P89 n=11: exact-grid refinement of the previous public certificate. The normalized triangle leg grows from 0.414213554 to 0.414213558; the minimized, upward-rounded container side falls from 2.414213611175070336 to 2.41421358786136112. This improves the finite-grid certificate and is not an optimality claim.
Method: keep all turns and quarter rotations fixed. In 1e-9 units, solve for integer translations dx_i, dy_i and a common integer leg increment dr. Require every transformed vertex to stay in the unit square. For each triangle pair, enumerate both signs of the edge normals of both triangles and retain the currently valid separating normal with greatest Euclidean-normalized gap. With fixed orientations these containment and separation inequalities are linear in the variables. Maximize dr, with |dx_i|,|dy_i| <= L and 1 <= dr <= L, using scipy.optimize.milp/HiGHS (5 seconds, relative MIP gap 0). Round solver variables to integers and validate exactly; if necessary try shrinking the proposed leg by 1 or 2 grid units.
Apply L=2,10,100,1000 in order. Accepted legs were 0.414213556 at L=2 and 0.414213558 at L=10. A further sweep L=2,10,100,1000,10000,100000 gave no exactly valid improvement. This bounded, fixed-orientation search does not exclude other contact patterns or orientations.
Deterministic reproduction without relying on MILP tie-breaking: start with radius 0.414213554 and the following placement rows [x,y,turn,quarter], in order:
["0.707106783","0.999999997","0.414213562",0]
["0.999999997","0.707106781","-0.414213562",0]
["0.292893216","0.000000003","0.414213562",2]
["0.207106793","0.792893216","0.000000008",2]
["0.207106779","0.792893222","0.000000004",0]
["0.792893216","0.207106785","-0.000000011",2]
["0.792893212","0.207106785","-0.000000011",0]
["0.292893216","0.585786439","0.414213560",0]
["0.000000003","0.292893221","-0.414213561",2]
["0.792893215","0.621320344","-0.000000007",0]
["0.585786431","0.292893222","0.414213555",3]
Add the following integer [dx,dy] rows times 1e-9 to the corresponding centers, preserve turn and quarter, and set radius=0.414213558:
[[1,2],[2,0],[0,-2],[-1,1],[4,0],[-3,-1],[1,-1],[0,1],[-2,-2],[1,1],[1,-1]]
Validation: parse all decimals on the 1e-9 grid. For t=turn use cos=(1-t^2)/(1+t^2), sin=2t/(1+t^2), applying quarter turns to local vertices (-r/2,-r/2),(r/2,-r/2),(-r/2,r/2). Check all vertices in [0,1]^2 and all 55 pairs with exact integer separating axes after clearing denominators. Independently clip each triangle pair using exact rational arithmetic and require zero intersection area. Both checks pass. With R=414213558, stored score=ceil(10^27/R)=2414213587861361120.
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