Global finite-depth stability and nonreal zeros of admissible Fourier transforms
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Abstract
We prove a global finite-jet stability theorem for the differential iteration T f=(f')^2-ff''. A uniformly positive reference curvature and bounded curvature-normalized higher derivatives give explicit amplitude budgets for positive, uniformly log-concave iterates through any prescribed finite depth. Intensified hyperbolic-cosh mixtures realize the required jets on the whole half-line, with all spatial tails controlled. Combining this realization with a credited finite-atomic Fourier multiplier gives a simultaneous separation theorem. For every finite generalized-Laguerre cutoff, finite root-coordinate iteration depth and 0<Y<H, one fully admissible kernel has an order-one Fourier transform satisfying all retained global Laguerre inequalities and all polynomial moment orders throughout 0≤ y≤ Y, yet has nonreal zeros inside |Im z|<H and negative physical polynomial squares at a later parameter. Parameters depend on the prescribed finite depths. The construction concerns general admissible kernels.
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Research details
- Subjects
- Mathematics · Complex Analysis
- Keywords
- admissible Fourier kernels, finite-jet stability, differential Laguerre iteration, log-concavity, generalized Laguerre inequalities, nonreal zeros
- Manuscript date
- 2026-10-25
- Reader reuse
- No additional permission granted
Scope
Explicit global finite-depth differential-iterate stability and admissible Fourier-kernel separations retaining prescribed finite Laguerre/root-depth tests while having nonreal zeros and later negative squares.
Disclosures
Research and manuscript preparation used AI assistance. This author preprint has not undergone external human refereeing. Earlier curvature-iteration methods, the independent six-atom seed and the author’s published lifting baseline are explicitly credited.
Reproduction
14-page manuscript and 11 source-package members. Compile main.tex with its included TeX files. Standard-library python verify_hierarchy_constants.py, also under -O, checks the explicit finite stability-budget and hierarchy constants; the global existence and separation arguments are analytic.