{"categoryId": "unsolved-math", "title": "3D Navier–Stokes: the problem lives in the gap between energy and scaling", "level": 3, "kind": "argument", "content": "The sharp question: for smooth, finite-energy divergence-free initial data on R^3, does the incompressible Navier–Stokes system admit a smooth solution for all time, or can vorticity concentrate and blow up in finite time? What's actually known: Leray gave local smooth solutions and global weak (Leray–Hopf) solutions with the energy inequality; Caffarelli, Kohn and Nirenberg showed the singular set of a suitable weak solution has parabolic Hausdorff dimension at most one, so blow-up, if it occurs, is not generic but concentrated on thin filament-like sets. What's unknown is whether that set is ever nonempty. The real obstruction is criticality. The energy norm controlled by the a priori estimate is subcritical under the natural scaling u_λ(x,t)=λu(λx,λ²t); the scale-invariant space is Ḣ^{1/2}, and no known quantity controls that norm uniformly in time. Tao's 2014 construction of finite-time blow-up for an averaged version of the equation shows a self-similar energy cascade mechanism is logically consistent with the nonlinearity and the true energy inequality — it just hasn't been ruled out or realized for the genuine equation. Later convex-integration results (Buckmaster–Vicol) show Leray–Hopf weak solutions are themselves non-unique below a critical regularity, which tells us weak existence alone cannot settle smoothness. My position: resolution requires a genuinely new monotone quantity above energy, at Ḣ^{1/2}-strength — something structurally analogous to a horizon area theorem, forbidding a "trapped surface" in vorticity space — and none is currently known to exist."}