== Theorem 1.1(a): M7 ==
  ok    10 triangles on vertices 0..6
  ok    M7 is a surface (edges in <=2 triangles, links paths/cycles, connected)
  ok    lk(0) is the path 2 1 4 5 3 6
  ok    lk(1) is the path 3 2 0 4
  ok    lk(2) is the path 0 1 3 5 4 6
  ok    lk(3) is the path 1 2 5 0 6 4
  ok    lk(4) is the path 1 0 5 2 6 3
  ok    lk(6) is the path 0 3 4 2
  ok    lk(5) is the cycle (0 3 2 4)
  ok    12 interior edges as printed
  ok    6 boundary edges 02,06,26,13,14,34
  ok    f-vector (7,18,10), no edge in 3 triangles
  ok    boundary circles 0 2 6 and 1 3 4: [(0, 2, 6), (1, 3, 4)]
  ok    chi(M7) = -1
  ok    M7 is non-orientable (orientation propagation fails)
  ok    M7 + 026 + 134 is a closed surface
  ok    capped surface: chi = 1, non-orientable (RP^2)
  ok    the 12 dual edges are exactly the printed ones
  ok    BFS layers from 012 as printed: {0: ['012'], 1: ['123', '014'], 2: ['235', '045'], 3: ['245', '035'], 4: ['246', '036'], 5: ['346']}
  ok    every dual edge joins consecutive layers
  ok    printed path 012,123,235,245,246,346 is a dual walk
  ok    diam M7 = 5, unique pair at distance 5: [('012', '346')]
  ok    X=012 has boundary edge 02, Y=346 has boundary edge 34
  ok    026 is a missing triangle with all edges present (M7 not flag)
== Theorem 1.1(b): chains of M7 with pendant triangles, 7 <= n <= 60 ==
  ok    n=12 (k=2, r=0): surface with 12 vertices, dual diameter 11 = n-1
  ok    all chains 7<=n<=60: surfaces with n vertices, d(X1,last)=n-3+k, diam = n-3+floor((n-2)/5)
    n k r #tri d(X,last) diam  (first rows)  [(7, 1, 0, 10, 5, 5), (8, 1, 1, 11, 6, 6), (9, 1, 2, 12, 7, 7), (10, 1, 3, 13, 8, 8), (11, 1, 4, 14, 9, 9), (12, 2, 0, 20, 11, 11)]
  ok    M7 + 0 pendant(s): n=7, diam=5=n-2
  ok    M7 + 1 pendant(s): n=8, diam=6=n-2
  ok    M7 + 2 pendant(s): n=9, diam=7=n-2
  ok    M7 + 3 pendant(s): n=10, diam=8=n-2
  ok    strip on 3 vertices: orientable disc, diam = n-3 = 0
  ok    strip on 4 vertices: orientable disc, diam = n-3 = 1
  ok    strip on 5 vertices: orientable disc, diam = n-3 = 2
  ok    strip on 6 vertices: orientable disc, diam = n-3 = 3
  ok    strip on 7 vertices: orientable disc, diam = n-3 = 4
  ok    strip on 8 vertices: orientable disc, diam = n-3 = 5
  ok    strip on 9 vertices: orientable disc, diam = n-3 = 6
  ok    strip on 10 vertices: orientable disc, diam = n-3 = 7
  ok    strip on 11 vertices: orientable disc, diam = n-3 = 8
== Theorem 1.1(c): G10 ==
  ok    the 20 listed cyclic triangles are the 20 triangles of G10
  ok    20 triangles, 10 vertices
  ok    G10 is a surface
  ok    the printed cyclic orders form a coherent orientation
  ok    G10 is orientable (propagation succeeds)
  ok    lk(0) is the path 2 1 3 4 5 6 9 8 7
  ok    lk(1) is the path 3 0 2 9
  ok    lk(2) is the path 0 1 9 4 5 8 3 6 7
  ok    lk(3) is the path 1 0 4 9 5 6 2 8
  ok    lk(6) is the path 8 7 2 3 5 0 9
  ok    lk(7) is the path 0 8 6 2
  ok    lk(8) is the path 3 2 5 9 0 7 6
  ok    lk(9) is the path 1 2 4 3 5 8 0 6
  ok    lk(4) is the cycle (0 3 9 2 5)
  ok    lk(5) is the cycle (0 4 2 8 9 3 6)
  ok    34 edges
  ok    8 boundary edges forming circles 0 2 7 and 1 3 8 6 9: [(0, 2, 7), (1, 3, 8, 6, 9)]
  ok    the second circle is 1-3-8-6-9 in this cyclic order
  ok    chi = -4
  ok    orientable, 2 boundary circles, chi=-4 -> genus 2
  ok    BFS layers from 012 as printed
  ok    26 dual edges; the only one inside a layer is 045-245: [('045', '245')]
  ok    printed geodesic: star of 0 followed by 678
  ok    diam G10 = 8, unique pair at distance 8: [('012', '678')]
  ok    012 has boundary edge 02 and 678 has boundary edge 68
== chains of G10 (orientable), 10 <= n <= 60 ==
  ok    all G10 chains 10<=n<=60: orientable surfaces, d = n-3+k, diam = n-3+floor((n-2)/8)
== second 8-vertex class of Theorem 1.2(b) ==
  ok    surface, 8 vertices, non-orientable, chi=-1, two boundary circles [(1, 4, 5), (2, 3, 7)]
  ok    dual diameter 6 (pairs [('023', '457')])
  ok    not isomorphic to M7 plus a pendant triangle
== Lemma 4.1 (edge sum) on random pairs of surfaces ==
  ok    Lemma 4.1 holds on 200 random gluings (surface, n, chi, orientability, bridge, distances)
== Holmes (EJC 2018): Figures 4, 6, 6+EHI and 7 ==
  ok    Figure 4 = image of M7 under 0..6 -> A,B,C,E,D,G,F
  ok    Figure 6 (Prop. 40) is a surface with dual diameter 6
  ok    Figure 6 + EHI (Prop. 41) is a surface with dual diameter 7
  ok    Figure 6 is isomorphic to M7 plus a pendant triangle on 34
  ok    Figure 7 is not a surface: edges in >= 3 triangles: ['CD', 'GH']
    Figure 7 dual diameter (as an (S2) complex): 9

SUMMARY: ALL CHECKS PASS
