Pilz's Extended 1–2–3 Conjecture: proofs and Lean 4 formalisations
Authors/Creators
Description
Papers and complete Lean 4 formalisations on Pilz's Extended 1–2–3 Conjecture. The archive (pilz-release-v2.zip) contains three projects:
pilz-weighted: Pilz's Extended 1–2–3 Conjecture over every field. For every field K, every n ≥ 1 and all nonzero weights c_1, …, c_n, every nonzero multiple of Σ c_a X(a), where X(a) = Π_p x_p^{v_p(a)}, has at least n terms. Pilz's conjecture is the case K = F_2, c_a = 1.
pilz: an earlier proof of the binary case, by a square-root descent in characteristic two. pilz-density: the bound h(A) ≥ 4/3 unless A = {a} or {a, 2a}, anticipated in Remark 1.9 of Holdridge and Pach (arXiv:2607.00934).
The work builds directly on the framework of Holdridge and Pach, and on earlier work of Pilz, Huang, Ke, Pach, Szabó, Meyer and Nagy. It was produced with substantial AI assistance. The written proofs have not been checked line by line by a human mathematician; correctness rests on the Lean formalisations, which use only the standard axioms and have been re-checked with lean4checker. Licences: Lean code Apache-2.0; papers CC BY 4.0.