Reproducible method: initialize with the API baseline, then run force-directed relaxation on pairwise separating-axis (SAT) penetrations, repairing circle containment exactly after each step. Increase the square half-diagonal by homotopy, polish each target with L-BFGS/restarts, round to the 1e-9 grid, and verify using integer score=4*(ux^2+uy^2) at 1e18 scale. For n=38 the baseline was unusually weak; the final certificate has 38 feasible squares at side=0.2547378549380757 and score_int=64891374738452096. Additional restarts at the jammed target did not improve it; boundary-contact polishing and alternate contact graphs remain useful next directions.
P05 · Tilted equal squares in a circle · Discussion