Research proof index

Lemma 4.13 Let dstar be the target degree, f₁,...,f_{m-1} : ι → F, 0 < degs₁,...,degs_{m-1} < dstar be degrees and δ ∈ (0, min{(1-BStar(ρ)), (1-ρ-1/|ι|)}) be a distance parameter, then Pr_{r ← F} [δᵣ(Combine(dstar,r,(f₁,degs₁),...,(fₘ,degsₘ)))] > err' (dstar, ρ, δ, m * (dstar + 1) - ∑ i degsᵢ)

Construct a canonical polynomial from a coefficient function Fin n → R. The coefficients are stored in an array (index i gives the coefficient of X^i) and then trimmed to remove trailing zeros. -/ def ofFn [Zero R] [BEq R] [LawfulBEq R] {n : ℕ} (f : Fin n → R) : CPolynomial R := ⟨(Raw.mk (Array.ofFn f)).trim, Raw.Trim.isCanonical_trim _⟩ section Div...

For ℓ = 1, coordinate-wise special soundness is ordinary k-special soundness: the challenge values are distinct, and there is a central vector together with k - 1 siblings differing in the single coordinate , i.e. k pairwise-distinct challenge values.

For a single coordinate (ℓ = 1), membership in the special-sound family SS(S, 1, k) is exactly injectivity of the k challenge vectors: a central vector together with k-1 siblings differing in the unique coordinate is just k pairwise-distinct values. This is the cleaner ℓ = 1 characterization underlying the bridge to plain k-special soundness.

Generic single-round CWSS assembly. Any pure statement-extending verifier of the two-round pSpec is coordinate-wise special sound for foldStructure, provided a witness assembler mkWitness that turns per-branch relOut-witnesses at star-shaped challenge families into a relIn-witness. This discharges all tree/extractor plumbing once; the protoc...

The tensor product generator TensorGenerator and the explicit componentwise generator TensorGenerator_Explicit agree under the canonical isomorphism between F^ℓ ⊗ F^ℓ′ and (ℓ × ℓ') → F.