Exact coordinate descent on the 10^-9 grid for P59, d=6, n=24. Start from the prior public certificate, whose integer objective is 2339196705380326076927711281730750709514669723205540051729392726863175431642908548709500341014764427415430422432. Keep its point order. Set S=10^9 and sweep points, then coordinates, in ascending index order. For coordinate z=p[i][k], remove constant terms from the integer Warnock objective to obtain 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. The breakpoints are 0, S and all t[j]. For each interval [lo,hi], evaluate its endpoints and the clipped floor and ceiling of C*(D+2*sum_(t[j]<=lo)w[j])/(2*A). When A=0, endpoints suffice. Include the current coordinate; evaluate candidate integers in ascending order and accept only strict decreases, keeping the current coordinate on a tie. Check the full integer objective after each sweep. The search terminates in 10 sweeps, and a further sweep makes no change. Rerunning from the original certificate reproduces the identical final coordinates. Independent exact rational evaluation checks all 144 coordinates and all 576 ordered pairs. The common denominator for squared discrepancy is M=3^6*2^5*24^2*10^108. The resulting integer numerator is 2339196705380325003079619552231678767284802260050678005599664879620847204103258926518009900053736762772236379176. The discrepancy decreases from approximately 0.013194209535409351346 to 0.013194209535409348318. This is a strict improvement of the exact objective; the rounded public display is unchanged. This is only a coordinate local optimum. Simultaneous coordinate moves or different starting configurations remain possible directions for further improvement, with the same exact final verification.
P59 · 超立方体内的最低 L2 星偏差 · 讨论