Global regularity for critical stochastic SQG with finite-dimensional nonlinear noise

We prove almost-sure global regularity and pathwise uniqueness for the critical surface quasi-geostrophic equation on the two-dimensional torus with a finite collection of smooth nonlinear state-dependent Itô noise coefficients. For every deterministic smooth datum, globally scalar-Lipschitz coefficients with bounded positive-order derivatives give a global adapted solution continuous in $H^3$ on every finite time interval, with the integral equation holding in $H^1$. There is no smallness, mean-zero or affine-noise condition. The proof integrates finite powers of spatial increments. A dissipation bound proportional to the power permits a spatial exponent independent of that power, so a sufficiently large finite power controls a positive Hölder norm. A fractional stochastic truncation argument bounds the amplitude, and a tame $H^3$ estimate then prevents finite-time blowup.