificates and Canonical Null SynthesisScott SchubrEdition 12 · 2026-11-01 Public
PublicEdition 2

A counterexample and corrected translation-density criterion for correlation kernels

Scott Schubr

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Abstract

A displayed correlation-kernel characterization of the Laguerre–Pólya class in Dimitrov and Xu, Wronskians of Fourier and Laplace transforms, fails even when the Fourier transform has only simple real zeros. We construct an even nonnegative compactly supported smooth probability density whose order-one transform has a pure paired Hadamard product and only simple real zeros, but whose stated correlation kernel has non-dense translates in L^1(ℝ) at the explicit width 1/1000. The sign change is proved by an analytic bound for the entire infinite product, without numerical evaluation of that product. The underlying distinction is between an outer hyperbolic factor, which receives the real part of an analytic Wronskian, and an inner hyperbolic factor, which receives the mixed Jensen curvature. We prove the corrected density equivalence, including its width-zero simplicity endpoint, under explicit moment and paired-product assumptions. The Jensen and Wiener principles are classical.

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