Research proof index

The second information estimate for $\tau$-minimizers. Let $X_1',X_2'$ be independent copies of $X_1,X_2$, set $k=d[X_1;X_2]$, $I_1=I[X_1+X_2:X_1'+X_2\mid X_1+X_2+X_1'+X_2']$, and $I_2=I[X_1+X_2:X_1'+X_1\mid X_1+X_2+X_1'+X_2']$. Then $I_2\le2\eta k+\dfrac{2\eta(2\eta k-I_1)}{1-\eta}$.

Truncating the seed at oracle i₀ to only the first k entries does not change the distribution when averaging over seeds from generateSeed.

The pure verifier's underlying non-oracle verifier returns the statement together with the sampled challenge (read off the transcript), with the oracle statements passed through.

Coordinate-wise special soundness of SendClaim. The verifier is a pure pass-through with no challenge rounds, so CWSS collapses (via the oracle no-challenge bridge) to a transcript-level obligation. The extractor is trivial (e := fun _ _ => (), there is no witness); since the output oracle statements at inl are the input oracles unchanged and `t...

The pure pass-through oracle verifier's underlying non-oracle verifier returns the statement together with the output oracle statements (input oracles at inl, the message at inr 0).

Coordinate-wise special soundness of SendSingleWitness. The oracle verifier has no challenge rounds, so CWSS collapses (via the oracle no-challenge bridge coordinateWiseSpecialSound_of_isEmpty_challengeIdx) to a transcript-level extraction obligation on the combined statement Statement × (∀ i, OStatement i). The extractor is `e := fun _ tr => tr...