No one has successfully connected proof (v) and prove (v) here, though the examples of the former are exhaustive in modern speech.
The difference is the two kinds of proving, one of which is vanishing from the language: the sense of testing something, practically, rather than by reason or evidence. A proof copy is when a type plate is prooved, the rising of dough prooves the culture's activity.
So these irregular noun-to-verb forms exist because proof has largely lost the second meaning.
