A. M7 = 012 014 035 036 045 123 235 245 246 346
   interior edges (12): 01 03 04 05 12 23 24 25 35 36 45 46
   boundary edges (6): 02 06 13 14 26 34
   lk(0) = path 2-1-4-5-3-6
   lk(1) = path 3-2-0-4
   lk(2) = path 0-1-3-5-4-6
   lk(3) = path 1-2-5-0-6-4
   lk(4) = path 1-0-5-2-6-3
   lk(5) = cycle 0-3-2-4
   lk(6) = path 0-3-4-2
   chi = -1 ; boundary circles: [[0, 2, 6], [1, 3, 4]] ; orientable: False
   dual diameter (Floyd-Warshall) = 5 ; far pairs: [('012', '346')]
   L0: 012
   L1: 014 123
   L2: 045 235
   L3: 035 245
   L4: 036 246
   L5: 346
   the 12 dual edges all join consecutive layers
   M7 + 026 + 134: closed surface, chi = 1, non-orientable (a 7-vertex RP^2)
B. Holmes, EJC 25(1) (2018) #P1.60, Figure 4 = image of M7 under 0A 1B 2C 3E 4D 5G 6F
C. gluing lemma verified on 150 random pairs from 167 random surfaces with boundary
   chains of M7 (strips for n < 7): dual diameter = n - 3 + floor((n-2)/5) exactly, 3 <= n <= 60
   chains of G10: orientable, dual diameter = n - 3 + floor((n-2)/8) exactly, 10 <= n <= 60
   12-vertex example: surface, chi = -3, 3 boundary circles, non-orientable, diameter 11, far pairs [('012', '8.9.11')]
D. G10 = 012 013 034 045 056 069 078 089 129 236 238 245 249 258 267 349 356 359 589 678
   boundary edges: 02 07 13 19 27 38 68 69 ; circles: [[0, 2, 7], [1, 3, 6, 8, 9]] ; chi = -4 ; orientable; genus = 2
   lk(0) = path 2-1-3-4-5-6-9-8-7
   lk(1) = path 3-0-2-9
   lk(2) = path 0-1-9-4-5-8-3-6-7
   lk(3) = path 1-0-4-9-5-6-2-8
   lk(4) = cycle 0-3-9-2-5
   lk(5) = cycle 0-4-2-8-9-3-6
   lk(6) = path 8-7-2-3-5-0-9
   lk(7) = path 0-8-6-2
   lk(8) = path 3-2-5-9-0-7-6
   lk(9) = path 1-2-4-3-5-8-0-6
   L0: 012
   L1: 013 129
   L2: 034 249
   L3: 045 245 349
   L4: 056 258 359
   L5: 069 238 356 589
   L6: 089 236
   L7: 078 267
   L8: 678
   26 dual edges; all join equal or consecutive layers; equal-layer edges: [('045', '245')]
   dual diameter = 8 with unique far pair [('012', '678')]
   chain n=12: (D, k, block sizes) = (11, 3, [7, 8, 3])
   chain n=17: (D, k, block sizes) = (17, 4, [7, 8, 8, 3])
   chain n=22: (D, k, block sizes) = (23, 5, [7, 8, 8, 8, 3])
   chain n=27: (D, k, block sizes) = (29, 6, [7, 8, 8, 8, 8, 3])
E. block argument checked from every start triangle of 173 surfaces (1123 cases)
F. the 4 complexes blocked in the uniqueness certificate are isomorphic to M7
   8-vertex surface 012 013 024 034 125 136 145 147 236 246 345: chi = -1, orientable = False, circles = 2, diameter 6
   8-vertex surface 012 013 023 124 135 246 256 257 346 347 356 457: chi = -1, orientable = False, circles = 2, diameter 6
   M7 with n - 7 pendant triangles on the edge 34: surface on n vertices with diameter n - 2, n = 8..11
ALL CHECKS PASSED
