My question concerns PL-homeomorphism and shellability.

I see from a previous question here that bistellar flips do not preserve shellability. However, the following two results are corollaries of Pachner's Theorem:

- Every simplicial PL-sphere is the boundary of a shellable simplicial ball.
- Bistellar equivalence implies PL-homeomorphism.

These results seem to be incompatible with the existence of non-shellable simplicial spheres, so I'm trying to find the flaw in my understanding.

If every simplicial PL-sphere is the boundary of a shellable ball, then how can we ever obtain non-shellable objects via bistellar flips? Is every triangulated sphere the boundary of a shellable object, while not being shellable itself?