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  1. Divisor tests and coefficient transfers for sextic character momentsScott SchubrEdition 1 · 2026-10-28 Public
  2. The Riemann Hypothesis through Arithmetic Weil PositivityScott SchubrEdition 4 · 2026-10-28 Public
  3. Signed Frequency Certificates and Constraint Costs for Odd Weil FormsScott SchubrEdition 1 · 2026-10-27 Public
  4. Uniform Resolvent Asymptotics for Growing Theta-Null Source FamiliesScott SchubrEdition 1 · 2026-10-27 Public
  5. Frequency-Source Resolvent Limits and Positive-Receiver InstabilityScott SchubrEdition 1 · 2026-10-27 Public
  6. Arithmetic cutoff bounds and sharp tail capacity for local Nyman–Beurling certificatesScott SchubrEdition 4 · 2026-10-27 Public
  7. Degree-eight positivity and polynomial obstructions for monotone Riemann theta averagesScott SchubrEdition 7 · 2026-10-27 Public
  8. Observation norms under double-exponential confinementScott SchubrEdition 2 · 2026-10-27 Public
  9. An explicit hyperbolicity threshold for xi Jensen polynomialsScott SchubrEdition 2 · 2026-10-27 Public
  10. Collision-uniform kernel asymptotics for Báez–Duarte zero packetsScott SchubrEdition 3 · 2026-10-27 Public
  11. Relative finite optimization of arithmetic approximation distancesScott SchubrEdition 2 · 2026-10-27 Public
  12. Growing null subtraction and weighted Weil zero detectionScott SchubrEdition 2 · 2026-10-27 Public
  13. Sharp logarithmic sampling thresholds and lacunary interpolation in the Báez–Duarte step spaceScott SchubrEdition 2 · 2026-10-27 Public
  14. Certified source elimination for localized odd Weil formsScott SchubrEdition 2 · 2026-10-27 Public
  15. Weighted polynomial localization and theta correction inertiaScott SchubrEdition 2 · 2026-10-27 Public
  16. Finite certificates for the lower edge of a weighted Weil operatorScott SchubrEdition 2 · 2026-10-27 Public
  17. Finite theta interpolation and eventual Laguerre failureScott SchubrEdition 2 · 2026-10-27 Public
  18. Sharp regularity criteria for arithmetic cutoff functionalsScott SchubrEdition 1 · 2026-10-26 Public
  19. Optimal energy for Dirichlet repairs in divisor bandsScott SchubrEdition 1 · 2026-10-26 Public
  20. Certified negative-band approximation for odd Weil formsScott SchubrEdition 1 · 2026-10-26 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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