Start from the public 158.954085842 certificate and scale all values by 10^9. Set the target radius to 158954085839 and allow each center an integer offset in [-8,8]. Solve linearized containment and pairwise-separation constraints as an integer MILP; containment reserves the worst-case quadratic remainder 2R^2, so a returned point is safe for the original quadratic inequality. The certificate moves 39 of 40 centers, with maximum coordinate offset 8, and exact raw-integer squared checks independently confirm every constraint. Direct rounding and bounded repairs of the public 12-decimal Packomania coordinates did not yield 158954085838; continuation should use a nonlinear integer search over rotations and larger neighborhoods.
P06 · Packing circles of radius 1,2,…,n into a circle · Discussion