Integral units of the Promislow group at small radius: dichotomy, counting, towers, and compactness

Abstract

Let P be the Promislow (Hantzsche-Wendt) group. The unit conjecture fails for K[P] over every field of positive characteristic and over C, but is open over Z -- Higman's original question. Radius <= 3 is closed over every field by Craven-Pappas, and the known counterexamples begin at radius 4, so radius 4 is where an integral statement must first differ from a statement over a field. This paper is a study of that radius. Reducing modulo 2 splits the problem. Where the residue is nontrivial we count the nontrivial units of F_2[P] supported in the radius-four ball -- exactly 52 -- and deduce that no unit of Z[P] supported in that ball has nontrivial reduction modulo 2, with no bound on the coefficients. On the same side, beyond that ball, we measure the reach of congruence methods by an increment tower on the radius of the correction w in u_1 + 2w (r_min(4) = 0, r_min(8) = 4, r_min(16) = 5, r_min(32) <= 6 for Gardam's residue), and a closed-form reformulation then decides whole congruence strata at once over Z_2: none of the seven known mod 64 strata lifts to mod 128, and the determinant classes c*(monomial) with c not congruent to 1 mod 8 are eliminated for units supported in the radius-four ball. Where the residue is trivial, no condition on u mod 2^K that is necessary for unit-hood can decide the question, at any radius -- though an exact equation reduced modulo a power of 2 still can, and we use that exception. In the twisted-unitary sector, a problem of Bartholdi has a positive answer; the witness modulo 4 appears in joint work with Nies, and here we prove the structure around it: a certified obstruction closing radius 4, its completeness among sigma*-invariant linear obstructions, and a witness modulo 8. A compactness theorem then reduces this sector at radius 4, on the residue class of +/-1, to one certified lower bound. On the characteristic-zero side, Gardam's sixteen complex units form a single gauge orbit, on which the Galois action is by gauge automorphisms, no product from an explicit finite menu of conjugates descends below Q(zeta_8), all sixteen are of Claramunt-Grabowski type 4, and the first levels of the type-2 certificate hierarchy are empty: the linear level over every ordered field, over Q_2 and over C, the xi-linear and quadratic levels over every ordered field and over Q_2. A torsion-free central extension of P, inside whose group ring the complex units lift to integral elements, is proposed as a new habitat for the rational question. Nothing here bears on the full conjecture for Z[P]: every unconditional statement about the existence of units carries a support restriction somewhere -- on the unit, on its mod-2 residue, or on a window fixed in advance. Positive results are verified by exact integer arithmetic. Completeness statements carry DRAT proofs checked by drat-trim where so marked; the grading of the reproducibility section has four levels, one of them for proofs that were verified and then not retained, and solver verdicts without proof objects are marked as such. This deposit contains the paper (79 pages), the complete ancillary bundle (126 scripts, 415 result files, 9 certificates, with an INDEX), and the LaTeX source. The ancillary bundle is self-contained: extracted into an empty directory and run with "python3 src/flagship_numbers_gate.py", it recomputes the headline numbers of the paper from the shipped files alone and checks, by the determinant criterion, that the 52 census supports really are units of F_2[P].