Unique mutation cycles of nucleus-extension quivers

We prove the uniqueness conjecture of Fomin and Neville for their long mutation cycles in all ranks $n\ge4$. More generally, we show that every nucleus-extension quiver has a unique simple cycle in its labeled mutation graph. These quivers allow an acyclic part of arbitrary size outside the nucleus. Our proof uses three mutation statuses determined by a closed nucleus and the full subquivers obtained by deleting one part of its nucleus decomposition at a time from the ambient quiver. We analyze transitions between these statuses to prove that every mutation leaving the prescribed cycle is performed at an exit.