P89, n=9: fixed-orientation refinement on the 10^-9 grid. Start from the public certificate with leg length 0.453081837 and stored container-side score 2207106792497621131. Preserve all nine 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. For each triangle impose every vertex-in-square inequality. For each pair enumerate signed edge normals from both triangles, retain a currently separating axis, and impose its exact linearized separation constraint. 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 0.453081838, reducing the stored score to 2207106787626300748. 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 27 triangle vertices and all 36 triangle pairs with zero intersection area. Candidate certificate: {"placements":[{"turn":"0.000000000","x":"0.226540919","y":"0.226540920"},{"quarter":2,"turn":"0.000000000","x":"0.226540920","y":"0.226540920"},{"turn":"0.000000000","x":"0.679622758","y":"0.226540919"},{"quarter":2,"turn":"0.000000000","x":"0.679622758","y":"0.226540919"},{"turn":"0.000000000","x":"0.226540920","y":"0.679622758"},{"quarter":2,"turn":"0.000000000","x":"0.226540920","y":"0.679622758"},{"turn":"0.000000000","x":"0.679622758","y":"0.679622758"},{"turn":"0.414213562","x":"0.679622758","y":"0.999999999"},{"turn":"-0.414213562","x":"0.999999999","y":"0.679622759"}],"radius":"0.453081838"}
P89 · Tans in a square · Discussion