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 R^ wf⋅R⊥w fails at the endpointH˙−1/2×H˙1/2→H˙−1/2 H^-1/2 H^1/2 H^-1/2H˙−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 (, ) C (∇θ, θ) yields a bounded renormalized productBren: H˙−1/2×H˙1/2→H˙−1/2Bₑ₄₍: H^-1/2 H^1/2 H^-1/2Bren: 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) ), =₀ (_ R^_ - C (_, _) ), 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) (, ) (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