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.

• The paper proves that the smallest non-unique-product set in the Promislow group within word-metric balls of radius 3–6 has exactly 14 elements, with 16 minimal witnesses and strong fiber-distribution rigidity.
• The study finds a symmetric minimum of 16 and a two-sided minimum of 22 for the Fibonacci group H4, while the Promislow group has a two-sided minimum of 24, showing that the two invariants rank these groups differently.
• Machine-checked solver certificates, exact arithmetic, and structural analysis establish ball-limited results, expose cocycle-based obstacles to ordering arguments, and reduce the global minimum question to an effective finite-diameter problem.

Context and motivation

Kaplansky's unit conjecture was disproved by Gardam in characteristic $2$ using the Promislow group $P$, the orientable Hantzsche–Wendt Bieberbach group of dimension three (Gardam, 2021). The combinatorial obstruction underlying such counterexamples is the failure of the unique product property (UPP): a finite set $A$ is non-UP if every element of the product set $A\cdot A$ admits at least two representations as $ab$. Promislow himself exhibited a $14$-element non-UP set in $P$ by computer search in 1988, and Nielsen–Soelberg later proved that over all torsion-free groups the minimum non-UP cardinality is exactly $8$, attained in a virtually-Heisenberg polycyclic group rather than in $P$. The paper under review undertakes a rigorous computational and structural study of how small a non-UP set can be inside $P$ itself, together with a companion study of the Fibonacci group $P$0. The author is explicit that neither the constraint-solver methodology nor the realization of $P$1 inside $P$2 is new; the contribution is the exact data and structure internal to these two groups.

Search model and verification discipline

The group $P$3 is realized as affine isometries of $P$4 with point group the Klein four-group of diagonal sign matrices and translation parts in $P$5, generated by torsion-free elements $P$6 satisfying the Hantzsche–Wendt relations. All arithmetic is exact integer arithmetic; balls are taken for the word metric of $P$7, with $P$8 up to $P$9.

Non-UP existence is encoded as a constraint satisfaction problem: Boolean variables select elements of a candidate universe, and each product value realized by exactly one ordered pair yields a forbidding clause. A crucial methodological point is that the symmetric non-UP property is not translation invariant: one-sided translation $A$0 is a twisted product, not a translate of $A$1. For the minimal witness, every nontrivial translate within $A$2 acquires between $A$3 and $A$4 unique products, while conjugation preserves non-UP-ness. Consequently anchoring $A$5 is a genuine restriction, and all reported results use the unanchored model.

The verification is layered. Positive claims are re-checked by an independent brute-force routine that never calls the solver. Non-existence claims rest on CP-SAT INFEASIBLE verdicts, but for the central symmetric claims this trust has been discharged twice over: each was re-derived by a DRAT-producing SAT solver with machine-checked unsatisfiability proofs via drat-trim, and the entire $A$6 symmetric minimality inside $A$7 was re-derived in the Glasgow Constraint Solver with VeriPB proofs checked independently. On instances small enough for full enumeration, the claims were re-established with no solver at all.

Ball-limited minimality in $A$8

The main computational result is sharp. An explicit non-UP $A$9-set of word-radius $A\cdot A$0 is given, with point-group distribution $A\cdot A$1 — notably it contains no identity element, which is why identity-anchored searches fail. Its $A\cdot A$2-cell product grid partitions into $A\cdot A$3 coincidence classes, every class of size $A\cdot A$4, and each class is a partial permutation matrix (a consequence of element distinctness). The solver sweep proves:

Radius	Sizes 8–13	Size 14
$A\cdot A$5	none (no non-UP set of any size)	none
$A\cdot A$6 through $A\cdot A$7	INFEASIBLE	yes

Thus for $A\cdot A$8 the minimum non-UP cardinality inside $A\cdot A$9 is exactly $ab$0. Any non-UP set in $ab$1 of size $ab$2–$ab$3, if one exists, lies outside the radius-$ab$4 ball. Whether $ab$5 is the global minimum in $ab$6 remains open.

Structural rigidity accompanies the bound. Of the $ab$7 possible fiber distributions for a $ab$8-set, exactly two admit a non-UP witness inside $ab$9 — $14$0 and its image under the swap $14$1. Moreover $14$2 has exactly $14$3 minimal non-UP $14$4-sets inside $14$5, forming four orbits under the ball-isometry group generated by the swap and inversion.

Two-sided minima, profiles, and staircases

The paper also computes the two-sided invariant $14$6, the least $14$7 with $14$8 non-UP, for which Nielsen–Soelberg give the universal lower bound $14$9. Within $P$0 of $P$1 the two-sided minimum is exactly $P$2, attained by distinct sets of size $P$3 each — strictly better than the symmetric pair ($P$4), with both sides fiber-balanced $P$5, in contrast to the lopsided symmetric witness. Hence $P$6.

