All-depth root-Laguerre positivity with nonreal Fourier zeros
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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. The construction concerns general admissible kernels.
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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 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. Newman’s existence framework and the earlier all-depth recurrence are credited.
Disclosures
Research and manuscript preparation used AI assistance. Classical Gaussian, canonical-product and Newman classification ingredients are credited.
Reproduction
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.