Fix k,c and original n>=20. Start with a<=cn degree-k equations, each listed with at most ceil(log2 n)^c monomials. After high-incidence branching and F_2 normalization, let I_i be the free-variable scope of test i. Keep every free variable declared; discard zero equations and return zero for a constant-one equation. If none remain, count all free assignments directly. Give each pair (i,v) a separate coordinate z_iv. Form its integer lift L_i and shared-truncation factor B_i(z_i,V), over B_(n+1,n+1) using ORIGINAL n. Multilinearize each V coefficient to obtain Bbar_i; define F=product_i Bbar_i in that ring. Different tests use disjoint coordinates. Only later set all copies z_iv equal to the same original bit x_v.