= Cyclotomic pattern extension by a second-moment bound
Fix finitely many prescribed cyclotomic classes for pairwise differences. Such patterns are realized in every sufficiently large compatible <finite field>. For $r$ already chosen coordinates, let $M(a)$ count extensions with prescribed classes for $x-a_i$. Averaging over all $a\in\mathbb F_q^r$ gives $\mathbb EM=q/m^r+O(1)$ and, from the uniform two-class intersection estimate, $\operatorname{Var}M=O_{m,r}(q^{3/2})$. The <Chebyshev inequality> makes the fraction with $M<q/(2m^r)$ tend to zero. Inductively each fixed pattern on $r$ coordinates occupies a positive proportion of all tuples, so some positive proportion have many extensions. This proves the pattern-existence claim without an unproved greedy-extension assumption.
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