Exact height-one probabilities of three-dimensional Abelian sandpiles, spanning-tree entropies of the snub-square and Cairo lattices, and elementary resistor-network reductions

Exact closed forms for the height-one probability P₁ of the Abelian sandpile, spanning-tree entropies and two-point resistances on a range of lattices, using the fact that all three are controlled by the same Kirchhoff transfer currents. Main results:• Three dimensions: P₁ on the simple-cubic, body-centred cubic, face-centred cubic and pyrochlore lattices, as polynomials in the corresponding Watson integral W and 1/(π²W). For example, P₁ ≈ 0.0545825 on the simple cubic lattice.• For bipartite lattices of girth at least six with symmetric second neighbours, P₁ = (z−2)^(z−1) / [z(z−1)^(z−1)], giving 2/27 for diamond.• Exact P₁ for ten two-dimensional lattices (king, 4.8.8, Union Jack, star, dice, Lieb, snub square, checkerboard, triakis, and honeycomb with next-nearest neighbours).• Spanning-tree entropy of the snub-square (Shastry–Sutherland) lattice, (10/3π)G + (1/3)ln(2+√3), and of its dual, the Cairo pentagonal lattice. A Clausen-function formula covers any dimer conductance.• Exact resistor-network formulas, including one elementary formula for the axis-bond resistance of any square lattice with diagonal bonds.• Appendix: a Hurwitz-zeta formula for the added mass of a ball touching the inside of a spherical tank. Closed forms were identified by integer-relation (PSLQ) searches at 25–60 digits and confirmed by exact finite-volume computations, independent numerical integration and Monte Carlo simulation. AI-use disclosure: this research, including choosing the problems, discovering the formulas, running all computations, and writing the manuscript, was carried out with extensive use of an AI system (Claude Code, by Anthropic) under the author's direction. It has not been peer reviewed. The author takes full responsibility for the content. Code: https://github.com/JacobGoodchild/findformula