In the literature [*], the "proof" is an actual proof, but that's not what the prover sends to the verifier. Rather, the verifier queries random pieces of the proof, eventually convincing themselves that with high probability, the prover knows a valid proof.
[*] At least in most of the older literature descending from the PCP literature. Modern papers sometimes abandon that distinction and just use "proof" and "argument" interchangeably.
[*] At least in most of the older literature descending from the PCP literature. Modern papers sometimes abandon that distinction and just use "proof" and "argument" interchangeably.