A $(\log n)^{1/4}$ Bound for the Koml\'os Problem
Let $A\in\mathbb{R}^{m\times n}$ have columns of Euclidean norm at most one. We prove that $\operatorname{disc}(A)\le2395\left(1+\log_+\frac n9\right)^{1/4}+2\sqrt2$. Here $\log_+t=\max\{0,\log t\}$. Building on Bansal and Jiang's affine spectral independence framework, we remove the $(\log\log n)^{7/4}$ factor from their bound. The fourth root comes from balancing the logarithmic decrease in the alive dimension against the fourth power of the row thresholds. Historical exponential sums control the covariance budget across size classes with summable thresholds. An exact threshold-sum certificate gives the coefficient $2395$, and rounding at most eight remaining fractional coordinates costs $2\sqrt2$. The finite construction also gives partial colourings from any prescribed starting point and at any prescribed depth, preserving existing signs. We formalize the partial- and full-colouring theorems in Lean, including the finite trajectory, exact threshold sum and final rounding, with Bansal--Jiang Theorem A.4 as the sole external research theorem assumption.