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 Tf = (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
Global curvature-normalized finite-depth stability, intensity-aware admissible cosh realization, and simultaneous finite-hierarchy separation for arbitrary prescribed finite cutoffs and an initial continuum of physical parameters. A different kernel may be needed at different depths.
Disclosures
Research and manuscript preparation used AI assistance. Earlier curvature-iteration methods, the independent six-atom seed and the author’s published lifting baseline are explicitly credited.
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The complete 11-member package contains the 14-page PDF, six editable TeX files, a Python standard-library exact checker, build instructions, citation metadata and a SHA-256 manifest. The isolated source rebuild reproduces every page’s extracted text and geometry; normal and optimized checks agree. Finite arithmetic controls supplement the analytic global proofs.