  Using heuristics from mirror symmetry, combinations of Gross, Hacking, Keel,
Kontsevich, and Siebert have given combinatorial constructions of canonical
bases of "theta functions" on the coordinate rings of various log Calabi-Yau
spaces, including cluster varieties. We prove that the theta bases for cluster
varieties are determined by certain descendant log Gromov-Witten invariants of
the symplectic leaves of the mirror/Langlands dual cluster variety, as
predicted in the Frobenius structure conjecture of Gross-Hacking-Keel. We
further show that these Gromov-Witten counts are often given by naive counts of
rational curves satisfying certain geometric conditions. As a key new technical
tool, we introduce the notion of "contractible" tropical curves when showing
that the relevant log curves are torically transverse.
