The one-sided unit count of F_2[P] at radius four, and an integral separation theorem

Abstract

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 F_2[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 F_2[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.