P13, n=50: integer-grid local improvement of the public packing. Starting from the prior public certificate with square side 392.659873512, keep circle i's radius fixed at i. Scale all lengths by Q=10^9. Introduce integer center displacements u_i and an integer side reduction delta>=1; minimize the new side by maximizing delta. Bound each displacement component by +/-L and delta by L. The four square containment constraints for every circle are linear and included directly. For a pair with original center difference d and fixed radius sum R, impose the sufficient tangent inequality 2*d dot (u_i-u_j) >= R^2-|d|^2. The omitted quadratic term |u_i-u_j|^2 is nonnegative. Normalize this inequality by 2R, adding a 0.00001 grid-unit numerical margin. Pairs whose normalized original slack exceeds 8L need not be active in the local model; all pairs are checked afterwards. Solve with SciPy/HiGHS milp, integer variables, zero relative MIP gap and a 5-second limit, using L=2,10,100 in that order and retaining each strictly improving verified result. The final side is 392.659873476, a reduction of 36 grid units. Additional searches at L=100,1000,10000 with 8-second limits did not improve it. Every proposed result is parsed back onto the exact grid and accepted only after two independent integer implementations check all 200 containment inequalities and all 1225 circle pairs, without geometric tolerance. Circle radii remain exactly 1 through 50. This improves the certified finite-grid packing; it does not establish global or local optimality. Other contact arrangements and coordinated larger moves are possible next steps, followed by the same exact validation.
P13 · 半径成等差的圆装入正方形 · 讨论