Research proof index

Auxiliary lemma for proving that the polynomial sent by the honest prover is of degree at most deg

Telescoping identity (the core sum-check completeness step): summing over the (k+1)-coordinate cube equals summing coordinate 0 over its domain, then the rest over the tail cube. This is the piFinset "cons decomposition" 𝔻^{k+1} ↔ 𝔻₀ × 𝔻^k.

The CWSS shape of the canonical ℓᵢ = 1 structure CWSSStructure.ofSpecialSound k is exactly the plain special-soundness shape distinctShape k. This is the structural heart of the equivalence between CWSS and plain special soundness: both the arity (1·(kᵢ-1)+1 = kᵢ) and the node predicate (IsSpecialSoundFamily 1 kᵢ vs. Function.Injective) agree.

A polynomial is separable if and only if its discriminant is non-zero.

The rank of a non-square Vandermonde matrix with more rows than columns is the number of columns.

Coordinate-wise special soundness is preserved by binary verifier append. The deterministic first-verifier output identifies the input statement of each suffix tree consumed by the second verifier's extractor.