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 R^ 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. H^-1/2 H^1/2 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) Tf (R^ 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 (logN) 1/2 (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) (, ) (θ, Ξ). Part II (companion): Renormalized and Enhanced Formulation of Critical Surface Quasi-Geostrophic Dynamics (Part II) — DOI: (to be added after publication).
Joseph Scott Penman (Wed,) studied this question.