Lemma 29 ([Polynomial avoidance]). Use independent edge draws as in Lemma 28 ([Finite product sampler]). For a fixed base walk, reuse its one voltage variable on every occurrence of an unoriented edge and use its inverse in the reverse …

The polynomial estimate is the maximal-atom version of the usual polynomial zero bound ; we include its short induction. If independent real-valued variables have atoms at most κ , a nonzero polynomial of total degree d vanishes with probability at most dκ . Induct on the number of variables. If its degree in the last variable is s, its leading coefficient is a nonzero polynomial in the preceding variables of degree at most d-s . By induction that coefficient vanishes with probability at most (d-s)κ ; otherwise the one-variable polynomial has at most s roots and costs at most sκ . Variables absent from the polynomial are discarded. A nonzero constant has zero vanishing probability, starting the induction. By the normal-form multiplication law just displayed, the horizontal coordinates of any word are integral linear combinations of the edge coordinates n_e,l_e , with coefficients equal to signed occurrence counts. If any such count is nonzero, one horizontal polynomial is nonzero. If all counts vanish, every central coordinate c_e cancels, and the central coordinate of the word is a polynomial of degree at most two in the n_e,l_e . Were that polynomial identically zero, the word would vanish for every assignment in , contrary to the hypothesis. Thus a nonzero polynomial of degree at most two is tested on independent horizontal variables. Their maximal atoms are bounded by C_b2^ -50H_ , since J_i/N_i=2^ -150H_i . Apply the polynomial bound. No conditioning on the central coordinate or on the matrix weight is performed. This proves the assertion also for an edge whose realized weight is zero.