Research proof index

For a nonzero multivariate polynomial f of total degree at most d, evaluated over a product of finite subsets of a field F, each of cardinality at least m, the number of roots is at most d / m multiplied by the side of the product set.

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...