Lemma 4. More than half of the |A_0|=M^ p^3 characters are admissible.

For each nonzero s∈P , translation partitions P into p^2 cycles of length p . A character fixed by s has constant exponent on each cycle, so exactly M^ p^2 characters are fixed. A union bound therefore bounds the fraction with nontrivial stabilizer by (p^3-1)M^ p^2-p^3 . Let c_r be the coordinate generator at r . Since χ_x(c)=λ(c_ -x ) , the diagonal entries of D_λ^ -1 range independently over K^* as λ ranges over all characters. For independent variables τ_x , Cauchy--Binet gives Q(τ):=(B^ T (τ_x)B) =∑_ J⊆P; |J|=h (B_J)^2∏_ x∈J τ_x. Here B_J consists of the indicated rows in their fixed order. Some maximal minor is nonzero because B has full column rank. The monomials for distinct J are distinct, so Q is nonzero of total degree h , even though the unweighted Gram matrix could be singular. We use the polynomial grid estimate of Schwartz~; see also the related identity-testing method of Zippel~. A nonzero polynomial of total degree at most a vanishes on at most an a/M fraction of a product grid whose coordinate sets each have size M . To prove this, induct on the number of variables. If the degree in the last variable is b , its leading coefficient has degree at most a-b . It vanishes at at most an (a-b)/M fraction of the earlier choices. For each remaining choice, the last-variable polynomial has at most b roots, contributing at most b/M . The sum proves the bound; the constant case starts the induction. Applying the estimate on (K^*)^ p^3 to Equation, the singular-pairing fraction is at most h/M . Both excluded fractions are strictly smaller than 1/4 : hM≤(p^3)/(p^4-1)<14, (p^3-1)M^ -p^2(p-1) ≤(p^3-1)/(M)<14. The estimates use p>600 ; for example, p^4-1-4p^3=p^3(p-4)-1>0 . Their union therefore excludes fewer than half of the characters.