FILTRATION (17 simplices): 2 0 3 4 34 1 02 13 04 23 14 24 12 124 123 024 234

step  simplex   boundary                 reduction (add earlier reduced columns until low is new)      result
  1   2         {}                                                                                      R=0: positive
  2   0         {}                                                                                      R=0: positive
  3   3         {}                                                                                      R=0: positive
  4   4         {}                                                                                      R=0: positive
  5   34        {3,4}                                                                                   low=4: PAIR (4,34)
  6   1         {}                                                                                      R=0: positive
  7   02        {2,0}                                                                                   low=0: PAIR (0,02)
  8   13        {3,1}                                                                                   low=1: PAIR (1,13)
  9   04        {0,4}                   +R(34)={3,4} -> {0,3}                                           low=3: PAIR (3,04)
 10   23        {2,3}                   +R(04)={0,3} -> {2,0} +R(02)={2,0} -> {}                        R=0: positive
 11   14        {4,1}                   +R(13)={3,1} -> {3,4} +R(34)={3,4} -> {}                        R=0: positive
 12   24        {2,4}                   +R(34)={3,4} -> {2,3} +R(04)={0,3} -> {2,0} +R(02)={2,0} -> {}  R=0: positive
 13   12        {2,1}                   +R(13)={3,1} -> {2,3} +R(04)={0,3} -> {2,0} +R(02)={2,0} -> {}  R=0: positive
 14   124       {14,24,12}                                                                              low=12: PAIR (12,124)
 15   123       {13,23,12}              +R(124)={14,24,12} -> {13,23,14,24}                             low=24: PAIR (24,123)
 16   024       {02,04,24}              +R(123)={13,23,14,24} -> {02,13,04,23,14}                       low=14: PAIR (14,024)
 17   234       {34,23,24}              +R(123)={13,23,14,24} -> {34,13,14} +R(024)={02,13,04,23,14} -> {34,02,04,23}  low=23: PAIR (23,234)

P (reduction = rank formula): (4,34) (0,02) (1,13) (3,04) (12,124) (24,123) (14,024) (23,234)
V_P (face pairs): (4,34) (0,02) (1,13) (12,124) (23,234)
non-incident pairs: (3,04) (24,123) (14,024)
essential: ['2']
closed V-path in V_P: None  => V_P is a gradient (acyclic)

critical cells of V_P (orig order): ['2', '3', '04', '14', '24', '123', '024']
   Morse boundary of 2    = []
   Morse boundary of 3    = []
   Morse boundary of 04   = ['2', '3']
   Morse boundary of 14   = []
   Morse boundary of 24   = ['2', '3']
   Morse boundary of 123  = ['14']
   Morse boundary of 024  = ['04', '24']
column reduction of the Morse boundary in the ORIGINAL order gives pairs: [('14', '123'), ('24', '024'), ('3', '04')]
non-incident pairs of P                                          : [('14', '024'), ('24', '123'), ('3', '04')]

C3a (V_P-pairs consecutive) complete search: NONE (nodes 382)
C3b (any f, any tie-breaking) complete search: NONE (nodes 19945)
