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
- 3 · Permanent link
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. Every root iterate also defines a strictly totally positive translation kernel of all orders, with explicit signed confluent determinants. The fixed kernel is φ(s)=exp(-e^(s^2)-999s^2). Its Fourier transform is entire of exact order one. Two whole-line rational Gaussian-moment enclosures certify the negative sixth cumulant, the sharper bound being κ_6<-1.8 × 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 admissible kernels.
Keywords Fourier kernels · Laguerre iteration · nonreal zeros · Gaussian damping · cumulants · log-concavity
Scope of the work
An explicit admissible kernel with nonreal Fourier zeros despite positive strictly log-concave root-Laguerre iterates at every depth and strictly totally positive translation kernels, certified by a negative sixth cumulant.
Reproduction notes
10-page manuscript and 12 source-package members. Compile main.tex with a compatible LaTeX engine. Standard-library Python runs verify_cumulant.py and verify_second_cumulant.py, also under -O, for two whole-line rational Gaussian-moment enclosures of the sixth cumulant.
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, Newman classification and Gumbel total-positivity ingredients are credited. Worldwide priority of the explicit conjunction is unconfirmed. The second certificate strengthens the public bound and proof reproducibility; it is not a second independent nonreal-zero theorem.
A stable scholarly record
Cite this edition
Scott Schubr. (2026). All-depth root-Laguerre positivity with nonreal Fourier zeros (Version 3). Luminary Research Archive. https://research.luminaryoddities.com/all-depth-root-laguerre-positivity-nonreal-fourier-zeros/v/3
BibTeX
@techreport{luminary_all_depth_root_laguerre_positivity_nonreal_fourier_zeros_v3,
title = {All-depth root-Laguerre positivity with nonreal Fourier zeros},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 3},
url = {https://research.luminaryoddities.com/all-depth-root-laguerre-positivity-nonreal-fourier-zeros/v/3}
}Loading discussion…