Two finer curves sharpen the picture. The profile $P$7 (least partner size against $P$8) shows that $P$9 forces balance: no two-sided witness in $8$0 has any side smaller than $8$1, so the lopsided shapes permitted by the universal bounds do not occur even at radius $8$2. The staircase $8$3 (least number of unique products over $8$4-sets) is non-monotone for $8$5: it rises to $8$6 at $8$7 before collapsing to $8$8 at $8$9. In both $P$0 and $P$1 the value $P$2 never occurs, so the two-unique-products property and the UPP fail simultaneously.

Why ordering arguments fail

The natural route from ball-limited bounds to a global bound would be an ordering/convexity argument maximizing a linear functional on translation parts, as succeeds in bi-orderable groups. This fails in $P$3 because the translation part is a cocycle, not a homomorphism: cross-fiber coincidences can make the globally maximal product non-unique. Empirically, for the minimal witness the functional-maximal product is non-unique for all $P$4 random directions tested, and even the maximal product within the identity fiber is made non-unique by cross-fiber coincidence in sampled cases. This is the precise obstruction preventing promotion of Theorem's radius-$P$5 bound to all of $P$6 by ordering arguments alone.

Decidability and the finite-diameter principle

Despite the obstruction, the question is decidable. Because the point matrices are diagonal, the integer realization equations decouple coordinatewise — reflecting the embedding $P$7 — into three systems with signed sparse coefficient matrices. Hadamard bounds on minors then yield an explicit constant $P$8 such that if $P$9 contains a non-UP $P$0-set, it contains one inside $P$1. Importantly, the small witness is re-realized from its coincidence pattern, not translated into the ball, since translation invariance fails.

Empirically, across $P$2 random consistent coincidence systems the solution lattices are generated by unit vectors and every pattern compresses into a unit coordinate box, so the pattern imposes essentially no diameter; the only real constraint is element distinctness, suggesting word-radius $P$3. The author conjectures $P$4, under which the existing radius-$P$5 computation would immediately prove that Promislow's $P$6 is the true minimum in $P$7. The practical solver ceiling is currently radius $P$8; radius $P$9 decomposes tractably per fiber distribution but full certification is a matter of solver time.

The contrasting case $P$00

The Fibonacci group $P$01, shown by Dietrich–Lee–Nies–Vinyals to fail the UPP via a two-sided witness with $P$02, is treated as a companion. The paper establishes an embedding into the index-$P$03 Heisenberg extension (faithfulness via Hirsch length additivity), and proves:

• Symmetric failure: $P$04 admits a non-UP $P$05-set of word-radius $P$06, and $P$07 is least over the radius-$P$08 ball — a formally stronger conclusion than the prior two-sided result.
• Two-sided optimum: the two-sided minimum over $P$09 is exactly $P$10 (split $P$11), improving $P$12 to the optimal value within that ball.
• Exact census: again exactly $P$13 minimal symmetric witnesses inside $P$14, matching the count for $P$15 — a numerical coincidence the author cannot explain.

The contrast is organized by an asymmetry gap $P$16. Over the stated balls, $P$17 for the Nielsen–Soelberg extremal group, $P$18 for $P$19, and $P$20 for $P$21; precisely this inversion orders the two groups oppositely under the two invariants ($P$22 symmetrically, $P$23 two-sidedly). Where $P$24's profile is a sharp cliff forcing balance, $P$25's is perforated — $P$26 despite $P$27 — and its staircase is flat at $P$28 until collapse at $P$29. As a sanity check, the integral Heisenberg group itself, being bi-orderable, shows no non-UP set up to size $P$30 in its radius-$P$31 ball: the failure in $P$32 is created entirely by the order-reversing generator.

Limitations and open questions

The paper is candid about scope. The headline values are ball-limited: whether $P$33 and $P$34 hold globally is open, pinned only to $P$35 and $P$36 respectively (and $P$37, $P$38 globally). The finite-diameter bound $P$39 is far from tight — roughly $P$40 at $P$41 against a true radius of $P$42 — and the conjecture $P$43 for $P$44 rests on empirical lattice evidence, not proof. The two-sided, profile, and staircase non-existence claims remain CP-SAT verdicts without externally checkable proof objects. The bump $P$45 and the profile spike $P$46 lack structural explanation, as does the unexplained census coincidence of $P$47 minimal witnesses in both groups. Size-$P$48 coincidence patterns are not classified, blocking a matching-census route to excluding small sizes. Finally, whether every group failing the UPP admits a symmetric witness, and what structural feature the asymmetry gap tracks, are left open.

Conclusion

This paper pins down, with layered machine-checked certificates, the exact minimum non-UP cardinality inside balls of $P$49 up to radius $P$50 — namely $P$51, Promislow's classical value — and computes the corresponding two-sided, profile, and staircase curves for both $P$52 and $P$53, revealing that the two invariants order the groups oppositely. Its structural contributions are the partial-permutation rigidity of coincidence classes, the forced fiber distributions of minimal witnesses, the cocycle obstruction defeating ordering arguments, and an effective finite-diameter principle making the global question decidable. The reduction of the open minimum to a concrete searchable diameter bound, conditional on a plausible conjecture about solution lattices, gives the question of whether $P$54 is truly minimal in $P$55 a well-defined computational target.