Cutting an irreducible manifold along incompressible tori preserves irreducibility

Let $S$ be a PL sphere in the interior of the selected cut component. Its image in $M$ is a PL sphere missing every cutting torus, so bounds a parametrized PL ball $B$ by [F1]. Each connected cutting torus lies wholly inside or wholly outside that sphere. If a torus lies inside $B$, its existing two-sided neighborhood can be shrunk into $\mathrm{Int}B$, by compactness and a common smaller product width. It remains a PL embedded torus there. [F3] supplies the compatible finite subdivisions needed for [F2], even if the original triangulation of $B$ was different from the one containing the torus.

Such an inside torus would have the PL compressing disk of [F2] entirely in $B\subset M$, contradicting its original incompressibility. Thus $B$ misses every cutting torus. The connected ball interior lies in the same complement component as its collar along $S$, so $B$ lifts to a PL ball in the specified cut component with boundary $S$. Every interior PL sphere there therefore bounds a PL ball. This is precisely PL irreducibility; no smooth-ball or arbitrary-topological-marking claim entered.