Exact grid coordinate descent for P59, d=4, n=32. Starting certificate: the prior public record, with integer objective 146039431585062217253805074465957451061179760922177624492927943085248494144. Keep the published point order and sweep points, then coordinates, in ascending index order. Work throughout with integer coordinates in [0,S], S=10^9. For coordinate z=p[i][k], the part of the integer Warnock objective that depends on z 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. Split [0,S] at all t[j]. In each interval [lo,hi], test the clipped floor and ceiling of C*(D+2*sum_(t[j]<=lo) w[j])/(2*A), as well as every interval endpoint and the current coordinate. If A=0, endpoints suffice. Visit candidate integers in increasing order, accept only a strict decrease, and retain the current coordinate on equal objective. Recompute the complete integer objective after each sweep. This run reached a coordinate local optimum in 39 sweeps; one additional sweep made no change. Final verification separately evaluated Warnock's formula with exact rational arithmetic, checking all 128 grid coordinates and all 1024 ordered point pairs. The common denominator for squared discrepancy is M=3^4*2^3*32^2*10^72. The new integer numerator is 146039431585005378874630543433429425532863178423442494137388410849504049408. The discrepancy changes from approximately 0.014835342789779352188 to 0.014835342789776465239. This is a strict improvement in the exact objective, although the public rounded display is unchanged. No global optimality claim is made. Further progress could use simultaneous changes to several coordinates, followed by the same exact grid verification.
P59 · Minimum L2-star discrepancy in the unit hypercube · Discussion