In finite group theory, the commuting probability — the chance that two randomly chosen elements commute — obeys a sharp gap theorem. No finite non-abelian group has commuting probability strictly between 5/8 and 1, except at exactly 3/4. Below 5/8, a dense spectrum of values appears. The gap is not a curiosity; it reflects the rigidity of near-commutativity. Groups that almost commute must have a very specific structure.
Mondal and Yadav show the same gap appears in skew left braces. A skew left brace is an algebraic structure with two group operations linked by a compatibility condition, originally introduced to study set-theoretic solutions of the Yang-Baxter equation. The setting is radically different from groups — the two operations interact non-trivially, the underlying structures are richer, and the algebraic constraints are distinct.
Yet the gap is identical. No non-trivial finite skew left brace has commuting probability strictly between 5/8 and 1, except at 3/4. The characterizations of structures achieving 3/4 and 5/8 mirror the group-theoretic results. The probability is preserved under isoclinism. The bound extends to compact Hausdorff topological skew left braces.
The structural point: the 5/8 gap is not a property of groups. It is a property of near-commutativity itself, invariant across algebraic contexts. When a structure “almost commutes” — regardless of whether it is a group, a brace, or presumably other algebraic systems — the combinatorics of near-commutativity force the same forbidden zone. The gap is structural, not categorical. The algebra tells you where it lives; the gap tells you what near-commutativity means.