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
- 2 · 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(−exp(s²) − 999s²). 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 κ₆ < −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 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 fixed fully admissible kernel has positive uniformly log-concave continuous root iterates at every depth, strict total positivity of all ordered root-translation minors with exact signed confluent determinants, and nonreal zeros of its order-one spatial Fourier transform. Two whole-line exact certificates support the same zero theorem; the second sharpens κ₆ < −10⁻¹⁸ to κ₆ < −1.8 × 10⁻¹⁸. Nonreal zeros persist for every damping M ≥ 999 by a classical preservation theorem. Root translation positivity is distinct from original spatial positive definiteness; its negative-tilt real extension is not an integrable PF density. No actual-theta/RH closure, worldwide-first claim, all-zero strip or initial physical-interval positivity is asserted.
Reproduction notes
The twelve-member package contains the ten-page PDF, six complete editable TeX sources, two independent Python standard-library exact rational verifiers, build instructions, citation metadata and an eleven-entry SHA-256 manifest. An isolated reader rebuild reproduces every page text and geometry. Both verifiers pass normally and optimized; a fresh reviewer reconstruction independently reproduces the second bound. The manuscript proves both whole-line remainders, the exact root hierarchy, all-order translation minors/confluent determinants, the Gaussian log-jet diagnostic and the sufficient M ≥ 999 damping range. The complete second certificate is now included publicly.
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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 2). Luminary Research Archive. https://research.luminaryoddities.com/all-depth-root-laguerre-positivity-nonreal-fourier-zeros/v/2
BibTeX
@techreport{luminary_all_depth_root_laguerre_positivity_nonreal_fourier_zeros_v2,
title = {All-depth root-Laguerre positivity with nonreal Fourier zeros},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 2},
url = {https://research.luminaryoddities.com/all-depth-root-laguerre-positivity-nonreal-fourier-zeros/v/2}
}Loading discussion…