2026/07/30 by Moe Tabei
Mathematics · #math.GR #msc:16S34 #msc:20C07 #msc:68V15
12 pages. Ancillary files: complete data (52 units, orbit representatives, box computation), all verification scripts, and the drat-trim verification transcript
arxiv created 2026/07/30 · arxiv updated 2026/08/04
Let P be the Hantzsche-Wendt (Promislow) group and B(4) the radius-four ball in its standard word metric. Dietrich, Lee, Nies and Vinyals determined the two-sided count: exactly 36 nontrivial units u of F2[P] with both supp(u) and supp(u-1) in B(4). We determine the one-sided count: exactly 52 nontrivial units with supp(u) in B(4) and no constraint on the inverse. The 16 new units have inverses supported at radius exactly 5; they form two orbits of size 8 under the symmetry group fixing the generating set, and all 52 units have support size 21 on both sides. Completeness is a single propositional unsatisfiability, certified by a DRAT proof checked with drat-trim. As an arithmetic consequence we prove: no unit of Z[P] with support in B(4) has nontrivial reduction modulo 2 -- with no bound on the coefficients or on the support of the inverse. Since F2[P] has 52 nontrivial units on that ball, this separates, in the untwisted setting, the integral group ring from its characteristic-two quotient at the first radius where the unit conjecture fails over a field.