Research proof index

Perfect completeness of the pure pass-through CheckClaim oracle reduction. Because the verifier no longer checks P at runtime (it is a pure pass-through, with P living in oracleRelOut), completeness needs the explicit hypothesis hP that every relIn input already satisfies P. Under hP, the prover forwards ⟨stmt, oStmt⟩ unchanged and t...

Coordinate-wise special soundness of CheckClaim. The verifier is a pure pass-through with no challenge rounds, so CWSS collapses (via the oracle no-challenge bridge coordinateWiseSpecialSound_of_isEmpty_challengeIdx) to a transcript-level obligation. The extractor is trivial (e := fun _ _ => (), there is no witness); since the pass-through outpu...

The pure pass-through oracle verifier's underlying non-oracle verifier returns the combined input statement unchanged.

The CheckClaim reduction satisfies perfect completeness with respect to the predicate as the input relation, and the output relation being always true.