====================================================================================================
S7 : 0 1 2 3 23 13 03
  simplices by number of vertices: {1: 4, 2: 3}
  pairing (column reduction == rank formula, asserted): (3,23) (2,13) (1,03)
  essential: ['0']
  V_P: ['{3<23}']
  non-incident: ['(2,13)', '(1,03)']
   step  5 23   d={2,3} -- -> pair (3,23)
   step  6 13   d={1,3} +R(23)->{1,2} -> pair (2,13)
   step  7 03   d={0,3} +R(23)->{0,2}; +R(13)->{0,1} -> pair (1,03)
  closed V-path: None
  directed cycle of G(V_P): None
  pair (2,13): 1 gradient path(s) ['13 > 3 < 23 > 2']
  pair (1,03): 0 gradient path(s) []
  OK   S7 pairing {(3,23),(2,13),(1,03)}
  OK   S7: vertex 0 essential
  OK   S7: V_P = {3<23}
  OK   S7: pair {2,13} has exactly the path 13>3<23>2
  OK   S7: no gradient path 03 -> 1
  OK   S7: G-reachable from 03 = {03,0,3,23,2}: ['0', '03', '2', '23', '3']
====================================================================================================
Pi7 (Kriebel a,b,c,d,cd,bd,ac with a,b,c,d=0,1,2,3) : 0 1 2 3 23 13 02
  simplices by number of vertices: {1: 4, 2: 3}
  pairing (column reduction == rank formula, asserted): (3,23) (2,13) (1,02)
  essential: ['0']
  V_P: ['{3<23}']
  non-incident: ['(2,13)', '(1,02)']
   step  5 23   d={2,3} -- -> pair (3,23)
   step  6 13   d={1,3} +R(23)->{1,2} -> pair (2,13)
   step  7 02   d={0,2} +R(13)->{0,1} -> pair (1,02)
  closed V-path: None
  directed cycle of G(V_P): None
  pair (2,13): 1 gradient path(s) ['13 > 3 < 23 > 2']
  pair (1,02): 0 gradient path(s) []
  OK   Pi7 pairing {(3,23),(2,13),(1,02)}
  OK   Pi7: no gradient path 02 -> 1
  OK   Pi7: G-reachable from 02 = {02,0,2}: ['0', '02', '2']
====================================================================================================
M17 : 4 5 3 0 35 1 14 34 05 01 13 03 013 04 035 014 034
  simplices by number of vertices: {1: 5, 2: 8, 3: 4}
  pairing (column reduction == rank formula, asserted): (3,35) (1,14) (5,34) (0,05) (03,013) (13,035) (04,014) (01,034)
  essential: ['4']
  V_P: ['{3<35}', '{1<14}', '{0<05}', '{03<013}', '{04<014}']
  non-incident: ['(5,34)', '(13,035)', '(01,034)']
   step  5 35   d={5,3} -- -> pair (3,35)
   step  7 14   d={4,1} -- -> pair (1,14)
   step  8 34   d={4,3} +R(35)->{4,5} -> pair (5,34)
   step  9 05   d={5,0} -- -> pair (0,05)
   step 10 01   d={0,1} +R(14)->{4,0}; +R(05)->{4,5}; +R(34)->{} -> positive
   step 11 13   d={3,1} +R(14)->{4,3}; +R(35)->{4,5}; +R(34)->{} -> positive
   step 12 03   d={3,0} +R(05)->{5,3}; +R(35)->{} -> positive
   step 13 013  d={01,13,03} -- -> pair (03,013)
   step 14 04   d={4,0} +R(05)->{4,5}; +R(34)->{} -> positive
   step 15 035  d={35,05,03} +R(013)->{35,05,01,13} -> pair (13,035)
   step 16 014  d={14,01,04} -- -> pair (04,014)
   step 17 034  d={34,03,04} +R(014)->{14,34,01,03}; +R(013)->{14,34,13}; +R(035)->{35,14,34,05,01} -> pair (01,034)
  closed V-path: None
  directed cycle of G(V_P): None
  pair (5,34): 1 gradient path(s) ['34 > 3 < 35 > 5']
  pair (13,035): 1 gradient path(s) ['035 > 03 < 013 > 13']
  pair (01,034): 2 gradient path(s) ['034 > 04 < 014 > 01', '034 > 03 < 013 > 01']
  OK   M17 pairing
  OK   M17: 5 vertices, 8 edges, 4 triangles
  OK   M17 V_P
  OK   M17: V_P acyclic, G acyclic
  OK   M17: {5,34} one path 34>3<35>5
  OK   M17: {13,035} one path 035>03<013>13
  OK   M17: {01,034} exactly two paths
====================================================================================================
M19 : 3 4 2 0 1 23 13 01 02 12 04 24 14 024 012 124 34 234 134
  simplices by number of vertices: {1: 5, 2: 9, 3: 5}
