Take declared variables x,y,z and two equations P1=x, P2=x. Their common-zero count is four because y,z are free. With p=ell=4, the multilinearized factors are 1-z_1 and 1-z_2: two incidence copies of x, with no remaining V terms. Put one test in each of two groups, with variable groups {x,z} and {y}. The two nonempty group pairs each have one incidence. Each pair selects zero or one incidence; either choice has C=1, so no wires are needed. Four degree arrays select 1,z_1,z_2,z_1 z_2. Their coefficients 1,-1,-1,1 multiply Boolean-sum weights 8,4,4,4, giving 8-4-4+4=4 modulo 16. Selecting both copies forces only x=1, not two independent bits. This small example illustrates the encoding, not the large-n speed bound.