Starting with the latter term $\gIJ=\sum\limits_{\wVecPrime\in\pw}\sketchPolarParam{\wVecPrime}$, by the definition of the image of $\sketchPolar$ and the property of associativity in addition, we can break the sum into
Setting the terms to $T_1=\sum\limits_{\substack{\wVecPrime\in\pw\st\\
\sketchPolarParam{\wVecPrime} = 0}} 1$ and $T_2 = \sum\limits_{\substack{\wVecPrime\in\pw\st\\
\sketchPolarParam{\wVecPrime} = 1}} -1$ and fixing $\buck$ to a specific value, gives a system of linear equations for each term. It is a known result for a consistent matrix multiplication that the number of solutions are $| \kDom |^{\numTup - rank(\matrixH')}$, where $\kDom$ is the set being considered. This gives us an exact calculation for both terms,
where the notation $\jpbit{y}$ denotes the polarity bit $\lenB$ value of the $\buck$ bucket identifier, specifically $\buck(b)$, such that $\buck(b)\in\{0, 1\}$.
Setting $T_3=\sum\limits_{\wVec\in\pw\st\sketchPolarParam{\wVec}=0}\kMapParam{\wVec}$, $T_4=\sum\limits_{\wVec\in\pw\st\sketchPolarParam{\wVec}=1}\kMapParam{\wVec}$ gives an exact calculation for each term given a fixed $\buck$: