Research proof index

[⊗_{i=0}^{ϑ-1}(1-r_i, r_i)] · [ - u₀ -; ...; - u_{2^ϑ-1} - ] - [⊗_{i=0}^{ϑ-2}(1-r_i, r_i)] · ([(1-r_{ϑ-1}) · U₀] + [r_{ϑ-1} · U₁])

If $(X_i)_{1 \leq i \leq m}$ is a $\tau$-minimizer, then $\sum_{i=1}^m d[X_i; X^0] \leq \frac{2m}{\eta} d[X^0; X^0]$.

For a multilinear t (each variable has degreeOf ≤ 1), substituting t into a univariate Q : L[X] via Polynomial.aeval yields a multivariate polynomial whose degree in each variable is bounded by Q.natDegree. Used by the structured sumcheck to bound the degree of Q(witness) in the round polynomial H = P · Q(t).

The per-variable / prismalinear degree-survival lemma: if a polynomial respects a per-variable degree bound b : Fin ℓ → ℕ, then fixing the first v variables to scalars produces a polynomial whose surviving Fin (ℓ-v) variables respect b restricted to their original suffix indices. Needed for SWIRL-style sumchecks where the multiplier has degree `|D...

Renaming by an equivalence e : σ ≃ τ transports a per-variable degree bound b to the pulled-back bound b ∘ e.symm on the target.

Currying via sumAlgEquiv preserves the per-variable bound on the outer (S₁) coordinates restricted to Sum.inl.