Fix a degree array. Each valid wire assignment chooses one incidence subset per group pair and therefore one squarefree monomial. Test-group entries multiply to its coefficient in F. Variable-group entries give its sum over the ORIGINAL free Boolean variables after all copies of each variable are identified. Invalid indices contribute zero. Every degree-at-most-K monomial belongs to exactly one array and one set of valid indices. Summing the contractions over arrays therefore equals sum_x F((z_iv=x_v),V) in the shared ring. Apply the coefficient-sum map afterward; add disjoint branch residues modulo 2^(n+1) to obtain the exact original common-zero count.