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
- 1 · 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 −18π²m². 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 −.98π²m² 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
The classifications concern intrinsic sectional curvature of fixed-phase affine Fourier-density models at uniform density. The four-frequency maximum concerns the aligned family {m, 2m, n, n+m}, n > 5m. Nonuniform certificates cover the complete closed cube [−1/100, 1/100]³ for aligned spectra {m, 2m, 3m} and {m, 2m, 4m}, including all positive integer dilations. No arbitrary four-mode phase or spectrum classification, ambient circle flatness, finite-bandwidth Sinkhorn curvature theorem, improved estimator, optimization rate, industrial performance or global convexity claim is made. Established Gauss, four-form and interval methods are credited.
Reproduction notes
The exact 15-member package includes the 13-page PDF, six LaTeX sources, README, standard-library Python reproduction programs, complete-box certificate data, dependency instructions and SHA256SUMS.txt covering the other 14 members. Run python -B reproducibility/check_results.py --output reproduced_results.json from the extracted package root. Finite checks supplement the general analytic proofs. The README explains source compilation, exact maximum-root isolation and regeneration of the full nonuniform box certificates.
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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 1). Luminary Research Archive. https://research.luminaryoddities.com/phase-transitions-effective-curvature-bounds-fourier-density-models/v/1
BibTeX
@techreport{luminary_phase_transitions_effective_curvature_bounds_fourier_density_models_v1,
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 1},
url = {https://research.luminaryoddities.com/phase-transitions-effective-curvature-bounds-fourier-density-models/v/1}
}Loading discussion…