The Triangular Lattice and Universal Optimality: human proofs, Lean 4 verification, numerical validation in Python and Rust, and physics from fluids to plasmas (a textbook for students)

A 105-page textbook for students that follows one theorem from its definitions to the laboratory: the triangular lattice has the least energy among all planar configurations of a given density, for every completely monotone repulsion (openai/math, Lean 4; AI-generated, not peer reviewed). Each result is shown twice, as a step-by-step human proof and as the Lean 4 statement that was kernel-checked, then tested numerically in Python and in Rust with the rusty-SUNDIALS CVODE solver. Every claim carries an Elenchus tier: A (kernel-checked Lean), L (literature), C (argument or proposal), X (floating-point exploration, never supporting a claim).

Chapters, each written independently from one fact sheet and then combined:

1. Foundations: lattices, Poisson summation, theta functions, completely monotone functions, the linear-programming bound; a review of an earlier draft with eight corrections.
2. Theorems, human and Lean side by side: universal optimality, Riesz optimality and finiteness for s > 2, every density, non-uniqueness (the lattice with a vacancy), and D4, open versus closed disks.
3. Numerics: a Python companion and the Rust/CVODE N-body study, with code listings pinned to a commit.
4. Physics, starting from fluid dynamics (2D Euler point vortices, vortex crystals, BEC vortex lattices) and opening to colloids, dusty plasmas and electrons on helium; what CVODE/IDA can and cannot do here.
5. Open problems and industry: the strong per-potential Cohn–Kumar conjecture, renormalized energies, the endpoint s = 2, dimensions 3, 8 and 24; student projects.

New in this book: D4 (Tier A). Upstream's lower energy and density-one condition are unchanged when closed disks are replaced by open disks, so universal optimality holds verbatim with open disks; the two identities need no local-finiteness hypothesis. Axiom footprints are {propext, Classical.choice, Quot.sound}. Scope: upstream's observables only (centred disks, N/(πR²) density, liminf energy); the density statement at densities other than one is Tier C.

Not done here, as the book states wherever they come up: the strong per-potential Cohn–Kumar conjecture and the endpoint s = 2 are open; renormalized energies are proved in upstream's manuscript (Tier L) but not formalised.

Erratum to doi:10.5281/zenodo.23293429: the shifted-force truncation at r_c = 8 changes absolute energies by about 1.8e-2 at kappa = 1, more than the triangular–square gap. The gap itself is nearly unchanged and all comparisons use the same truncated potential, so the observation stands for that potential, which is an approximation of Yukawa outside the theorem's class.

Review: an independent whole-book audit by a separate model instance (accept with corrections, all 20 applied; scripts re-run with identical output). The book has not been reviewed by human experts; red-team and citizen-science review are invited. The work was produced by AI agents (Claude) under the author's direction; the author is responsible for the content. The bundle holds the LaTeX sources, the three Python scripts, the Lean files with the axiom audit, the D1–D4 records and the Rust crate.