PART A. Conjecture 2, 7 simplices (the two filtrations; the path variant is the one of Kriebel)
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STAR7 (ours): 7 simplices
filtration : 0 1 2 3 23 13 03

step  simplex   boundary                 reduction (add earlier reduced columns until low is new)      result
  1   0         {}                                                                                      R=0: positive
  2   1         {}                                                                                      R=0: positive
  3   2         {}                                                                                      R=0: positive
  4   3         {}                                                                                      R=0: positive
  5   23        {2,3}                                                                                   low=3: PAIR (3,23)
  6   13        {1,3}                   +R(23)={2,3} -> {1,2}                                           low=2: PAIR (2,13)
  7   03        {0,3}                   +R(23)={2,3} -> {0,2} +R(13)={1,2} -> {0,1}                     low=1: PAIR (1,03)

P          : (3,23) (2,13) (1,03)
V_P        : (3,23)
non-incident: (2,13) (1,03)
essential  : ['0']
numbers of gradient paths tau -> sigma: {('2', '13'): 1, ('1', '03'): 0}
   directed paths in the modified Hasse diagram G from 13 to 2 exist: True
   directed paths in the modified Hasse diagram G from 03 to 1 exist: False
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PATH7 (Kriebel: a,b,c,d,cd,bd,ac = 0,1,2,3,23,13,02): 7 simplices
filtration : 0 1 2 3 23 13 02

step  simplex   boundary                 reduction (add earlier reduced columns until low is new)      result
  1   0         {}                                                                                      R=0: positive
  2   1         {}                                                                                      R=0: positive
  3   2         {}                                                                                      R=0: positive
  4   3         {}                                                                                      R=0: positive
  5   23        {2,3}                                                                                   low=3: PAIR (3,23)
  6   13        {1,3}                   +R(23)={2,3} -> {1,2}                                           low=2: PAIR (2,13)
  7   02        {0,2}                   +R(13)={1,2} -> {0,1}                                           low=1: PAIR (1,02)

P          : (3,23) (2,13) (1,02)
V_P        : (3,23)
non-incident: (2,13) (1,02)
essential  : ['0']
numbers of gradient paths tau -> sigma: {('2', '13'): 1, ('1', '02'): 0}
   directed paths in the modified Hasse diagram G from 13 to 2 exist: True
   directed paths in the modified Hasse diagram G from 02 to 1 exist: False

PART B. Conjecture 2: pairs with two and with three gradient paths (smallest examples found)
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MP17: 17 simplices
filtration : 4 5 3 0 35 1 14 34 05 01 13 03 013 04 035 014 034
P          : (3,35) (1,14) (5,34) (0,05) (03,013) (13,035) (04,014) (01,034)
V_P        : (3,35) (1,14) (0,05) (03,013) (04,014)
non-incident: (5,34) (13,035) (01,034)
essential  : ['4']
   pair {5,34}: paths ['34 3 35 5']
   pair {13,035}: paths ['035 03 013 13']
   pair {01,034}: paths ['034 04 014 01', '034 03 013 01']
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MP19: 19 simplices
filtration : 3 4 2 0 1 23 13 01 02 12 04 24 14 024 012 124 34 234 134
P          : (2,23) (1,13) (0,01) (4,04) (24,024) (12,012) (14,124) (34,234) (02,134)
V_P        : (2,23) (1,13) (0,01) (4,04) (24,024) (12,012) (14,124) (34,234)
non-incident: (02,134)
essential  : ['3']
   pair {02,134}: paths ['134 34 234 24 024 02', '134 14 124 24 024 02', '134 14 124 12 012 02']

PART C. Theorem 1 fails: 17 simplices, closed V-path
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T1_17: 17 simplices
filtration : 0 1 2 3 4 03 04 12 13 14 23 24 34 124 234 034 134

step  simplex   boundary                 reduction (add earlier reduced columns until low is new)      result
  1   0         {}                                                                                      R=0: positive
  2   1         {}                                                                                      R=0: positive
  3   2         {}                                                                                      R=0: positive
  4   3         {}                                                                                      R=0: positive
  5   4         {}                                                                                      R=0: positive
  6   03        {0,3}                                                                                   low=3: PAIR (3,03)
  7   04        {0,4}                                                                                   low=4: PAIR (4,04)
  8   12        {1,2}                                                                                   low=2: PAIR (2,12)
  9   13        {1,3}                   +R(03)={0,3} -> {0,1}                                           low=1: PAIR (1,13)
 10   14        {1,4}                   +R(04)={0,4} -> {0,1} +R(13)={0,1} -> {}                        R=0: positive
 11   23        {2,3}                   +R(03)={0,3} -> {0,2} +R(12)={1,2} -> {0,1} +R(13)={0,1} -> {}  R=0: positive
 12   24        {2,4}                   +R(04)={0,4} -> {0,2} +R(12)={1,2} -> {0,1} +R(13)={0,1} -> {}  R=0: positive
 13   34        {3,4}                   +R(04)={0,4} -> {0,3} +R(03)={0,3} -> {}                        R=0: positive
 14   124       {12,14,24}                                                                              low=24: PAIR (24,124)
 15   234       {23,24,34}                                                                              low=34: PAIR (34,234)
 16   034       {03,04,34}              +R(234)={23,24,34} -> {03,04,23,24} +R(124)={12,14,24} -> {03,04,12,14,23}  low=23: PAIR (23,034)
 17   134       {13,14,34}              +R(234)={23,24,34} -> {13,14,23,24} +R(124)={12,14,24} -> {12,13,23} +R(034)={03,04,12,14,23} -> {03,04,13,14}  low=14: PAIR (14,134)

