Cross-Boundary Moment Kernels: One-Crossing Geometry, Total Positivity, Size Bias, and Curvature

This preprint develops a hierarchy of cross-boundary moment determinants associated with nonnegative full-support weights on (0,∞). It establishes a universal one-crossing geometry, strict total positivity of order two, truncated Gram decompositions, multiplicative size-bias relations, an inter-order equivalence principle, and a curvature-pairing identity. An explicit Gamma family provides sharp crossing thresholds and counterexamples showing that log-concavity or positive moment-ratio curvature alone do not determine crossing order. The main mathematical results are independently formalized in Lean 4/mathlib; the accompanying software release is archived at DOI 10.5281/zenodo.22046413.