Global well-posedness of the dynamical sine-Gordon model up to $6\pi$

We prove the global well-posedness of the dynamical sine-Gordon model up to the third threshold, i.e., for parameters ${\beta }^2<6\pi$. The key novelty in our approach is the introduction of the so-called resonant equation, whose solution is entirely deterministic and completely captures the size of the solution to the dynamical sine-Gordon model. The probabilistic fluctuations in the dynamical sine-Gordon model are then controlled using uniform estimates for modified stochastic objects.