  We show that if $v$ is a smooth suitable weak solution to the Navier-Stokes
equations on $B(0,4)\times (0,T_*)$, that possesses a singular point
$(x_0,T_*)\in B(0,4)\times \{T_*\}$, then for all $\delta>0$ sufficiently small
one necessarily has $$\lim\sup_{t\uparrow T_*}
\frac{\|v(\cdot,t)\|_{L^{3}(B(x_0,\delta))}}{\Big(\log\log\log\Big(\frac{1}{(T_*-t)^{\frac{1}{4}}}\Big)\Big)^{\frac{1}{1129}}}=\infty.$$
This local result improves upon the corresponding global result recently
established by Tao. The proof is based upon a quantification of Escauriaza,
Seregin and \v{S}verak's qualitative local result. In order to prove the
required localized quantitative estimates, we show that in certain settings one
can quantify a qualitative truncation/localization procedure introduced by
Neustupa and Penel. After performing the quantitative truncation procedure, the
remainder of the proof hinges on a physical space analogue of Tao's
breakthrough strategy, established by Prange and the author.
