bipartite: True
denominator parameter k=2..20: [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20]
subtrees: 39  Conforti family: 7
    ['qw', 'qz']  leaves ['w', 'z']
    ['pm', 'pz', 'tm']  leaves ['t', 'z']
    ['qm', 'qw', 'tm']  leaves ['t', 'w']
    ['qm', 'qz', 'tm']  leaves ['t', 'z']
    ['pm', 'pz', 'qm', 'qw']  leaves ['w', 'z']
    ['qm', 'qw', 'qz', 'tm']  leaves ['t', 'w', 'z']
    ['pm', 'pz', 'qm', 'qw', 'tm']  leaves ['t', 'w', 'z']
maximal members == {T1,T2}: True ; all members inside T1 or T2: True ; every edge covered: True
explicit combos exact (incl. P, F_k value, coords >= -1), k=3..300: True
k=2 exact min F_k over S cap box[-3,3]^3 (z free) = 2 ; F_k(x^(k)) = 2
k=3 exact min F_k over S cap box[-3,3]^3 (z free) = 3 ; F_k(x^(k)) = 8/3
k=4 exact min F_k over S cap box[-3,3]^3 (z free) = 4 ; F_k(x^(k)) = 18/5
k=5 exact min F_k over S cap box[-3,3]^3 (z free) = 5 ; F_k(x^(k)) = 32/7
k=6 exact min F_k over S cap box[-3,3]^3 (z free) = 6 ; F_k(x^(k)) = 50/9
k=7 exact min F_k over S cap box[-3,3]^3 (z free) = 7 ; F_k(x^(k)) = 72/11
certificate_k3.json: 39 trees verified: True ; flags agree with my family: True ; covers all subtrees: True
F_k facet (affine rank 5 + lineality), k=3..40: True
