Research Article · Preprint Mathematics
Phase transitions and effective curvature bounds in Fourier density models
- Primary subject
- Mathematics
- Subject subfields
- Analysis and Spectral Theory
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 2 · Permanent link
Overview
Abstract
We study the intrinsic Wasserstein metric on affine Fourier-density models on the unit circle. For three distinct frequencies, with one arbitrary fixed phase per frequency, every sectional curvature at the uniform density is strictly negative unless the frequencies have ratio 1:2:3. In that exceptional family we determine the exact attained curvature range and a phase-dependent sign criterion: the same frequency magnitudes can have mixed curvature or strictly negative curvature solely through relative phase. For the aligned four-frequency family {m,2m,n,n+m}, n>5m, we determine the exact attained maximum sectional curvature at every frequency ratio. Two sharp algebraic thresholds separate mixed signs from strict negativity and then mark saturation of the maximum at -18π^2m^2. Finally, exact rational Fourier integration with a bounded infinite reciprocal-series remainder certifies nonuniform closed parameter cubes: the {m,2m,3m} model has both signs throughout [-.01,.01]^3, whereas every sectional curvature of the {m,2m,4m} model is less than -.98π^2m^2 there. The phase and four-frequency classifications are analytic; the explicit neighborhood bounds are computer-assisted and accompanied by reproducible exact-arithmetic certificates.
Keywords Wasserstein geometry · sectional curvature · Fourier density models · phase transitions · exact curvature bounds · interval certificates
Scope of the work
Complete fixed-phase three-frequency curvature classification, an attained exact maximum throughout the stated resonant four-frequency family, and exact certified nonuniform cubes for the two benchmark Fourier density models.
Reproduction notes
The paper has 13 pages. The editable-source archive contains 15 files. Compile manuscript/main.tex. The standard-library reproducibility/check_results.py reconstructs coordinate checks, maximum enclosures and full density-box certificates; the supplied source scripts regenerate each component.
The manuscript
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Author disclosures
AI tools assisted research, derivation, proof checking and manuscript preparation. This is an author-published preprint; internal assistant reviews are not external human peer review, and worldwide novelty remains provisional. This substantial follow-up credits and preserves Frequency interactions and curvature in Fourier density models, edition 1.
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Cite this edition
Scott Schubr. (2026). Phase transitions and effective curvature bounds in Fourier density models (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/phase-transitions-effective-curvature-bounds-fourier-density-models/v/2
BibTeX
@techreport{luminary_phase_transitions_effective_curvature_bounds_fourier_density_models_v2,
title = {Phase transitions and effective curvature bounds in Fourier density models},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 2},
url = {https://research.luminaryoddities.com/phase-transitions-effective-curvature-bounds-fourier-density-models/v/2}
}Loading discussion…