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arXiv · 2609.17559

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

Moe Tabei
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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 $\ge2$ 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.

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BibTeXRIS

Moe Tabei. 2026-09-21. Localizing the Gardam unit: the support geometry of units in F_2[P] and its non-unique-product relatives. https://arxiv.org/abs/2609.17559

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