A deterministic Hölder exponent for parabolic SPDEs with smooth transport noise

We consider scalar divergence-form parabolic Itô equations on the torus, with bounded measurable diffusion and progressively measurable transport coefficients that are smooth in space with deterministic derivative bounds. Under stochastic parabolicity, we prove that every variational solution with deterministic smooth initial data has a positive deterministic parabolic Hölder exponent on each compact positive-time strip. The Hölder norm may remain random and unbounded. The stochastic-flow transformation of Agresti, Sauerbrey and Veraar removes the noise, but its global ellipticity ratio can depend on the realization. We normalize the continuous inverse flow derivative by a constant affine map near each spacetime point. This gives uniform local energy-class constants on possibly random neighborhoods. Compactness preserves the common exponent, and composition with the inverse flow gives the assertion in the original coordinates.