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 𝔽2[P], P the Promislow group, with support of size $21$; Gardam asked whether $21$ is least possible. Over 𝔽2 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 ≥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 H4=F(3,4) and the Nielsen--Soelberg groups are swept with certificates; G3 is Gardam's amalgam S, whose unit we localize into its radius-$4$ ball. H4 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 𝔽2[P] computable in principle.
Related
- The information geometry of product-reference discrete diffusion: Interaction growth complexity and optimal scheduling
- Evaluation of TF-IDF and Word2Vec Representations Using Support Vector Machine for Multiclass Product Category Classification of Tokopedia Reviews
- A Machine-Checked Proof that Gardam's Ã2 Lattice Does Not Have Unique Products
- A Product Principle for Harmonic Schwarz Lemmas: Boxes, Polydiscs, and Metric Geometry
- Uniform Three-Fold Product Expansion of Algebraic-Geometry Codes over a Fixed Field
BibTeX
@misc{tabei2026localizing,
title = {{Localizing the Gardam unit: the support geometry of units in F_2[P] and its non-unique-product relatives}},
author = {Moe Tabei},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17559},
eprint = {2609.17559},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17559},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Localizing the Gardam unit: the support geometry of units in F_2[P] and its non-unique-product relatives
AU - Moe Tabei
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.17559
UR - https://arxiv.org/abs/2609.17559
AB - Gardam's counterexample to the Kaplansky unit conjecture is a unit of 𝔽_2[P], P the Promislow group, with support of size $21$; Gardam asked whether $21$ is least possible. Over 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 ≥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 𝔽_2[P] computable in principle.
ER -