Research Article · Preprint Mathematics
All-depth root-Laguerre positivity with nonreal Fourier zeros
- Primary subject
- Mathematics
- Subject subfields
- Complex Analysis
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 1 · Permanent link
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Overview
Abstract
We give an explicit admissible Fourier kernel for which every continuous root-coordinate Laguerre iterate is positive and uniformly strictly log-concave, although its Fourier transform has nonreal zeros. The fixed kernel is φ(s) = exp(−exp(s²) − 999s²). Its Fourier transform is entire of exact order one, and a whole-line rational Gaussian-moment enclosure proves that the normalized sixth cumulant is less than −10⁻¹⁸. The real-zero canonical product would require that cumulant to be nonnegative. Exact quadratic-tilt invariance explains why the entire root hierarchy misses this obstruction. We also give a general Gaussian-localization formula for higher cumulants, including an effective finite-jet error bound. Newman’s Gaussian-damping classification already implies existence of a bad tilt for the underlying double-exponential family; the present conclusion is an explicit certified all-depth counterexample. It concerns general kernels and gives no conclusion about the Riemann theta kernel.
Keywords Fourier kernels · Laguerre iteration · nonreal zeros · Gaussian damping · cumulants · log-concavity
Scope of the work
One explicit fixed fully admissible kernel has positive uniformly log-concave continuous root iterates at every depth and an order-one Fourier transform with nonreal zeros, certified by exact whole-line moment bounds. No actual-theta or RH conclusion, worldwide-first claim, confinement of all zeros to a strip, or inherited initial physical-interval positivity. Newman’s existence framework and the earlier all-depth recurrence are credited.
Reproduction notes
The nine-member package contains the eight-page PDF, four complete editable TeX sources, a Python standard-library exact rational verifier, build instructions, citation metadata and a SHA-256 manifest. An isolated rebuild reproduces all eight page texts and geometry. Normal and optimized checks agree; a separate independent fourth-order enclosure confirms the sign. The manuscript supplies the global integral bounds and all-depth induction.
The manuscript
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Author disclosures
Research and manuscript preparation used AI assistance. This author preprint has not undergone external human refereeing. Classical Gaussian, canonical-product and Newman classification ingredients are credited; worldwide priority of the explicit conjunction is unconfirmed.
A stable scholarly record
Cite this edition
Scott Schubr. (2026). All-depth root-Laguerre positivity with nonreal Fourier zeros (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/all-depth-root-laguerre-positivity-nonreal-fourier-zeros/v/1
BibTeX
@techreport{luminary_all_depth_root_laguerre_positivity_nonreal_fourier_zeros_v1,
title = {All-depth root-Laguerre positivity with nonreal Fourier zeros},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 1},
url = {https://research.luminaryoddities.com/all-depth-root-laguerre-positivity-nonreal-fourier-zeros/v/1}
}Loading discussion…