Theta bases and log Gromov-Witten invariants of cluster varieties

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