Frequency interactions and curvature in 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}, whose sharp attained range is π²m²[−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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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
The curvature classification concerns the intrinsic pullback Wasserstein/Otto metric of aligned-cosine affine density models at uniform density. The complete classification is for three distinct positive frequencies, with separate sufficient spacing results in arbitrary finite dimension and explicit four-mode counterexamples. The sharp 1+√2 constant concerns universal Fourier curvature-operator diagonality. The corrected Sinkhorn comparisons certify the stated finite bandwidths and fixed two-parameter box.
Disclosures
AI tools assisted research, derivation, proof checking and manuscript preparation. This is an author-published preprint. 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 exact 28-member package contains the 13-page paper, four complete LaTeX sources, reader instructions, exact Fourier-geometry checks, interval-certificate programs and mathematical data, dependency instructions, and SHA-256 checksums. The README distinguishes the analytic proofs of infinite families from supplementary finite enumeration and explains source compilation and certificate reproduction.