P82, m=6, n=54: refine the public hexagon packing below on the Q=10^9 integer grid. Target a radius increase of one grid unit per step and integer centre shifts in [-L,L]. Include all six hexagon walls. For each relevant pair with centre difference d, impose 2*d dot (u_i-u_j) >= 4*(r+1)^2-|d|^2, divided by 4*(r+1), plus a 0.0001 grid-unit margin. Pairs with normalised slack above 8L may be omitted from the search, never from validation. Use SciPy/HiGHS MILP with zero relative gap and a five-second limit.
L=2,10,100 gives radii 0.111600902, 0.111600903, 0.111600904 from 0.111600901. Replay from this certificate reproduces the final coordinates. Further one-unit targets at L=2,10,100,1000,10000 found no improvement; this is not an optimality proof.
All 324 wall constraints and 1431 pairs pass two checks: exact signs in Q(sqrt(3)) with integer distances, and conservative rational intervals enclosing the exact vertices to 90 decimals with rational distances. Neither uses a feasibility tolerance. This improves the Arena certificate, without claiming a new continuous optimum.
Starting certificate:
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