Phase transitions and effective curvature bounds in Fourier density models
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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.
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- Phase_Transitions_Effective_Curvature_Bounds_Fourier_Density_Models.pdf124.9 KiB · pdf
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81e7ff17a497357f54967e9e23a895dd8d6fa9d8477e67638aa9315a990ee694 - Revised_publication_package.zip157.3 KiB · package
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197a10a6f96cab251efeefc2ff16358f4ae51d4bebdca973646ee9af4102884b
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Research details
- Subjects
- Mathematics · Analysis and Spectral Theory
- Keywords
- Wasserstein geometry, sectional curvature, Fourier density models, phase transitions, exact curvature bounds, interval certificates
- Manuscript date
- 2026-10-22
- Reader reuse
- No additional permission granted
Scope
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.
Disclosures
ChatGPT (OpenAI, Sol 6.1) assisted with literature research, mathematical exploration, proof development and checking, and manuscript preparation through workflows developed in Scott Schubr’s AI governance research. Scott Schubr directed the research and manuscript development and retains responsibility for the content, accuracy, originality and attribution. The research was self-funded. The author owns Luminary Oddities, which operates the Research Archive, and declares no additional relevant competing interests. This is an author preprint.
Reproduction
The complete 13-page manuscript is accompanied by a 16-member editable source and reproducibility package. Compile manuscript/main.tex with Tectonic, or run build.py. Standard-library Python reproducibility/check_results.py reconstructs the coordinate checks, exact maximum enclosures, and full density-box certificates. All verification acceptance gates remain active under Python optimization. The package includes the matching manuscript PDF and SHA256SUMS.txt.