Random distances on disphenoids and moments on Platonic surfaces

We determine the intrinsic distance distribution of two independent area-uniform points on every nondegenerate Euclidean disphenoid with congruent acute triangular faces. The survival function is the square of an explicit radial-area formula for a cyclic lattice hexagon, whose side distances are the face side lengths. The regular tetrahedron is a specialization, with an explicit mean and a finite-sum formula for every even moment. A second theorem gives a fixed-alphabet calculation principle for valid finite minima of algebraic unfolding quadratics. Polynomial target integration, rational parametrizations of the source cells, and a fixed source polar divisor that splits linearly in the first integration variable lead to logarithms and dilogarithms at algebraic arguments independent of the moment order. The octahedral moment-generated space is exactly the rational span of 1, log 2, log 3, and Cl2(π/3)/√3. Three further face-pair sequences each generate a space spanned by 1 and a single logarithm. Exact mean-distance reductions reveal elliptic boundary curves and motivate questions about minimal alphabets and cancellation of their periods. 2020 Mathematics Subject Classification: Primary 60D05; Secondary 52A22, 33E05, 14F40. The deposit contains the paper (PDF) and a supplementary ZIP with the exact certificates, replay scripts, and numerical tables cited in the appendices.