Localizing the Gardam unit: the support geometry of units in F_2[P] and its non-unique-product relatives

Abstract

Gardam's counterexample to the Kaplansky unit conjecture is a unit of ${\mathbb{F}}_2[P]$, $P$ the Promislow group, with support of size $21$; Gardam asked whether $21$ is least possible. Over ${\mathbb{F}}_2$ the unit equation is a parity condition on a pair of supports, searchable over word balls. We upgrade two statements to machine-checked certificates: no support pair of sizes $\ge 2$ lies in the radius-$3$ ball (a DRAT proof of a ball form of the Craven--Pappas theorem), and every nontrivial unit with supports in the radius-$4$ ball has total support at least $42$, confirming Gardam's expectation in ball-limited form. The relatives $H_4=F(3,4)$ and the Nielsen--Soelberg groups are swept with certificates; $G_3$ is Gardam's amalgam $S$, whose unit we localize into its radius-$4$ ball. $H_4$ resists the twisted-unitary ansatz through radius $6$ for every non-identity dihedral twist. An effective localization principle makes the least support of a nontrivial unit of ${\mathbb{F}}_2[P]$ computable in principle.

arXiv comment11 pages. v2: the radius-6 twisted-unitary sweep of F(3,4) now covers all four dihedral twists; two references added. Ancillary files: code and certificates. Reproducibility archive: doi:10.5281/zenodo.22866715