Logarithmic Hausdorff Measure of the Singular Set for Navier-Stokes Solutions in $L^\infty_t L^{3,\infty}_x$

We consider suitable weak solutions of the three-dimensional incompressible Navier--Stokes equations under the critical scaling assumption \(u \in L^\infty(0,T; L^{3,\infty}(\mathbb{R}^3))\). We prove a refined \(\varepsilon\)-regularity criterion with a logarithmic threshold and use it to show that the singular set \(\mathcal{S}\) has finite logarithmic Hausdorff measure with respect to the gauge \(h_\theta(t) = t|\log t|^{-\theta}\), and even vanishing measure for any gauge \(h_\gamma(t) = t|\log t|^{-\gamma}\) with \(\gamma > \theta\). The exponent \(\theta\) depends explicitly on the weak-\(L^3\) norm \(M = \|u\|_{L^\infty_t L^{3,\infty}_x}\). This result provides a quantitative refinement of the classical Caffarelli--Kohn--Nirenberg partial regularity theory at the scaling-invariant Lorentz endpoint \(L^{3,\infty}\).