In [Jim Posen's note](https://hackmd.io/AZ7BNUfuT9y_V39sFo8Xqw), the protocol ends with the verifier querying
$$
v_{i,j} = M_i(j || r_0, \ldots, r_{\nu -k - 1} )
$$
for all $j\in B_k$.

Using these, the verifier computes the evaluations
$$
\hat{v}_i = \hat{M}_i(r_Y, r_0,\ldots,r_{\nu -k - 1}) = \sum_{j \in D} v_{i,j} \cdot L_{j,D}(r_Y),
$$
and is able to check that $C(\hat{v}_0, \ldots, \hat{v}_{n-1})$ equals the final Sumcheck sum.

If we assume that the values $v_{i,j}$ are provided as-is, we can check that they are correct using a single query to $M_i$. Moreover, this avoids an extra invocation of the Sumcheck protocol, though it comes at the expense of more verifier work.

The verifier sends a new random element $r_e \in \mathbb{F}$ and evaluates the $2^k$ Lagrange polynomials over $B_k$ at $\vec{r}_e = ( r_e, r_e^2, \ldots, r_e^{2^{k-1}})$. The evaluations $L_{j, B_k}(\vec{r}_e)$, for each $\ j \in B_k$ can be computed using $O(k2^k)$. This allows the verifier to compute an evaluation $v_i$ of $M_i$ at $(\vec{r}_e|| r_0, \ldots, r_{\nu -k - 1})$. The following shows how the same evaluation can be derived by the verifier:

$$
\begin{aligned}
v_i
&= M_i(\vec{r}_e|| r_0, \ldots, r_{\nu -k - 1}) \\
&= \sum_{j \in B_k} M_i(j || r_0, \ldots, r_{\nu -k - 1} ) \cdot L_{j, B_k}(\vec{r}_e) \\
&= \sum_{j \in B_k} v_{i,j} \cdot L_{j, B_k}(\vec{r}_e)\\
\end{aligned}
$$

Alternatively, we can minimize the amount of verifier work by invoking a slightly more optimized Sumcheck instance.

Reusing the same random value $r_e$, we define the $k$-variate polynomial $\mathsf{pow}_{r_e}(X_0,\ldots,X_{k-1})$ such that $\mathsf{pow}_{r_e}(j) = r_e^j$ for all $j \in B_k$. The verifier computes
$$
\sigma_i = \sum_{j=0}^{2^k-1} r_e^j \cdot v_{i,j}$$

which is the claimed sum for the Sumcheck instance

$$
\mathsf{pow}_{r_e}(X_0,\ldots,X_{k-1}) \mathsf{eq}(r_0, \ldots, r_{\nu -k - 1}; X_{k}, \ldots, X_{\nu-1}) \cdot M_i(X_0,\ldots,X_{\nu-1})
$$

The following shows that the sum is correct:
$$
\begin{aligned}
\sigma_i &=
\sum_{j_1 \in B_k}\sum_{j_2 \in B_{\nu-k}} \mathsf{pow}_{r_e}(j_1) \mathsf{eq}(r_0, \ldots, r_{\nu -k - 1}; j_2) \cdot M_i(j_1 || j_2) \\
&= \sum_{j_1 \in B_k} \mathsf{pow}_{r_e}(j_1) \sum_{j_2 \in B_{\nu-k}} \mathsf{eq}(r_0, \ldots, r_{\nu -k - 1}; j_2) \cdot M_i(j_1 || j_2) \\
&= \sum_{j_1 \in B_k} \mathsf{pow}_{r_e}(j_1) \cdot M_i(j_1 || r_0, \ldots, r_{\nu -k - 1}) \\
&= \sum_{j_1 \in B_k} r_e^{j_1} \cdot M_i(j_1 || r_0, \ldots, r_{\nu -k - 1})\\
&= \sum_{j_1 \in B_k} r_e^{j_1} \cdot v_{i,j}
\end{aligned}
$$

While this introduces $\nu$ additional rounds of interaction, the verifier only performs $O(2^k + \nu)$ field operations.