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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.