Research Article · Preprint Mathematics
Curvature classification and certified Sinkhorn metrics for Fourier density models
- Primary subject
- Mathematics
- Subject subfields
- Geometry and Optimal Transport
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 2 · Permanent link
Overview
Abstract
We classify the intrinsic sectional-curvature sign at the uniform density for every three-frequency aligned-cosine affine density model on the unit circle. All sectional curvatures are strictly negative except for the spectrum {m,2m,3m}, which has the sharp attained range π^2m^2[-25-√(1649),-25+√(1649)]. A finite Fourier Gauss formula identifies the normal frequency interactions responsible for this exception. In arbitrary finite dimension, successive frequency ratios exceeding 1+√(2) force a diagonal Fourier curvature operator and give an explicit attained negative curvature range; the constant is optimal for this universal diagonality assertion. Two four-frequency examples show that the triple criterion does not extend by inspecting subtriples and that a positive curvature-operator direction need not represent positive sectional curvature. As a computational application, we certify corrected heat-kernel Sinkhorn metrics and their parameter derivatives through order two on a fixed two-parameter box at five bandwidths. The smallest bandwidth has a uniform relative metric-error improvement greater than 870.37. We prove bandwidth-uniform magnitude bounds for the cancellation-preserving interval iteration and provide exact and interval reproducibility code. The curvature theorems concern intrinsic finite-dimensional model geometry.
Keywords Wasserstein geometry · sectional curvature · Fourier density models · frequency interactions · Sinkhorn metrics · interval certificates
Scope of the work
Complete aligned three-frequency intrinsic curvature-sign classification, sharp higher-dimensional spacing criteria and counterexamples, with certified corrected Sinkhorn metric values and derivatives on the stated two-parameter box and five bandwidths.
Reproduction notes
The paper has 13 pages. The editable-source archive contains 28 files. Compile main.tex with its included geometry, application and bibliography files. Standard-library geometry/certificate postprocessors check the supplied intervals; full interval producers use the pinned mpmath dependency.
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 new companion retains and credits the published Sharp convergence of heat-kernel Sinkhorn metrics and Leading bias and positive correction of heat-kernel Sinkhorn metrics.
References and related publications
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Cite this edition
Scott Schubr. (2026). Curvature classification and certified Sinkhorn metrics for Fourier density models (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/frequency-interactions-curvature-fourier-density-models/v/2
BibTeX
@techreport{luminary_frequency_interactions_curvature_fourier_density_models_v2,
title = {Curvature classification and certified Sinkhorn metrics for Fourier density models},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 2},
url = {https://research.luminaryoddities.com/frequency-interactions-curvature-fourier-density-models/v/2}
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