  pairing (column reduction == rank formula, asserted): (2,23) (1,13) (0,01) (4,04) (24,024) (12,012) (14,124) (34,234) (02,134)
  essential: ['3']
  V_P: ['{2<23}', '{1<13}', '{0<01}', '{4<04}', '{24<024}', '{12<012}', '{14<124}', '{34<234}']
  non-incident: ['(02,134)']
   step  6 23   d={3,2} -- -> pair (2,23)
   step  7 13   d={3,1} -- -> pair (1,13)
   step  8 01   d={0,1} +R(13)->{3,0} -> pair (0,01)
   step  9 02   d={2,0} +R(01)->{3,2}; +R(23)->{} -> positive
   step 10 12   d={2,1} +R(13)->{3,2}; +R(23)->{} -> positive
   step 11 04   d={4,0} +R(01)->{3,4} -> pair (4,04)
   step 12 24   d={4,2} +R(23)->{3,4}; +R(04)->{} -> positive
   step 13 14   d={4,1} +R(13)->{3,4}; +R(04)->{} -> positive
   step 14 024  d={02,04,24} -- -> pair (24,024)
   step 15 012  d={01,02,12} -- -> pair (12,012)
   step 16 124  d={12,24,14} -- -> pair (14,124)
   step 17 34   d={3,4} +R(04)->{} -> positive
   step 18 234  d={23,24,34} -- -> pair (34,234)
   step 19 134  d={13,14,34} +R(234)->{23,13,24,14}; +R(124)->{23,13,12}; +R(012)->{23,13,01,02} -> pair (02,134)
  closed V-path: None
  directed cycle of G(V_P): None
  pair (02,134): 3 gradient path(s) ['134 > 34 < 234 > 24 < 024 > 02', '134 > 14 < 124 > 24 < 024 > 02', '134 > 14 < 124 > 12 < 012 > 02']
  OK   M19 pairing
  OK   M19: 5 vertices, 9 edges, 5 triangles
  OK   M19: V_P = first eight pairs, acyclic
  OK   M19: {02,134} exactly three paths ['134 > 34 < 234 > 24 < 024 > 02', '134 > 14 < 124 > 24 < 024 > 02', '134 > 14 < 124 > 12 < 012 > 02']
====================================================================================================
T17 : 0 1 2 3 4 03 04 12 13 14 23 24 34 124 234 034 134
  simplices by number of vertices: {1: 5, 2: 8, 3: 4}
  pairing (column reduction == rank formula, asserted): (3,03) (4,04) (2,12) (1,13) (24,124) (34,234) (23,034) (14,134)
  essential: ['0']
  V_P: ['{3<03}', '{4<04}', '{2<12}', '{1<13}', '{24<124}', '{34<234}', '{14<134}']
  non-incident: ['(23,034)']
   step  6 03   d={0,3} -- -> pair (3,03)
   step  7 04   d={0,4} -- -> pair (4,04)
   step  8 12   d={1,2} -- -> pair (2,12)
   step  9 13   d={1,3} +R(03)->{0,1} -> pair (1,13)
   step 10 14   d={1,4} +R(04)->{0,1}; +R(13)->{} -> positive
   step 11 23   d={2,3} +R(03)->{0,2}; +R(12)->{0,1}; +R(13)->{} -> positive
   step 12 24   d={2,4} +R(04)->{0,2}; +R(12)->{0,1}; +R(13)->{} -> positive
   step 13 34   d={3,4} +R(04)->{0,3}; +R(03)->{} -> positive
   step 14 124  d={12,14,24} -- -> pair (24,124)
   step 15 234  d={23,24,34} -- -> pair (34,234)
   step 16 034  d={03,04,34} +R(234)->{03,04,23,24}; +R(124)->{03,04,12,14,23} -> pair (23,034)
   step 17 134  d={13,14,34} +R(234)->{13,14,23,24}; +R(124)->{12,13,23}; +R(034)->{03,04,13,14} -> pair (14,134)
  closed V-path: 24<124 14<134 34<234
  directed cycle of G(V_P): 14 -> 134 -> 34 -> 234 -> 24 -> 124 -> 14
  OK   T17 pairing
  OK   T17: vertex 0 essential
  OK   T17: only non-incident pair (23,034)
  OK   T17: {14<134},{34<234},{24<124} in V_P
  OK   T17: 34<134, 24<234, 14<124 are facet relations
  OK   T17: 14->134->34->234->24->124->14 is a directed cycle of G(V_P)
  OK   T17: 5 vertices, 8 edges, 4 triangles
====================================================================================================
B17 : 4 1 0 14 3 13 01 2 23 02 04 12 24 124 024 123 012
  simplices by number of vertices: {1: 5, 2: 8, 3: 4}
  pairing (column reduction == rank formula, asserted): (1,14) (3,13) (0,01) (2,23) (24,124) (12,024) (04,123) (02,012)
  essential: ['4']
  V_P: ['{1<14}', '{3<13}', '{0<01}', '{2<23}', '{24<124}', '{02<012}']
  non-incident: ['(12,024)', '(04,123)']
   step  4 14   d={4,1} -- -> pair (1,14)
