How small can a counterexample to the Non-Cancelling-Intersections conjecture be?

Abstract

Wilhelm has refuted the Non-Cancelling-Intersections conjecture of Amarilli, Monet and Suciu by showing that for every prime p ≥ 10⁵ some marking m of the lines of 𝔽ₚ² makes the lattice P_(p,m), of p³+2p²+2 elements, admit no winning dot-algebra tree; the smallest lattice his argument produces has 1 000 110 003 900 047 elements, and he asks how small a counterexample can be. We answer with exact integer arithmetic on his own first moment. First, his chain of inequalities, evaluated instead of simplified, already closes at p=5273, and 5273 is the least such prime; the threshold 10⁵ comes from a single lossy step. The chain is moreover not monotone in p: it fails again at p=5309, exactly where ⌈√(2p) ⌉ jumps. Second, the same first moment used to its full strength — the exact hypergeometric hit probability, a sharper Cauchy–Schwarz count of lines, and linear-programming duality over the trace profile — closes at p=59, giving a lattice of 212 343 elements with no winning dot-algebra tree, smaller by a factor 4 709 879 788.36…. Here 59 is least, and again the bound is not monotone: it fails at 61 and closes again at 67. From below, every lattice with at most 8 elements admits a winning dot-algebra tree, so the smallest counterexample has between 9 and 212 343 elements. All the statements above are machine-checked in Lean 4.

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