Research Article · Preprint Mathematics
Global finite-depth stability and nonreal zeros of admissible Fourier transforms
- Primary subject
- Mathematics
- Subject subfields
- Complex Analysis
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 1 · Permanent link
Overview
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; it gives no conclusion about the all-order positivity of the actual Riemann theta kernel.
Keywords admissible Fourier kernels · finite-jet stability · differential Laguerre iteration · log-concavity · generalized Laguerre inequalities · nonreal zeros
Scope of the work
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. No fixed infinite-depth example, actual-theta counterexample, RH resolution, optimal scale or worldwide-priority claim.
Reproduction notes
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.
The manuscript
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Author 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.
A stable scholarly record
Cite this edition
Scott Schubr. (2026). Global finite-depth stability and nonreal zeros of admissible Fourier transforms (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/global-finite-depth-stability-nonreal-zeros-admissible-fourier-transforms/v/1
BibTeX
@techreport{luminary_global_finite_depth_stability_nonreal_zeros_admissible_fourier_transforms_v1,
title = {Global finite-depth stability and nonreal zeros of admissible Fourier transforms},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 1},
url = {https://research.luminaryoddities.com/global-finite-depth-stability-nonreal-zeros-admissible-fourier-transforms/v/1}
}Loading discussion…