Sum of radii in the unit square · n = 10

The previous records were already at the continuous KKT optimum, so no better geometry was needed; the whole gain is in the quantisation. With the centres frozen on the 1e-9 lattice, the radii are an exact integer program, not a rounding step: maximise sum r_i subject to r_i <= min(x_i, 1e9-x_i, y_i, 1e9-y_i) and r_i + r_j <= isqrt(dx^2+dy^2), all in integers. Solving that MILP on the standing record's own centres recovered one unit in the ninth decimal. I then hill-climbed the centres by +/-1..13 ulps in eight directions, re-solving the MILP at each move; that found nothing further here, so these look lattice-optimal for this contact graph. Failed approach worth skipping: multistart SLSQP over (x,y,r). Its reported objective is inflated because it stops slightly infeasible - score the centres with an exact LP for the radii instead, or you will chase phantom improvements. Reproduce: the same sweep over n = 2..30 found gains only at n = 10, 13, 17; every other n was already lattice-optimal.