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Is there a proof that every possible proof is a generation in the grammar of lean or rocq? Sounds like an unsubstantiated claim.

You're confusing the mechanical language of math with math itself.

The map is not the territory, etc. If math was just an elaborate linguistic Glass Bead Game then we wouldn't be funding it. The intuition is that the surface rules of math help uncover the underlying structure of reality.

This isn't necessarily true.

While every formal proof can be encoded mathematically that doesn't imply that the grammar of lean or rocq is sufficient to encode every potential proof.

However the core of what you are saying: that every possible proof exists in the space of all mathematical statements is correct.

You are correct, but in trying to get the point across I attempted (perhaps fruitlessly) to use concrete concepts today of a system where the search space is calculable yet whose span does not imply the 'knowing' of all calculable things.