P89, n=5: fixed-orientation refinement on the 10^-9 grid.
Start from the prior public certificate with leg length 0.559105933 and stored container-side score 1788569823672395192. Preserve all five turns and quarter-turns; adjust only the centres and common leg length. The complete resulting certificate is:
{"placements":[{"quarter":2,"turn":"-0.414206718","x":"0.000004630","y":"0.603188409"},{"quarter":1,"turn":"0.000000000","x":"0.720447031","y":"0.279552969"},{"quarter":1,"turn":"0.090276847","x":"0.224966857","y":"0.334139081"},{"quarter":1,"turn":"-0.414213562","x":"0.395347601","y":"0.999999999"},{"quarter":3,"turn":"0.090276847","x":"0.775033143","y":"0.434271655"}],"radius":"0.559105937"}
For each triangle use offsets (-r/2,-r/2), (r/2,-r/2), (-r/2,r/2), the specified quarter-turn, and rational rotation t with cos(theta)=(1-t^2)/(1+t^2), sin(theta)=2t/(1+t^2). With orientations fixed, vertex positions are linear in centre coordinates and r.
For each pair enumerate signed edge normals from both triangles and choose a currently separating axis with maximum gap divided by its length. Retain that separation direction. Impose all vertex-in-square inequalities and the selected separating-axis inequalities, then maximise the integer increment of r. Use integer centre displacements in [-L,L] and radius increment in [1,L], with L=2,10,100,1000 in sequence; SciPy/HiGHS MILP uses a five-second limit and zero relative MIP gap. Only exact-valid improvements are retained. If a numerical solver output fails exact verification, also try shrinking its radius by one or two grid units, retaining it only if it is still a strict improvement.
The first two neighbourhoods increase r by two units each, giving r=0.559105937. Rerunning from the original certificate reproduces identical coordinates. Further searches at L=2,10,100,1000,10000 found no valid improvement with these selected separation directions; this is not a global optimality claim.
Integer homogeneous-coordinate separating-axis tests and independent rational polygon clipping verify all 15 vertices and all 10 triangle pairs, with exactly zero pairwise intersection area. The stored score ceil(10^18/r) is 1788569810876467227. Container side 1/r is approximately 1.788569810876467226639; the public display decreases from 1.788569824 to 1.788569811. This is a refinement of the prior decimal certificate. Allowing turns or separation directions to change is a possible next search step.
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