Universality Boundary for Critical SQG via Paraproduct Obstruction (Part I)

We show that the critical surface quasi-geostrophic (SQG) equation cannot lie in the parabolic unique-continuation (UC) universality class: the bilinear coupling B(f,w)=f⋅R⊥wB(f,w)=f\cdot R^\perp wB(f,w)=f⋅R⊥w governing the linearized evolution fails at the critical endpoint mapping H˙−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˙−1/2. Using Bony’s paraproduct decomposition, we prove that this failure is entirely localized to the low–high channel Tf(R⊥w)T_f(R^\perp w)Tf(R⊥w), which exhibits a marginal logarithmic divergence, while the high–low and resonant channels remain bounded. We construct an explicit normalized sequence realizing the (log⁡N)1/2(\log N)^{1/2}(logN)1/2 blow-up and prove a no-go theorem: any closure mechanism relying on the endpoint bilinear estimate (such as Carleman or frequency-rigidity arguments) cannot apply at criticality. This identifies a canonical, scheme-independent marginal obstruction and places critical SQG outside the parabolic UC universality class. This paper is Part I of a two-part series.The companion paper, Part II, shows that the logarithmic obstruction is canonical and renormalizable, leading to a bounded renormalized bilinear form and a deterministic enhanced formulation of SQG in an enlarged state space (θ,Ξ)(\theta,\Xi)(θ,Ξ). Part II (companion): Renormalized and Enhanced Formulation of Critical Surface Quasi-Geostrophic Dynamics (Part II) — DOI: (to be added after publication).