Take a decreasing sequence of integer sets A₁ ⊃ A₂ ⊃ A₃ ⊃ ... converging to a limit set A. For each set, form the h-fold sumset — all possible sums of h elements. The natural question: does the h-fold sumset of the limit equal the intersection of the h-fold sumsets of the approximating sets?
For small h, the answer is typically yes. The limit operation and the sumset operation commute. You might expect this to be uniform — either the equality holds for all h, or it breaks at some threshold and stays broken.
Marques and Nathanson (arXiv:2603.14510) show the truth is stranger. For any threshold h₀ ≥ 2, there exist sequences where the equality holds for h = 1, 2, ..., h₀ - 1 but fails at h₀. This much you might anticipate — the commutativity breaks down as sums get more complex.
But there also exist sequences where the equality holds for h = 1 and h = h₀ while failing for all intermediate values h = 2, 3, ..., h₀ - 1. The gap is not at the boundary. It is in the middle. The sumset operation and the limit operation commute at a specific fold number, fail for all smaller folds above 1, and then commute again.
This means the commutativity is not a monotone property. It does not break once and stay broken. It can skip, landing on isolated values of h while missing the ones between. The arithmetic structure of the limit set determines which fold numbers work, and this determination is not ordered by complexity.
The through-claim: the interaction between two natural operations — taking limits and forming sums — can be intermittent in a way that has no topological explanation. The set where they agree is not an interval but an arbitrary subset of the integers. Simple operations, composed, produce behavior that is arithmetically structured but not geometrically ordered.