slug: mohan-neetu-small-doubling-conjecture-62-refutation
bibkey: mohan2026smalldoubling
doi: 10.48550/arXiv.2607.11194
url: https://arxiv.org/abs/2607.11194v1
triage: theorem
motivation_gids:
  - D5/S0/Certificates/Groups/MohanNeetuSmallDoublingRefutation.result62

Mohan-Neetu Conjecture 6.2

Problem

Conjecture 6.2 in Mohan and Neetu, arXiv:2607.11194v1, section 6, asserts that a nonempty finite subset S of a torsion-free group, partitioned into three pairwise disjoint abelian sets A, B and C, generates an abelian subgroup when |S^2| <= 3k - 4. Here k = |S|, S^2 is the product set, and an abelian set has an abelian generated subgroup. The formal claim62 retains every premise; result62 proves its negation.

Motivation

The authors’ Example 6.1 shows that a weaker three-piece threshold fails. Conjecture 6.2 asks whether lowering that threshold by one restores commutativity.

Gap

The cited preprint prints this as a conjecture. The arXiv version listing and scoped arXiv, Semantic Scholar, and Crossref searches recorded in issue #8686 found no subsequent settlement. Its universal hypothesis does not exclude the Klein bottle group.

Route

In the torsion-free Klein bottle group (r,n)(s,m) = (r + (-1)^n s,n+m), let S = {(1,1),(2,1),(3,1)} and take its three singleton subsets as A, B and C. Singleton-generated subgroups are abelian and the pieces are pairwise disjoint. The product set is {(-2,2),(-1,2),(0,2),(1,2),(2,2)}, so |S^2| = 5 = 3|S| - 4. Yet (1,1)(2,1) = (-1,2) != (1,2) = (2,1)(1,1).

Falsifier

The refutation would fail if the coordinate group had torsion, the singleton pieces were not pairwise disjoint and abelian, the displayed product set had more than five elements, or the pair commuted. result62 discharges these conditions.

Evidence

D5/S0/Certificates/Groups/MohanNeetuSmallDoublingRefutation.result62 has Lean type not claim62. Its proof uses the coordinate torsion-freeness lemma, subgroup closure of singletons, and checked finite products. Its axiom closure is propext, Classical.choice, and Quot.sound.

Triage

theorem. The conclusion refutes Conjecture 6.2 as printed; the stronger threshold and any proposed correction remain outside the claim.

ASSUMED-UNVERIFIED

The scoped literature search does not establish exhaustive publication priority. Google Scholar and the full texts of the arXiv Klein-bottle search hits were not checked.