Direct fold obstruction for dropping a prefix

Suppose a predicate accepts some $u$ and rejects some $v$. Dropping its initial accepted list prefix maps both the empty list and the singleton $u$ to the empty list, but maps the extensions by $v$ to different lists. A direct list fold cannot do this: its combining function receives the same head and the same folded-tail summary in both cases. This is an instance of the fold factorisation criterion. An auxiliary list fold retaining more information followed by projection is not ruled out.