   step  6 13   d={1,3} -- -> pair (3,13)
   step  7 01   d={1,0} -- -> pair (0,01)
   step  9 23   d={3,2} -- -> pair (2,23)
   step 10 02   d={0,2} +R(23)->{0,3}; +R(13)->{1,0}; +R(01)->{} -> positive
   step 11 04   d={4,0} +R(01)->{4,1}; +R(14)->{} -> positive
   step 12 12   d={1,2} +R(23)->{1,3}; +R(13)->{} -> positive
   step 13 24   d={4,2} +R(23)->{4,3}; +R(13)->{4,1}; +R(14)->{} -> positive
   step 14 124  d={14,12,24} -- -> pair (24,124)
   step 15 024  d={02,04,24} +R(124)->{14,02,04,12} -> pair (12,024)
   step 16 123  d={13,23,12} +R(024)->{14,13,23,02,04} -> pair (04,123)
   step 17 012  d={01,02,12} +R(024)->{14,01,04}; +R(123)->{13,01,23,02} -> pair (02,012)
  closed V-path: None
  directed cycle of G(V_P): None
  pair (12,024): 2 gradient path(s) ['024 > 24 < 124 > 12', '024 > 02 < 012 > 12']
  pair (04,123): 0 gradient path(s) []
  OK   B17 pairing
  OK   B17: vertex 4 essential
  OK   B17 V_P
  OK   B17: V_P acyclic and G(V_P) acyclic
  OK   B17 non-incident pairs
  OK   B17: 5 vertices, 8 edges, 4 triangles
  F3 cosets d024 + a*d124 + g*d012 + d*d123:
    a=0 g=0 d=0: {02,04,24}
    a=0 g=0 d=1: {02,04,12,13,23,24}
    a=0 g=1 d=0: {01,04,12,24}
    a=0 g=1 d=1: {01,04,13,23,24}
    a=1 g=0 d=0: {02,04,12,14}
    a=1 g=0 d=1: {02,04,13,14,23}
    a=1 g=1 d=0: {01,04,14}   <- no 02, no 24
    a=1 g=1 d=1: {01,04,12,13,14,23}   <- no 02, no 24
  OK   F3: 02,24 absent only for (a,g)=(1,1); then R={01,04,14}+d{12,13,23}; 12 in R forces d=1
  OK   F2: 04 lies in the boundary of 024 only (triangles of B17)
  OK   F1: cofaces of 02 are 02,012,024; of 24 are 24,124,024
  OK   F1: arrow 02 -> 012 in G(V_P)
  OK   F1: arrow 012 -> 12 in G(V_P)
  OK   F1: arrow 024 -> 02 in G(V_P)
  OK   F1: arrow 24 -> 124 in G(V_P)
  OK   F1: arrow 124 -> 12 in G(V_P)
  OK   F1: arrow 024 -> 24 in G(V_P)
====================================================================================================
A17 : 2 0 3 4 34 1 02 13 04 23 14 24 12 124 123 024 234
  simplices by number of vertices: {1: 5, 2: 8, 3: 4}
  pairing (column reduction == rank formula, asserted): (4,34) (0,02) (1,13) (3,04) (12,124) (24,123) (14,024) (23,234)
  essential: ['2']
  V_P: ['{4<34}', '{0<02}', '{1<13}', '{12<124}', '{23<234}']
  non-incident: ['(3,04)', '(24,123)', '(14,024)']
   step  5 34   d={3,4} -- -> pair (4,34)
   step  7 02   d={2,0} -- -> pair (0,02)
   step  8 13   d={3,1} -- -> pair (1,13)
   step  9 04   d={0,4} +R(34)->{0,3} -> pair (3,04)
   step 10 23   d={2,3} +R(04)->{2,0}; +R(02)->{} -> positive
   step 11 14   d={4,1} +R(13)->{3,4}; +R(34)->{} -> positive
   step 12 24   d={2,4} +R(34)->{2,3}; +R(04)->{2,0}; +R(02)->{} -> positive
   step 13 12   d={2,1} +R(13)->{2,3}; +R(04)->{2,0}; +R(02)->{} -> positive
   step 14 124  d={14,24,12} -- -> pair (12,124)
   step 15 123  d={13,23,12} +R(124)->{13,23,14,24} -> pair (24,123)
   step 16 024  d={02,04,24} +R(123)->{02,13,04,23,14} -> pair (14,024)
   step 17 234  d={34,23,24} +R(123)->{34,13,14}; +R(024)->{34,02,04,23} -> pair (23,234)
  closed V-path: None
  directed cycle of G(V_P): None
  pair (3,04): 1 gradient path(s) ['04 > 4 < 34 > 3']
  pair (24,123): 2 gradient path(s) ['123 > 23 < 234 > 24', '123 > 12 < 124 > 24']
  pair (14,024): 0 gradient path(s) []
  OK   A17 V_P
  OK   A17: V_P acyclic
  OK   A17 non-incident pairs
  B17 critical cells: ['4', '04', '12', '024', '123']
   d_M(04) = 0
   d_M(12) = 0
   d_M(024) = 04
   d_M(123) = 12
  OK   Morse boundary printed above (expected d_M 024 = 04, d_M 123 = 12)
====================================================================================================
ALL CHECKS PASSED
