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  1. Uniform Resolvent Asymptotics for Growing Theta-Null Source FamiliesScott SchubrEdition 1 · 2026-10-27 Public
  2. Frequency-Source Resolvent Limits and Positive-Receiver InstabilityScott SchubrEdition 1 · 2026-10-27 Public
  3. Arithmetic cutoff bounds and sharp tail capacity for local Nyman–Beurling certificatesScott SchubrEdition 4 · 2026-10-27 Public
  4. An explicit hyperbolicity threshold for xi Jensen polynomialsScott SchubrEdition 2 · 2026-10-27 Public
  5. Collision-uniform kernel asymptotics for Báez–Duarte zero packetsScott SchubrEdition 3 · 2026-10-27 Public
  6. Sharp logarithmic sampling thresholds and lacunary interpolation in the Báez–Duarte step spaceScott SchubrEdition 2 · 2026-10-27 Public
  7. Polynomial cores and inertia under Carleman moment boundsScott SchubrEdition 2 · 2026-10-27 Public
  8. Finite-projection holomorphic stabilization of contractible affine hypersurfacesScott SchubrEdition 3 · 2026-10-27 Public
  9. Compact capture and protected bridges on the Koras–Russell cylinderScott SchubrEdition 2 · 2026-10-27 Public
  10. Dimension-dependent holomorphic certificate degrees on zero-sum hyperplanesScott SchubrEdition 2 · 2026-10-27 Public
  11. Admissible kernel lifts with positive endpoint forms and nonreal Fourier zerosScott SchubrEdition 2 · 2026-10-27 Public
  12. Global finite-depth stability and nonreal zeros of admissible Fourier transformsScott SchubrEdition 2 · 2026-10-27 Public
  13. Weighted polynomial localization and theta correction inertiaScott SchubrEdition 2 · 2026-10-27 Public
  14. Critical-tail localization and growing theta correction inertiaScott SchubrEdition 2 · 2026-10-27 Public
  15. Typical-value anchors for polynomial approximation and theta correction inertiaScott SchubrEdition 2 · 2026-10-27 Public
  16. All-depth root-Laguerre positivity with nonreal Fourier zerosScott SchubrEdition 3 · 2026-10-27 Public
  17. Polynomial synthesis and zero ideals for the Riemann xi-functionScott SchubrEdition 2 · 2026-10-27 Public
  18. A counterexample and corrected translation-density criterion for correlation kernelsScott SchubrEdition 2 · 2026-10-27 Public
  19. Finite theta interpolation and eventual Laguerre failureScott SchubrEdition 2 · 2026-10-27 Public
PublicEdition 2

Finite theta interpolation and eventual Laguerre failure

Scott Schubr

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Abstract

We construct positive finite reweightings of the actual integer-square theta atoms which reproduce any prescribed finite set of modular endpoint jets, converge to the complete theta density in every fixed source derivative norm, and nevertheless eventually violate the first Laguerre inequality. The first missed odd jet has an explicit strictly positive Gaussian-tail formula. More generally, every nonzero finite signed compact-scale mixture of these atoms has a nonzero odd endpoint jet, finitely many real transform zeros and infinitely many nonreal zeros. For the exact split into a finite surrogate and its complete arithmetic remainder, we determine the leading cancellation of both diagonal Laguerre densities and their cross term, uniformly on bounded horizontal strips including the real axis. Each diagonal has only finitely many absolute moments, although their complete combination decreases rapidly. Explicit source-seminorm bounds give a sufficient negativity onset. The positive interpolation mechanism is classical; the conclusions concern its explicit realization and global-sign obstruction in the actual theta atom family.

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