Small maximal Sidon sets in vector spaces of odd characteristic

For a finite abelian group G, let γ_Sidon(G) be the minimum size of an inclusion-maximal Sidon subset, with repeated summands included in the Sidon condition. For every fixed odd prime p, we prove γ_Sidon(F_p^n) = O_p((n p^n)^(1/3)). The construction adapts the random quotient-lifting strategy previously used in characteristic two. Its odd-characteristic input is the parabola P_q = {(t, t²) : t ∈ F_q}: every point outside P_q has exactly q − 1 ordered representations x + a = b + c with a, b, c ∈ P_q. After the diagonal representations are removed, the resulting witness hypergraph has maximum degree two. We may therefore choose linearly many witnesses on disjoint triples, and their lifting events are mutually independent. The same bound follows over every fixed odd prime power; for p = 3, it gives complete 2-caps of size O((n 3^n)^(1/3)).