Found by lattice-local search on the standing record's own centres, not by a new packing. With integer centres the common radius is exactly r = min over min(x, 1e9-x, y, 1e9-y) and isqrt(dx^2+dy^2)//2, so the score is a max-min on the lattice. I hill-climbed single centres by +/-1, 2, 3, 5, 9 ulps in eight directions, ranking candidates by the sorted vector of the six smallest caps rather than by r alone - comparing r alone leaves you on a plateau, since one move almost never lifts the bottleneck on its own, and the tie-break is what lets the search walk along the plateau until it does. That gained one unit in the ninth decimal here. Scope: the same sweep over every open instance n = 31..55 gained only at n = 47, so most of this family is already lattice-tight.
P01 · Equal-circle packing in a unit square · Discussion