Quadrupole-Depletion Closure: Scale-Invariant Gate and Rigidity for 3D Incompressible Navier--Stokes Global Regularity

This record deposits a closure manuscript for the 3D incompressible Navier--Stokes global regularity problem on T³ (and, with standard decay assumptions, on R³). The argument is built around the critical L³ identity and a structural property of the pressure Hessian operator ∂ⱼ∂ₖΔ⁻¹. Navier--Stokes is\[∂_t v + (v·∇)v = -∇ p + ν Δ v, ∇· v = 0,\]with pressure determined by\[-Δ p = ∂_j∂_k(v_j v_k), p = R_jR_k(v_j v_k),\]where Rⱼ are the Riesz transforms. Multiplying the equation by $|v|v$ yields the critical identity\[d/dt\|v(t)\|_3^3 + 3ν D_3(v(t)) = -3∫ |v|\,v·∇ p\,dx,\]where\[D_3(v) = ∫(|v|\,|∇ v|^2 + (v·∇ v)^2/|v|)dx.\]Global regularity follows if the pressure work term is strictly dominated by dissipation, i.e.\[|∫ |v|\,v·∇ p\,dx|≤ c\,D_3(v) c<ν,\]because then \|v(t)\|₃ is nonincreasing and the endpoint criterion v∈ L^∞ₜ L³ₓ prevents blow-up. The core mechanism is a commutator formulation of pressure work together with an exact quadrupole cancellation. The physical-space principal value kernel of ∂ⱼ∂ₖΔ⁻¹ is, up to a universal constant,\[Kⱼₖ(z)=p.v.\,{3 z_j z_k-δⱼₖ}{4π|z|^3}, z=z/|z|.\]Its angular factor is traceless symmetric and lies purely in the spherical harmonic channel =2, hence it has zero spherical mean. In a symmetric two-point increment representation, this rigid =2 structure forces an angular-derivative gain that cancels the borderline radial scale weight $dr/r$ in the near field. The resulting log-free square-function estimate reduces the global pressure obstruction to a single scale-invariant depletion functional built from tangential speed-gradient energy:\[Θ(v)=y^30<ρ≤ L1/ρ∫B(y,ρ){|P_{{y-x}}∇|v|(x)|^2}{|v(x)|}\,dx,\]where P_y-x denotes orthogonal projection onto the tangent plane of the sphere centered at y through x. The quadrupole commutator bound takes the form\[|∫ |v|\,v·∇ p\,dx|≤ √K_0\,Θ(v)\,D_3(v),\]with K₀ universal. Therefore, whenever Θ(v)<ν²/K₀, the strict domination c<ν holds and \|v(t)\|₃ is monotone. The manuscript then develops the closure needed to exclude a first singular time. A localized speed-entropy identity is derived from the speed equation for $u=|v|$ and the entropy density ulog(u/r₀)-u, producing the coercive term\[ν∫ φ^2\,|∇ u|^2/u\,dx\]exactly, for a cutoff φ adapted to a ball B(y,ρ). All far-field annulus contributions and cutoff commutators are isolated into an explicit tail module, bounded by a small multiple of the coercive term plus an explicit local budget (including a controlled outer logarithm 1+log(L/ρ)). The key new step is a scale-invariant gate: if the depletion threshold is crossed at some (t₀,y,ρ), then an associated Reynolds-type calibration parameter satisfies a universal nondegeneracy bound, and a quantitative dissipation payment must occur on the parabolic time window of size ρ²/ν. Concretely, if\[1/ρ∫B(y,ρ){|P_{{y-x}}∇ u(x,t_0)|^2}{u(x,t_0)}\,dx ≥ ν^2/K_0,\]then there exist universal constants η>0 and c_*>0 such that\[∫t_0t_0+c_*ρ^2/ν∫B(y,2ρ)|∇ v(x,t)|^2\,dx\,dt≥η\,ν∫B(y,ρ)|∇ u(x,t_0)|^2/u(x,t_0)\,dx.\]Finally, a rigidity alternative is proved: near a putative first singular time T^*, either there are enough threshold crossings to force infinite total dissipation (contradicting the finite dissipation budget on finite time intervals for suitable solutions), or else the threshold set has zero measure near T^*, implying Θ(v(t))<ν²/K₀ almost everywhere near T^* and therefore v∈ L^∞ₜ L³ₓ there, which precludes blow-up by the endpoint criterion. This closes the global regularity argument within the quadrupole-depletion framework. This deposit is intended to be read as the closure companion to the earlier reduction manuscript (v1.1): the first paper establishes the quadrupole commutator reduction and the Θ-control framework, while this closure manuscript supplies the scale-invariant gate and the rigidity argument that excludes a first singular time.