Least sizes of non-unique-product sets: the Promislow group and a Heisenberg-type candidate

Abstract

Let P be the Promislow group, the orientable Hantzsche-Wendt Bieberbach group of dimension 3, which underlies Promislow's classical non-unique-product set and Gardam's disproof of the unit conjecture. A finite set A is non-UP if A.A contains no uniquely represented element; such sets are the combinatorial obstruction in Kaplansky's zero-divisor and unit problems. We make a fully verified computational and structural study of non-UP sets inside P. In an exact integer model we (i) exhibit an explicit non-UP set of 14 elements of minimal word-radius 3 with its complete coincidence pattern; (ii) prove by an exact constraint solver, run to a proof of infeasibility, that the minimum size of a non-UP subset of the radius-r ball is exactly 14 for 3 <= r <= 6; and (iii) isolate the structural reasons why these ball-limited bounds cannot be promoted to all of P by ordering arguments alone. Exploiting that P embeds in D_infinity^3, we prove an effective finite-diameter principle: if a non-UP n-set exists at all, one exists in the ball of explicit radius D(n) <= 4^n poly(n), so P's minimum is effectively decidable. We conjecture D(n) = O(n^{1/3}), under which our radius-6 computation would already prove that 14 is the minimum non-UP cardinality in P; whether 14 is this minimum remains open. We also compute the two-sided minimum, the least |A|+|B| with A.B non-UP: within radius 3 it equals 24, so it lies in [16,24], the lower bound being the Nielsen-Soelberg theorem. As a companion case we treat the Fibonacci group H_4 = F(3,4): it fails the UPP symmetrically with least symmetric size exactly 16 over the radius-4 ball, while its two-sided minimum over the radius-3 ball is 22. The constraint-solver methodology is not new; our contribution is the P-internal data and structure.