Quadrupole-Depletion Route to L³ Regularity for 3D Incompressible Navier--Stokes

This upload contains a self-contained research manuscript proposing a single, unified proof program for global regularity of the three-dimensional incompressible Navier--Stokes equations, organized around the critical L³ dissipation identity and an explicit algebraic cancellation in the pressure Hessian. We consider the incompressible Navier--Stokes system (on T³ or R³ with decay) \ₜ v + (v) v = - p + v, v = 0, viscosity >0. The pressure is determined by\- p = ⱼₖ (vⱼ vₖ), = RⱼRₖ (vⱼ vₖ), Rⱼ are Riesz transforms. The starting point is the critical L³ identitydt\|v (t) \|₃³ + 3 D₃ (v (t) ) = -3 |v|\, v p\, dx, ₃ (v): = (|v|\, | v|² + (v v) ²|v|) \, dx. regularity follows if the pressure work satisfies\| |v|\, v p\, dx| c\, D₃ (v) some c0\}). localized entropy identity is derived by testing the local speed equation against a cutoff weight and (u/r₀), which produces the coercive term q exactly. A tube/ball packing mechanism is then proposed to convert largeness of at a scale into a quantitative dissipation payment on that scale (up to a slow logarithmic factor related to the viscous core scale). (B) Coherent tubes imply depletion. Under a coherent-tube regime (vorticity-direction coherence plus a circulation lower bound of the form u || on high-vorticity sets), one obtains a Carleson/Morrey bound that forces (v) to be small, giving an explicit conditional global regularity theorem with a falsifiable scale-selection rule for the threshold parameter. The manuscript concludes with three concrete remaining tasks for an unconditional proof: (i) a fully detailed localized quadrupole-commutator estimate (with tail control suitable for scale extraction) ; (ii) elimination of the remaining slow logarithm in the packing route via a scale-invariant gate in hypothetical blow-up regimes; (iii) a rigidity alternative showing that any attempt for to cross the threshold forces dissipation incompatible with global L³ / energy budgets.