Research Article · Preprint Mathematics
Finite theta interpolation and eventual Laguerre failure
- Primary subject
- Mathematics
- Subject subfields
- Complex Analysis · Number Theory
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 2 · Permanent link
Overview
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.
Keywords theta function · Laguerre inequality · positive interpolation · Fourier cosine transform · arithmetic approximation · nonreal zeros
Scope of the work
Positive finite integer-square theta reweightings match prescribed finite modular jets and eventually violate the first Laguerre inequality; signed compact-scale mixtures have infinitely many nonreal transform zeros.
Reproduction notes
The 9-page manuscript and editable LaTeX sources are supplied in this 8-member package. README.md gives build instructions. verify.py provide the finite mathematical controls specified in the manuscript alongside its analytic proofs.
The manuscript
Read the paper
The complete manuscript, as published in this edition.
If the reader is unavailable on your device, open or download the PDF directly.
Author disclosures
AI assistants supported derivation, literature comparison, exposition and independent checks. The author is responsible for the mathematical statements and presentation.
A stable scholarly record
Cite this edition
Scott Schubr. (2026). Finite theta interpolation and eventual Laguerre failure (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/finite-theta-data-global-laguerre-inequalities/v/2
BibTeX
@techreport{luminary_finite_theta_data_global_laguerre_inequalities_v2,
title = {Finite theta interpolation and eventual Laguerre failure},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 2},
url = {https://research.luminaryoddities.com/finite-theta-data-global-laguerre-inequalities/v/2}
}Loading discussion…