Research proof index

The ReduceClaim oracle reduction satisfies perfect completeness for any relation. Proof strategy mirrors the non-oracle reduction_completeness: the prover deterministically returns the mapped output, the verifier deterministically computes mapStmt, and the positive-probability output is exactly the mapped element which lies in relOut by hRel.

Coordinate-wise special soundness of the ReduceClaim oracle reduction. As in the non-oracle case, the verifier is pure with no challenge rounds, so CWSS collapses to a transcript-level obligation discharged by the witness pull-back mapWitInv and the compatibility hRel (identical to the RBR knowledge soundness hypothesis, mapStmt replaced by `m...

The ReduceClaim reduction satisfies perfect completeness for any relation.

Coordinate-wise special soundness of ReduceClaim. The verifier is pure with no challenge rounds, so CWSS collapses (via the no-challenge bridge) to a transcript-level obligation. Given the witness pull-back mapWitInv and the compatibility hRel (the same hypothesis as for RBR knowledge soundness), the extractor `e stmtIn := mapWitInv stmtIn witOu...

allVerdicts has the same distribution as allOutputs with a projection retaining the verifier output as the first component; it differs only by a pure post-map.

The identity / trivial reduction is perfectly complete.