All data in the Sato-Tate group database was computed using rigorous algorithms that do not depend on any unproved hypotheses.
The completeness of the list of 52 rational Sato-Tate groups of weight 1 and degree 4 in the database is the main theorem of [arXiv:1110.6638, 10.1112/S0010437X12000279, MR:2982436]. As noted in this paper, there are 3 additional subgroups of $\mathrm{USp}(4)$ that satisfy the Sato-Tate axioms in weight 1 but do not arise as Sato--Tate groups of abelian surfaces; these are not included in the database.