An Angularly Corrected Maximum-Vorticity Continuation Criterion for the 3D Navier–Stokes Equations

How fast can the strongest vortex in a fluid intensify? For the 3D Navier–Stokes equations, a classical estimate tracks the maximum of vorticity and bounds its growth by the local stretching rate, traditionally discarding the entire viscous term at the maximum point, because its sign is all that is easily available there. This note records that the viscous defence at a vorticity maximum in fact splits into two channels that need not be discarded together: diffusion of the peak's amplitude, and dissipation caused by spatial variation of the vorticity direction. Retaining both yields an exact identity: the right-hand growth rate of the maximum vorticity equals a "full" effective rate, and dropping only the amplitude channel yields a robust angular upper envelope. The resulting continuation criteria state that a solution stays smooth as long as the time-accumulated effective rate remains finite. The consequences are stated as a two-channel diagnostic: finite-time blow-up would require stretching to exceed the combined angular and amplitude damping, non-integrably in time, at the maximum points, forcing a very specific geometry (a maximum that is simultaneously intense, directionally underdamped, and second-order flat at the strain scale). A quasi-maximal version makes every quantity directly measurable in DNS data, and an exact Beltrami solution provides an analytic validation test. The ingredients are classical: the magnitude–direction equation is Constantin's (1994); the full rate appeared in Bustamante's 2012 workshop presentation; the maximum-set envelope is Agélas's (2019), both located by multi-model adversarial review after two independent novelty sweeps returned empty, and both verified against sources. The contribution is the assembled criterion hierarchy, the diagnostics, and the measurement layer. The note was developed under a documented adversarial review methodology (four review rounds, all identities hand-verified); no claim on the Millennium problem is made.