Axisymmetric Navier--Stokes with Swirl: Regularity and Blow-Up Constraints

For smooth finite-energy axisymmetric Navier–Stokes solutions with swirl, we give a continuation criterion in terms of the signed cumulative work of the actual post-cut source-generated meridional current, with quartic viscous dissipation counted once. In the stated strong Sobolev class the criterion is necessary and sufficient and requires neither a global circulation bound nor spatial moments. We also prove cumulative inward-compression criteria, collect quantitative necessary conditions at a first singular time, and give a complete-history mixed-residual test for noncontinuation. Auxiliary results quantify quartic dissipation, energy-funded source bounds, and causal source–receiver identities. An explicit smooth whole-space datum has positive newborn source production.