Optimal lattices for a three-body power-law energy: steep-decay limits, computer-assisted proofs and the minima of the modular graph functions Cₛ,ₛ,ₛ

We study which Bravais lattice of fixed density minimises the three-body energy $T_\nu(\Lambda)=\sum'_{x,y\in\Lambda}(|x|\,|y|\,|x-y|)^{-\nu}$, the sum over all lattice triangles through the origin of the product of their side lengths to the power $-\nu$. In two dimensions $T_{2s}$ coincides, up to a factor, with the two-loop modular graph function $C_{s,s,s}(\tau)$. We prove that in the steep-decay limit $\nu\to\infty$ the minimisers converge to the rectangular lattice with aspect ratio $\sqrt{(\sqrt{17}-1)/2}$ in $d=2$, by solving exactly the associated max–min problem for the smallest triangle product, and to the body-centred cubic lattice in $d=3$, by a computer-assisted argument. For finite exponents we give computer-assisted proofs, carried out in rigorous interval arithmetic, that $C_{2,2,2}$, $C_{3,3,3}$ and $C_{4,4,4}$ attain their minimum only at the square point $\tau=i$ (whereas $C_{1,1,1}$ is minimal at the hexagonal point), that the square lattice is the unique minimiser also for $\nu=5,7$, that the hexagonal lattice is the unique minimiser for $\nu=3.5$, that the hexagonal and square energies cross in $(3.918364,\,3.918366)$, and that the square lattice loses local minimality at some $\nu_2\in(8.604,\,8.608)$. The elementary planar results are formalised in Lean 4. Numerically, in three dimensions BCC is the best lattice found for all tested exponents $3.2\le\nu\le12$, and hcp is never optimal at the tested exponents. Code and certificates: https://doi.org/10.5281/zenodo.22975868. The computations were carried out with the assistance of the AI system Claude (Anthropic).