Phi-Renormalization for Axisymmetric-with-Swirl Navier–Stokes: A Conditional Reduction of Global Regularity

We study axisymmetric-with-swirl solutions of the 3D incompressible Navier–Stokes equations, where the 1/r⁴ centrifugal forcing on the axis r=0 obstructs standard energy methods and classically requires weighted Hardy inequalities. Writing Φ = u_θ/r, the renormalization identity r⁻⁴∂_z(Γ²) = ∂z(Φ²) — a direct consequence of Γ = r²Φ — relocates the axis difficulty out of the swirl-vorticity forcing and into the advection coefficient u^r/r, with no Hardy inequality. For the Q₁-augmented (linear fractional hyperdissipative) system this yields global C^∞ regularity at each fixed ε>0 via the Lions threshold. The global regularity of the axisymmetric-with-swirl class is thereby reduced to a single explicit, conditional criterion: ε-independence of the constant C(ε) = 2·sup_t‖u^r_ε/r‖{L^∞} controlling the radial component near the axis. The paper makes no claim of unconditional global regularity; it isolates and states precisely the one remaining obstruction.