Logarithmic Hausdorff Measure of the Singular Set for Navier-Stokes Solutions in L^ₜ L^3, ₓ

We consider suitable weak solutions of the three-dimensional incompressible Navier--Stokes equations under the critical scaling assumption \ (u L^ (0, T; L^3, (R³) ) \). We prove a refined \ (\) -regularity criterion with a logarithmic threshold and use it to show that the singular set \ (S\) has finite logarithmic Hausdorff measure with respect to the gauge \ (h_ (t) = t| t|^-\), and even vanishing measure for any gauge \ (h_ (t) = t| t|^-\) with \ (> \). The exponent \ (\) depends explicitly on the weak-\ (L³\) norm \ (M = \|u\|₋^䂻 ₋^₃, ₓ\). This result provides a quantitative refinement of the classical Caffarelli--Kohn--Nirenberg partial regularity theory at the scaling-invariant Lorentz endpoint \ (L^3, \).