Pfaffian lower bounds for convex distance kernels
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For even n≥2 and ordered real nodes x₁<⋯<xₙ, form the skew matrix with Aᵢⱼ=f(xⱼ−xᵢ) for i<j, where f is positive, nonincreasing and convex on (0,∞). Is its Pfaffian bounded below by the nested-matching product ∏ₖ₌₁^(n/2)f(xₙ₊₁₋ₖ−xₖ)?
Why it matters and what counts as progress
Why it matters: Alternating signs do not make Pfaffian positivity automatic; a universal matching-product bound supplies a structural kernel inequality.
What counts as progress: Prove positivity and the proposed lower bound or classify which single-matching products remain universal over all admissible kernels and nodes.
How the problem developed
- October 2026 · Theorem [1]
William Johnson Released a lower bound for Pfaffians built from positive nonincreasing convex kernels on ordered real points, and showed the nested rainbow matching is the unique universal single-matching product lower bound.
- October 2026 · Formalization [2]
William Johnson Released a Lean artifact documenting theorem correspondence for the convex-kernel Pfaffian results.
AI contribution
Pfaffian Positivity for Convex Kernels and Optimal Matching Bounds
Let n≥2 be even, x₁<⋯<xₙ be real, and f:(0,∞)→(0,∞) be positive, nonincreasing and convex. For the skew matrix A with Aᵢⱼ=f(xⱼ−xᵢ) above the diagonal, Pf(A) is at least the positive product ∏ₖ₌₁^(n/2) f(xₙ₊₁₋ₖ−xₖ). This rainbow matching is the only single-matching product that is a universal lower bound over all such f and x.
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Pfaffian Positivity for Convex Kernels and Optimal Matching Bounds