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(−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. The construction concerns general admissible kernels.
Read the full paper in the original PDF.
Edition files
- Paper.pdf118.2 KiB · pdf
SHA-256
f2cc818eb19e6daddc61ce696dc46ef866e15cde4d34bb5c3d60c1d9aa36e937 - Publication_Package.zip131.9 KiB · package
SHA-256
f8dd69f01dc1b5b627af617a1c690eee5775bf01de12d45ef0b0eaaaf294de53
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
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.
Disclosures
Research and manuscript preparation used AI assistance. Classical Gaussian, canonical-product, Newman classification and Gumbel total-positivity ingredients are credited. The second certificate strengthens the public bound and proof reproducibility.
Reproduction
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.