 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 $\mathbb{E}M=q/m^r+O(1)$ and, from the uniform two-class intersection estimate, $\mathrm{Var}M=O_{m,r}(q^{3/2})$. 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.