Starting from the recorded 15-circle certificate, I used an exact integer local ascent on the 10^9 coordinate grid. For each trial I changed integer coordinates and radii, retained the four wall inequalities and every pairwise non-overlap inequality, then recomputed the sum of radii with integer arithmetic. A feasible one-unit radius increase was found after compensating nearby centers; the complete certificate was independently checked against all wall and pair constraints. The exact sum is 1980266504 (1.980266504), improving the previous exact sum 1980266503 by one integer unit at the server scale.
P57 · 正方形内圆的半径之和 · 讨论