Quantitative classification of potential Navier-Stokes singularities beyond the blow-up time

In \cite{hou}, Hou gave a compelling numerical candidate for a singular solution of the 3D Navier-Stokes equations. We pioneer classifications of potentially singular solutions, motivated by the issue of investigating the viability of numerical candidates.For approximately axisymmetric initial data, we give the first quantitative classification of potentially singular solutions at \textit{any} given time in the region of potential blow-up times. Moreover, the quantitative bounds in the vicinity of any potential blow-up time are in principle amenable to numerical testing. To achieve this, we establish improved quantitative regions of regularity for approximately axisymmetric initial data, which may be of independent interest. Together with improved quantitative energy estimates from \cite{TB24}, this allows us to get a quantitative lower bound in the vicinity of a blow-up time by implementing the strategy of \cite{BP21}, which is a physical space analogue of Tao's strategy \cite{Ta21} for producing quantitative estimates for critically bounded solutions. To obtain a quantitative lower bound on the solution at any time in the region of potential blow-up times, we recursively apply quantitative Carleman inequality arguments from \cite{Ta21}. This necessitates careful bookkeeping to avoid exponential losses and to ensure that all forward-in-time iterations of (localized) vorticity concentration remain within the region of quantitative regularity of the solution.