Power windows in the Nielsen–Soelberg group G₁: the eleventh staircase cell, and the balls that were not computed
Abstract
Nielsen and Soelberg proved that a finite subset A of a torsion-free group whose square A · A contains no uniquely represented element has |A| ≥ 8, and exhibited two groups attaining the bound; the first, here G₁, is a two-generator group that is virtually class-2 nilpotent, with a torsion-free nilpotent normal subgroup of index 32. A recent quantitative study of these extremal groups computes, inside the word balls Bᵣ of the defining generating set of G₁, the unique-product staircase u_(Bᵣ)(n): the least number of uniquely represented elements of A · A over all n-element A ⊆ Bᵣ. It finds u_(B₅)(n)=2 for 2 ≤ n ≤ 10 and leaves the cells n=11 and n=12 open, its solver returning only the upper bound 15 against a proved lower bound of 2. We settle the first of them: u_(B₅)(11)=2. The instrument is a lemma that appears in neither that paper nor its companion: n ≥ 2 consecutive powers of an element of infinite order have exactly two unique products, the multiplicative reading of the elementary fact that an n-term arithmetic progression has exactly two sums of multiplicity one. It is sharp here in a precise sense: the longest run of consecutive powers of a single element of G₁ inside Bᵣ, normalised to contain the identity, is exactly 11 for r=5 and 13 for r=6. So the method reaches n=11 in B₅ and no further — exactly where the published flat row ends — and beyond it we narrow u_(B₅)(12) from [2,15] to [2,3]. In the two larger balls, whose staircases are not computed in the source at all, it gives u_(B₆)(n) ≤ 2 for 2 ≤ n ≤ 13 and u_(B₇)(n) ≤ 2 for 2 ≤ n ≤ 15, the latter beside u_(B₇)(8)=0. We also record a parity obstruction: for an inverse-closed set in a torsion-free group the number of unique products is even, so a set realising the value 1 — the object of the source's remaining open cell u_(B₆)(8) ∈ {1,2} — cannot be inverse-closed. Every count the theorems below rest on is machine-checked in Lean 4; the auxiliary computations that are not are labelled where they occur.
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Claim ledger
Stated results
TN1candidate2026-09-03
u_(B(5))(11) <= 2 in the Nielsen-Soelberg group G1: the eleven consecutive powers x⁻⁵,...,x⁵ lie in the radius-5 ball and their square has exactly two uniquely represented elements. With the lower bound 2 that arXiv:2607.19687v2 Prop. 5.1 reports, this DECIDES its first open staircase cell: u_(B(5))(11) = 2, extending its flat-2 row from 2 <= n <= 10 to 2 <= n <= 11. The source's own solver returned only u <= 15 here.
TN2candidate2026-09-03
u_(B(5))(12) <= 3 in G1: an explicit 12-element subset of the radius-5 ball (the 11-term power window plus one further element) whose square has exactly three uniquely represented elements. This narrows the source's reported u_(B(5))(12) in [2,15] to [2,3]; whether 2 is attained stays open, and the power-window route provably cannot reach it (see TN6).
TN3candidate2026-09-03
u_(B(6))(n) <= 2 for every 2 <= n <= 13 in G1, by the prefixes of the 13-term power window xᵏ: |k| <= 6 of the 933-element radius-6 ball. arXiv:2607.19687v2 computes no staircase outside B(5) at all.
TN4candidate2026-09-03
u_(B(7))(n) <= 2 for every 2 <= n <= 15 in G1, by the prefixes of the 15-term power window of the 1935-element radius-7 ball. At n = 8 the bound is not tight – B(7) holds the Nielsen-Soelberg witness with u = 0 – so the two facts pin the shape of the B(7) staircase from both sides at the critical size.
TN5known data2026-09-03
The word balls of G1 in the metric of x⁺⁻¹, y⁺⁻¹ have sizes |B(0)|,...,|B(7)| = 1, 5, 17, 53, 153, 401, 933, 1935, by breadth-first search in the model. This reproduces the sequence arXiv:2607.19687v2 section 3 prints.
TN6candidate2026-09-03
The longest run of consecutive powers of a single element of G1 inside a ball is exactly 11 in B(5) and 13 in B(6) (longest exponent interval containing 0; fuel 20 each direction), attained by the four generators x⁺⁻¹, y⁺⁻¹. Hence the power-window bound u_B(n) <= 2 stops exactly at n = 11 in B(5) – precisely where the source's flat-2 row stops – and at n = 13 in B(6).
TN7known data2026-09-03
The model satisfies both defining relators of G1 and its coset table is a group table of order 32; the Nielsen-Soelberg witness A1 is non-UP (u = 0) with words of length <= 7 and is not inside B(6); and the source's own u_(B(5))(8) = 2 minimiser has u = 2. The compute-first-values gate, kernel-bound.
TN8prose2026-09-03
If G is torsion-free and A = A⁻¹ is finite with |A| >= 2, then the number of uniquely represented elements of A.A is EVEN: (a,b) -> (b⁻¹,a⁻¹) is a fixed-point-free involution of that set (a fixed point would be 2-torsion, and 1 itself has the |A| >= 2 representations a.a⁻¹). Consequence: any set attaining u = 1 – the object the source's open cell u_(B(6))(8) in 1,2 asks for – must be NON-symmetric.
This ledger entry is reported in prose and is not bound to a Lean theorem.Provenance
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- Machina Mathematica
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- Korea Superintelligence Labs
- Source context
- A group G has the unique product property (UPP) if for all finite nonempty A, B ⊆ G some element of A·B has exactly one representation ab with a ∈ A, b ∈ B; a single finite set A is non-UP when A·A has no uniquely represented element. Non-UP sets are the combinatorial obstruction behind Kaplansky's zero-divisor and unit problems for group rings of torsion-free groups.
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- 2026-09-07 03:53 UTC
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