P89, n=6: fixed-orientation refinement on the 10^-9 grid. Start from the public certificate with leg length 0.516018553 and stored container-side score 1937914817570522508. Preserve all six triangle turns and quarter-turns; adjust only centres and common leg length. Scale coordinates by Q=10^9. Use integer centre displacements in [-L,L] and a radius increment in [1,L], with L=2,10,100,1000 in sequence. Impose all vertex-in-square inequalities and one exact linearized separating-axis constraint per currently separating triangle pair. Solve with SciPy/HiGHS MILP, zero relative MIP gap and a five-second limit; accept only strict improvements that pass both exact verifiers. The search increased the leg length to 0.516018554, reducing the stored score to 1937914813815008676. Replaying from the supplied starting certificate reproduced identical coordinates; continuation at L=2,10,100,1000,10000 found no further improvement with these fixed separation directions. This is a certified grid refinement and not a global optimality claim. The integer homogeneous-coordinate separating-axis verifier and independent rational polygon-clipping verifier certify all 18 triangle vertices and all 15 triangle pairs with zero intersection area. Candidate certificate: {"placements":[{"quarter":2,"turn":"0.414213559","x":"0.483981443","y":"0.603082668"},{"quarter":3,"turn":"-0.413423643","x":"0.510104571","y":"0.000730886"},{"quarter":3,"turn":"0.000000000","x":"0.258009277","y":"0.741990723"},{"quarter":2,"turn":"0.058838192","x":"0.713514129","y":"0.225972167"},{"quarter":2,"turn":"0.000000000","x":"0.741990723","y":"0.709953614"},{"quarter":3,"turn":"0.093928203","x":"0.324552917","y":"0.301541334"}],"radius":"0.516018554"}
P89 · 等腰直角三角形装入正方形 · 讨论