== Petersen family: 7 graphs with [6, 7, 7, 8, 8, 9, 10] vertices, 15 edges each
== 1. direct enumeration of all pairs of disjoint cycles, M1 against M2
4-simplex              f=(5, 10, 10, 5) cycles=37 disjoint pairs=0 |lk| histogram={}  M1 symmetric & M1=M2(two diagrams): yes  diagrams embedded: True/True  global signs: None/None  partition search agrees: True  example: None  (0.0s)
prism over tetrahedron f=(8, 16, 14, 6) cycles=170 disjoint pairs=52 |lk| histogram={0: 52}  M1 symmetric & M1=M2(two diagrams): yes  diagrams embedded: True/True  global signs: None/None  partition search agrees: True  example: None  (0.0s)
3-3 duoprism           f=(9, 18, 15, 6) cycles=312 disjoint pairs=78 |lk| histogram={0: 78}  M1 symmetric & M1=M2(two diagrams): yes  diagrams embedded: True/True  global signs: None/None  partition search agrees: True  example: None  (0.0s)
3-4 duoprism           f=(12, 24, 19, 7) cycles=1531 disjoint pairs=723 |lk| histogram={0: 723}  M1 symmetric & M1=M2(two diagrams): yes  diagrams embedded: True/True  global signs: None/None  partition search agrees: True  example: None  (0.5s)
pyramid over cube      f=(9, 20, 18, 7) cycles=472 disjoint pairs=87 |lk| histogram={0: 87}  M1 symmetric & M1=M2(two diagrams): yes  diagrams embedded: True/True  global signs: None/None  partition search agrees: True  example: None  (0.0s)
stacked, 8 vertices    f=(8, 22, 28, 14) cycles=1123 disjoint pairs=231 |lk| histogram={0: 231}  M1 symmetric & M1=M2(two diagrams): yes  diagrams embedded: True/True  global signs: None/None  partition search agrees: True  example: None  (0.1s)
== 2. C(6,4): complementary triangles
C(6,4)                 f=(6, 15, 18, 9) cycles=197 disjoint pairs=10 |lk| histogram={0: 9, 1: 1}  M1 symmetric & M1=M2(two diagrams): yes  diagrams embedded: True/True  global signs: 1/-1  partition search agrees: True  example: ((0, 2, 4), (1, 3, 5), 1)  (0.0s)
   10 pairs of complementary triangles (C, D, lk, C is a 2-face, D is a 2-face):
     ((0, 1, 2), (3, 4, 5), 0, True, True)
     ((0, 1, 3), (2, 4, 5), 0, True, True)
     ((0, 1, 4), (2, 3, 5), 0, True, True)
     ((0, 1, 5), (2, 3, 4), 0, True, True)
     ((0, 2, 3), (1, 4, 5), 0, True, True)
     ((0, 2, 4), (1, 3, 5), 1, False, False)
     ((0, 2, 5), (1, 3, 4), 0, True, True)
     ((0, 3, 4), (1, 2, 5), 0, True, True)
     ((0, 3, 5), (1, 2, 4), 0, True, True)
     ((0, 4, 5), (1, 2, 3), 0, True, True)
   pairs with |lk| = 1: 1; other pairs: lk = 0
== 3. octahedral prism: two disjoint 6-cycles with lk = +-1
octahedral prism       f=(12, 30, 28, 10) cycles=13100 disjoint pairs=8483 |lk| histogram={0: 8477, 1: 6}  M1 symmetric & M1=M2(two diagrams): yes  diagrams embedded: True/True  global signs: 1/-1  partition search agrees: True  example: ((0, 1, 5, 3, 2, 6), (4, 8, 9, 7, 11, 10), -1)  (4.3s)
   pairs of disjoint 6-cycles by |lk|: {0: 358, 1: 6}  example: ((0, 1, 5, 3, 2, 6), (4, 8, 9, 7, 11, 10), -1)
   coordinates of the example: [(1, 0, 0, 0), (1, 0, 0, 1), (0, 1, 0, 1), (-1, 0, 0, 1), (-1, 0, 0, 0), (0, -1, 0, 0)] [(0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 1, 1), (0, -1, 0, 1), (0, 0, -1, 1), (0, 0, -1, 0)]
   same pair by crossings in the Schlegel diagram: -1
== 4. further linked examples (direct enumeration where feasible)
C(7,4)                 f=(7, 21, 28, 14) cycles=1172 disjoint pairs=175 |lk| histogram={0: 154, 1: 21}  M1 symmetric & M1=M2(two diagrams): yes  diagrams embedded: True/True  global signs: 1/1  partition search agrees: True  example: ((0, 1, 3, 5), (2, 4, 6), 1)  (0.1s)
hypersimplex(2,5)      f=(10, 30, 30, 10) cycles=15703 disjoint pairs=6596 |lk| histogram={0: 6590, 1: 6}  M1 symmetric & M1=M2(two diagrams): yes  diagrams embedded: True/True  global signs: -1/1  partition search agrees: True  example: ((0, 1, 7, 9, 6), (2, 3, 8, 4, 5), -1)  (4.3s)
cross-polytope         f=(8, 24, 32, 16) cycles=2766 disjoint pairs=691 |lk| histogram={0: 688, 1: 3}  M1 symmetric & M1=M2(two diagrams): yes  diagrams embedded: True/True  global signs: 1/-1  partition search agrees: True  example: ((0, 2, 1, 3), (4, 6, 5, 7), -1)  (0.3s)
== 5. 4-cube (16 vertices): partition search and confirmation by M2
   pair found: [8, 9, 11, 3, 7, 6, 4, 12] [14, 10, 2, 0, 1, 5, 13, 15] lk(M1) = -1  lk(M2) = 1  lengths 8 8
   pairs of disjoint cycles of length <= 6: 2116, with lk != 0: 0
   pairs of complementary 8-cycles by |lk|: {0: 288, 1: 12}
== 6. 24-cell: random search
   pair found: lengths 7 7 lk(M1) = -1  lk(M2) = -1
ALL VALIDATION ASSERTIONS PASSED
