All-depth root-Laguerre positivity with nonreal Fourier zeros
Loading original first page…
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.
Read the full paper in the original PDF.
Edition files
- main.pdf118.5 KiB · pdf
SHA-256
011d93f5884222ecf61c42e904315402ca15f37387bc4c04bedcd3b5cceefcb6 - Publication_Package.zip132.5 KiB · package
SHA-256
110ddf93234e48720d594b1e8f6a49d5388137e7dd95f810ee7e3b80cf38a9af
Available editions
Research details
- Subjects
- Mathematics · Complex Analysis
- Keywords
- Fourier kernels, Laguerre iteration, nonreal zeros, Gaussian damping, cumulants, log-concavity
- Manuscript date
- 2026-10-25
- Reader reuse
- No additional permission granted
Scope
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.
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.
Reproduction
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.