Renormalized and Enhanced Formulation of Critical Surface Quasi-Geostrophic Dynamics (Part II)

Part I shows that critical SQG does not lie in the parabolic unique-continuation universality class: the bilinear couplingf⋅R⊥wf\cdot R^\perp wf⋅R⊥w fails at the endpointH˙−1/2×H˙1/2→H˙−1/2\dot H^{-1/2}\times \dot H^{1/2}\to \dot H^{-1/2}H˙−1/2×H˙1/2→H˙−1/2due to a logarithmically divergent low–high paraproduct. We prove that this obstruction is canonical and renormalizable. The divergent part is given by a universal, scheme-independent symbol, and subtraction of a canonical counterterm C(∇θ,θ)C(\nabla\theta,\theta)C(∇θ,θ) yields a bounded renormalized productBren:H˙−1/2×H˙1/2→H˙−1/2B_{\mathrm{ren}}:\dot H^{-1/2}\times \dot H^{1/2}\to \dot H^{-1/2}Bren:H˙−1/2×H˙1/2→H˙−1/2. We define the canonical second-order distribution Ξ=lim⁡ε→0(∇θε⋅R⊥θε−C(∇θε,θε)),\Xi=\lim_{\varepsilon\to0}\big(\nabla\theta_\varepsilon\cdot R^\perp\theta_\varepsilon - C(\nabla\theta_\varepsilon,\theta_\varepsilon)\big),Ξ=ε→0lim(∇θε⋅R⊥θε−C(∇θε,θε)), prove its mollifier-independent existence, and show that the critical SQG dynamics close in the enhanced phase space (θ,Ξ)(\theta,\Xi)(θ,Ξ), in direct analogy with Lévy area, Wick products, and regularity structures. This yields the first fully explicit, deterministic, scheme-independent renormalization and enhancement for a critical nonlocal active scalar. This is Part II of a two-part series.Companion paper (Part I): Universality Boundary for Critical SQG via Paraproduct Obstruction — DOI: 10.5281/zenodo.18405432