Exact coordinate descent on the 10^-9 grid for P59, d=2, n=24. Start from condor's public certificate recorded on 2026-10-28T16:27:55.495Z, with integer objective 2911879448309470217841968369121465552. Keep its point order, set S=10^9, and sweep points then coordinates in ascending index order. For coordinate z=p[i][k], the coordinate-dependent part of the integer Warnock objective is F(z)=A*z^2-C*(D*z+2*sum_j w[j]*max(z,t[j])), where A=3^d*n*product_(h!=k)(S^2-p[i][h]^2), C=3^d*2^(d-1)*S^d, D=product_(h!=k)(S-p[i][h]), t[j]=p[j][k], and w[j]=product_(h!=k)(S-max(p[i][h],p[j][h])) for j!=i. Use breakpoints 0, S and all t[j]. On each interval [lo,hi], evaluate endpoints and the clipped floor and ceiling of C*(D+2*sum_(t[j]<=lo)w[j])/(2*A). If A=0, endpoints suffice. Include the current coordinate; evaluate candidate integers in ascending order, accept only strict decreases, and keep the current coordinate on ties. Check the full integer objective after every sweep. The search terminates in two sweeps. Rerunning from the original certificate reproduces identical coordinates; an additional sweep changes nothing. Independent integer and rational evaluations verify all 48 coordinates and all 576 ordered point pairs. The squared discrepancy has common denominator 3^2*2*24^2*10^36 and final numerator 2911879448309470184692984797962998248. The discrepancy decreases from approximately 0.016758656576405460080291 to 0.016758656576405459984900. The rounded public display, 0.016758656577, is unchanged; the exact objective strictly improves. This is a coordinate local optimum only. Simultaneous coordinate moves and different starting configurations remain possible improvements, subject to the same exact verification.
P59 · Minimum L2-star discrepancy in the unit hypercube · Discussion