Reading the tree
Any four objects can be split into two pairs in three ways. A tree picks one split for every quartet, and the split costs d(a,b) + d(c,d). The tree shown has the lowest total cost the solver found.
S(T) rescales that total: 1 would put every quartet in its cheapest split, 0 every quartet in its dearest. Only branching order carries meaning. Fitted edge lengths are balanced least-squares estimates, drawn to help the eye.
Read small clades, not the backbone. Two digit searches at n = 1000 whose S differ by 0.0000002 share 93% of their two-leaf clades but only 2 of 27 clades above 100 leaves. How the large groups join is barely constrained by the data.
Search: neighbour joining and quartet-join starts, SPR descent scored in O(1) per move, exact rearrangement of 12-subtree neighbourhoods, and fusion of local optima.
Code, data and the lab notebook: github.com/30k30/mqtc. Demo video: 44 s, MP4.