Report: Lean 4 proof establishes W(2;3,21) = 416
A report says W(2;3,21) = 416 has been proved with a Lean 4 kernel-checked proof, resolving a conjecture Kullmann made in 2010. The theorem covers 209 cases and 38,661,140 leaf formulas. Each refutation was checked by cake_pr, the LRAT checker verified in CakeML, and bound by digest to its Lean term. The proof was produced with Hupyy’s neuro-symbolic Large Proof Model; the report says nothing counted unless accepted by the Lean kernel or cake_pr.
The reported result provides a machine-checked resolution of a combinatorial conjecture that remained on DeepMind’s Formal Conjectures list earlier in the year, while demonstrating the use of a neuro-symbolic proof model subject to formal validation.
w(2;3,21) = 416 - today proved with a Lean 4 kernel- checked proof (preprint + 13.6 GB evidence package).
Every red-blue colouring of {1, ..., 416} contains a red 3-term or a blue 21-term arithmetic progression, and some colouring of {1, ..., 415} contains neither.
Kullmann conjectured the value in 2010; DeepMind's Formal Conjectures list still carried it as open this year.
The proof is a Lean 4 theorem over 209 cases and 38,661,140 leaf formulas, each refutation checked by cake_pr (the LRAT checker verified in CakeML) and bound by digest to its Lean term.
It was produced with LPM (Large Proof Model), a neuro-symbolic system from Hupyy; nothing counted unless the Lean kernel or cake_pr accepted it.
Paper: https://doi.org/10.5281/zenodo.23776306
Evidence package (Lean sources, checker records, replay kit): https://doi.org/10.5281/zenodo.23749222
To check: download it and run 'shasum -a 256 -c SHA256SUMS'
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