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Research Article · Preprint Mathematics

Frequency interactions and curvature in Fourier density models

Scott Schubr

  • Mathematics
  • Geometry and Optimal Transport
Primary subject
Mathematics
Subject subfields
Geometry and Optimal Transport
Paper type
Research Article
Manuscript stage
Preprint
Manuscript
Archived
Edition
1 · Permanent link
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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}, 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; the finite-band certificates do not assert a new all-bandwidth convergence rate or a complexity bound.

Keywords Wasserstein geometry · sectional curvature · Fourier density models · frequency interactions · Sinkhorn metrics · interval certificates

Scope of the work

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, not optimal negative-curvature spacing. No arbitrary-phase, general nonuniform-density or arbitrary-spectrum classification is asserted. The corrected Sinkhorn comparisons certify the stated finite bandwidths and fixed two-parameter box; they do not establish a new all-bandwidth convergence rate, complexity bound or industry/runtime claim.

Reproduction notes

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.

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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

  1. Scott Schubr. Sharp convergence of heat-kernel Sinkhorn metrics, edition 1.
  2. Scott Schubr. Leading bias and positive correction of heat-kernel Sinkhorn metrics, edition 1.

A stable scholarly record

Cite this edition

Scott Schubr. (2026). Frequency interactions and curvature in Fourier density models (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/frequency-interactions-curvature-fourier-density-models/v/1

BibTeX
@techreport{luminary_frequency_interactions_curvature_fourier_density_models_v1,
  title = {Frequency interactions and curvature in Fourier density models},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 1},
  url = {https://research.luminaryoddities.com/frequency-interactions-curvature-fourier-density-models/v/1}
}

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