Research proof index

A prime number theorem in arithmetic progressions, in Chebyshev-weighted form. If $q\ge1$, $a<q$, and $\gcd(a,q)=1$, then $\sum_{\substack{p\le x\\p\equiv a\pmod q}}\log p\sim\frac{x}{\varphi(q)}$ as $x\to\infty$.

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.

Chernoff symmetric bound