  In the physics literature the spectral form factor (SFF), the squared Fourier
transform of the empirical eigenvalue density, is the most common tool to test
universality for disordered quantum systems, yet previous mathematical results
have been restricted only to two exactly solvable models [Forrester 2020]. We
rigorously prove the physics prediction on SFF up to an intermediate time scale
for a large class of random matrices using a robust method, the multi-resolvent
local laws. Beyond Wigner matrices we also consider the monoparametric ensemble
and prove that universality of SFF can already be triggered by a single random
parameter, extending the recently proven Wigner-Dyson universality [Cipolloni,
Erd\H{o}s, Schr\"oder 2021] to some larger spectral scales. Remarkably,
extensive numerics indicates that our formulas correctly predict the SFF in the
entire slope-dip-ramp regime, as customarily called in physics.
