We turn to computing the exact values of $\sum\limits_{\wVec\in\pw}\sketchJParam{\sketchHashParam{\wVec}}\cdot\sketchPolarParam{\wVecPrime}.$ Starting with the term $\gIJ=\sum\limits_{\wVecPrime\in\pw}\sketchPolarParam{\wVecPrime}$, by the definition of $\sketchPolar$ and the property of associativity in addition, we can break the sum into
Setting the terms to $T_1=\sum_{\substack{\wVecPrime\in\pw\st\\
\sketchPolarParam{\wVecPrime} = 0}} 1$ and $T_2 = \sum_{\substack{\wVecPrime\in\pw\st\\
\sketchPolarParam{\wVecPrime} = 1}} -1$ and fixing $\buck$ to a specific value, gives a system of linear equations for each term. It is a known result for a consistent multiplication that the number of solutions are $| \kDom |^{\numTup - rank(\matrixH')}$. This gives us an exact calculation for both terms,
\begin{align*}
T_1 \in\{0, 2^{\numTup - rank(\matrixH')}\},\\
T_1 \in\{0, 2^{\numTup - rank(\matrixH')}\}.
\end{align*}
\subsection{Algorithm for $\gIJ$}
\begin{algorithmic}
\If{$\matrixH' \cdot\wVec= j^{(0)}$ is consistent}
\If{$\matrixH' \cdot\wVec= j^{(1)}$ is consistent}