Exact counterexamples to local common-neighbour rigidity in three dimensions

Several routes to a three-dimensional crystallization theorem pass through a local implication of the following shape: if two particles at approximately unit distance each have approximately twelve neighbours at approximately unit distance, then they have at least four common neighbours—the value realised by every bond of every Barlow structure. The exact form, in which “approximately” is replaced by “exactly” and the ambient set is 1-separated, is Conjecture 2.2 of Flatley and Theil, and is open. We give explicit, exactly verified counterexamples to the robust form of that implication below unit separation. For specified separation thresholds and bond windows, both endpoints of a bond have degree twelve while the common-neighbour count is two or three; the obstruction persists for nearly exact incident bonds and on the exact-contact manifold, where all 23 incident distances are exactly 1. An elementary geometric argument bounds the count by five, and five is attained—even by an exact-contact configuration at separation at least 1. Finally, we exhibit exact-contact degree-twelve two-stars with four common neighbours that are not congruent to Barlow fragments. None of these examples refutes the exact Flatley–Theil conjecture at unit separation.