Do all languages in $P$ have polynomial proofs that they are in $P$?

A proof for a language $ L$ belonging to a complexity class $ C$ that is accepted by a mathematical journal can be framed as there existing a verifier $ V$ that accepts this proof as the first part of its input and the language as the second. The verifier (referee) verifies this language is a member (a word) in the language representing the complexity class.

$ Verifier$ : (Proof for $ L$ in $ C$ , $ L$ )$ –>$ [0,1]

Do all languages in $ P$ have a proof of the fact that they are in $ P$ that can be verified in polynomial time? Given a language, determining if an arbitrary $ L$ is in $ P$ or not is undecidable; however, given a proof for a language being in $ P$ , can that language be verified to be in $ P$ in polynomial time?