Curvature classification and certified Sinkhorn metrics for Fourier density models
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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.
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- Frequency_Interactions_Curvature_Fourier_Density_Models.pdf126.4 KiB · pdf
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Research details
- Subjects
- Mathematics · Geometry and Optimal Transport
- Keywords
- Wasserstein geometry, sectional curvature, Fourier density models, frequency interactions, Sinkhorn metrics, interval certificates
- Manuscript date
- 2026-10-21
- Reader reuse
- No additional permission granted
Scope
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.
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.
Reproduction
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.