Stochastic nonlinear wave equation with rougher than white noise

We study the singular stochastic wave equation on ${\mathbb{T}}^2$, 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 $\alpha <\frac{3}{8}$. This extends [GKO18] beyond white noise and strengthens the quadratic-case result [OO21] ($\alpha <\frac{1}{2}$). Our argument develops new trilinear estimates in Bourgain spaces together with sharp, case-specific cubic counting estimates.