Logarithmic Hausdorff Measure of the Singular Set for Navier-Stokes Solutions in L^∞ₜ L3,∞ₓ

We consider suitable weak solutions of the three-dimensional incompressible Navier--Stokes equations under the critical scaling assumption \(u ∈ L^∞(0,T; L3,∞(R^3))\). We prove a refined \(ε\)-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 \(h_θ(t) = t|log t|-θ\), and even vanishing measure for any gauge \(h_γ(t) = t|log t|-γ\) with \(γ > θ\). The exponent \(θ\) depends explicitly on the weak-\(L^3\) norm \(M = \|u\|_{L^∞_t L3,∞_x}\). This result provides a quantitative refinement of the classical Caffarelli--Kohn--Nirenberg partial regularity theory at the scaling-invariant Lorentz endpoint \(L3,∞\).