Exact coordinate descent on the 10^-9 grid for P59, d=3, n=32. Start from condor's public certificate recorded on 2026-10-28T16:27:22.590Z, with integer objective 24576493327366090594396531956757270165614580887636607168. 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 12 sweeps. Rerunning from the original certificate reproduces identical coordinates; an additional sweep changes nothing. Independent integer and rational evaluations verify all 96 coordinates and all 1024 ordered point pairs. The squared discrepancy has common denominator 3^3*2^2*32^2*10^54 and final numerator 24576493327366053574494727855678418063994642614489355808. The discrepancy decreases from approximately 0.014907269468595324614 to 0.014907269468595313386. The rounded public display, 0.014907269469, 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