Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Geometry and Optimal Transport
- Paper type
- Research Article
- Manuscript stage
- Preprint
We study the quartics hλ = 4e₁e₃ + λa²b², where the elementary symmetric polynomials are taken in four variables. We prove that their real-stability interval is exactly [0, 64/27]. Throughout its positive part, the parameter is a genuine complex projective modulus, and the full linear automorphism group of the positive hyperbolicity cone consists of positive scalings and the independent interchanges of the two distinguished variable pairs. These cones are consequently pairwise linearly nonisomorphic and nonhomogeneous, including at the endpoint. The projective surfaces have eight singularities of type A₁ and one of type A₃ in the interior; one additional A₁ point appears at the endpoint. Correspondingly, the dual cone has exactly four Lorentz facets in the interior and exactly five at the endpoint. The arguments give an exact classification for this coefficient deformation of the specialized Vámos quartic, using classical hyperbolicity, singularity, and tangent-cone methods.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Geometry and Optimal Transport
- Paper type
- Research Article
- Manuscript stage
- Preprint
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.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Geometry and Optimal Transport
- Paper type
- Research Article
- Manuscript stage
- Preprint
We identify the leading bias of the exact population heat-kernel Sinkhorn tensor on a flat torus and construct a compact, strictly positive correction with fourth-order smooth-input accuracy. The leading quadratic form is minus one forty-eighth of the Wasserstein Hessian of Fisher information, and an explicit nonconstant density exhibits both bias signs. The correction preserves the completed tangent form space and square-root range, with equivalent norms; full L² accuracy remains first order. An interval-certified nonconstant two-parameter example encloses continuum residual, differential-form and quadrature errors. For prescribed analytic density families sufficiently close to uniform, we construct fourth-order finite metric approximations, including normalization and every fixed parameter-derivative order. In each fixed dimension, per-coordinate Fourier resolution and iteration counts are logarithmic in inverse bandwidth; total modes and quadrature nodes grow as the dimension-th power of that logarithm. This is a resolution bound, not a running-time or optimality claim.
Edition 1
Scott Schubr
- Primary subject
- Mathematics
- Subject subfields
- Geometry and Optimal Transport
- Paper type
- Research Article
- Manuscript stage
- Preprint
We study the Sinkhorn metric associated with the periodic heat kernel at a smooth strictly positive probability density. We prove convergence to the inverse weighted Laplacian in full L² operator norm at the optimal rate O(h). The sharper rate from Hʳ to L² is O(h¹⁺ʳᐟ²), 0 ≤ r ≤ 2; every exponent is optimal already at constant density. Parameter-differentiated estimates yield second-order convergence of smooth finite-dimensional pullback metrics and their intrinsic Christoffel symbols and curvature under nondegeneracy. A population normalization residual ρ gives projected tensor error O(h + ρ/h), with an obstruction at ρ comparable to h. We also give an explicit projected scale approximation whose tensor differs from the exact tensor by O(h²). The cost depends on the heat scale; fixed Euclidean squared-cost problems with boundaries are outside the scope.