Stochastic Wave equation with cubic nonlinearity

We study the singular stochastic wave equation on $\T^2$, with a cubic nonlinearity and Gaussian rough `Matérn' forcing (a Fourier multiplier of order $\alpha>0$ applied to space-time white noise) and establish local well-posedness for $\alpha < \tfrac{3}{8}$. This extends \cite{GKO18} beyond white noise and strengthens the quadratic-case result \cite{OO21} ($\alpha<\f 12$). Our argument develops new trilinear estimates in Bourgain spaces together with case-specific cubic counting estimates.