P          : (3,03) (4,04) (2,12) (1,13) (24,124) (34,234) (23,034) (14,134)
V_P        : (3,03) (4,04) (2,12) (1,13) (24,124) (34,234) (14,134)
non-incident: (23,034)
essential  : ['0']
closed V-path (lower cells): ['14', '34', '24', '14']
V-path checked: 14 < 134 > 34 < 234 > 24 < 124 > 14
directed cycle in the modified Hasse diagram G: 34 -> 234 -> 24 -> 124 -> 14 -> 134 -> 34

PART D. Conjecture 3 fails with V_P acyclic: B17
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B17: 17 simplices
filtration : 4 1 0 14 3 13 01 2 23 02 04 12 24 124 024 123 012

step  simplex   boundary                 reduction (add earlier reduced columns until low is new)      result
  1   4         {}                                                                                      R=0: positive
  2   1         {}                                                                                      R=0: positive
  3   0         {}                                                                                      R=0: positive
  4   14        {4,1}                                                                                   low=1: PAIR (1,14)
  5   3         {}                                                                                      R=0: positive
  6   13        {1,3}                                                                                   low=3: PAIR (3,13)
  7   01        {1,0}                                                                                   low=0: PAIR (0,01)
  8   2         {}                                                                                      R=0: positive
  9   23        {3,2}                                                                                   low=2: PAIR (2,23)
 10   02        {0,2}                   +R(23)={3,2} -> {0,3} +R(13)={1,3} -> {1,0} +R(01)={1,0} -> {}  R=0: positive
 11   04        {4,0}                   +R(01)={1,0} -> {4,1} +R(14)={4,1} -> {}                        R=0: positive
 12   12        {1,2}                   +R(23)={3,2} -> {1,3} +R(13)={1,3} -> {}                        R=0: positive
 13   24        {4,2}                   +R(23)={3,2} -> {4,3} +R(13)={1,3} -> {4,1} +R(14)={4,1} -> {}  R=0: positive
 14   124       {14,12,24}                                                                              low=24: PAIR (24,124)
 15   024       {02,04,24}              +R(124)={14,12,24} -> {14,02,04,12}                             low=12: PAIR (12,024)
 16   123       {13,23,12}              +R(024)={14,02,04,12} -> {14,13,23,02,04}                       low=04: PAIR (04,123)
 17   012       {01,02,12}              +R(024)={14,02,04,12} -> {14,01,04} +R(123)={14,13,23,02,04} -> {13,01,23,02}  low=02: PAIR (02,012)

P          : (1,14) (3,13) (0,01) (2,23) (24,124) (12,024) (04,123) (02,012)
V_P        : (1,14) (3,13) (0,01) (2,23) (24,124) (02,012)
non-incident: (12,024) (04,123)
essential  : ['4']
V_P is acyclic; the modified Hasse diagram G has no directed cycle.
   critical cell 4    Morse boundary (odd number of V_P-paths): []
   critical cell 04   Morse boundary (odd number of V_P-paths): []
   critical cell 12   Morse boundary (odd number of V_P-paths): []
   critical cell 024  Morse boundary (odd number of V_P-paths): ['04']
   critical cell 123  Morse boundary (odd number of V_P-paths): ['12']

Coset of the reduced column of 024 (hand proof, step 2): d024 + a*d124 + g*d012 + d*d123
   (a,g,d)=(0,0,0): {02,04,24}                        contains neither 02 nor 24: False  contains 12: False
   (a,g,d)=(0,0,1): {13,23,02,04,12,24}               contains neither 02 nor 24: False  contains 12: True
   (a,g,d)=(0,1,0): {01,04,12,24}                     contains neither 02 nor 24: False  contains 12: True
   (a,g,d)=(0,1,1): {13,01,23,04,24}                  contains neither 02 nor 24: False  contains 12: False
   (a,g,d)=(1,0,0): {14,02,04,12}                     contains neither 02 nor 24: False  contains 12: True
   (a,g,d)=(1,0,1): {14,13,23,02,04}                  contains neither 02 nor 24: False  contains 12: False
   (a,g,d)=(1,1,0): {14,01,04}                        contains neither 02 nor 24: True   contains 12: False
   (a,g,d)=(1,1,1): {14,13,01,23,04,12}               contains neither 02 nor 24: True   contains 12: True

B17: total orders L (faces first) with pairing(L) = P: 1852
     of them with 12 before both 02 and 24 (F1 of the hand proof, a consequence of the definition of a total refinement): 0
     of them with all V_P pairs consecutive (the strict reading): 0

PART E. second Conjecture 3 example A17 (computer only)
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A17: 17 simplices
filtration : 2 0 3 4 34 1 02 13 04 23 14 24 12 124 123 024 234
P          : (4,34) (0,02) (1,13) (3,04) (12,124) (24,123) (14,024) (23,234)
V_P        : (4,34) (0,02) (1,13) (12,124) (23,234)
non-incident: (3,04) (24,123) (14,024)
essential  : ['2']
A17: total orders L (faces first) with pairing(L) = P: 302 ; with all V_P pairs consecutive: 0

PART F. positive control: the filtration 1 2 3 0 03 02 01 (STAR7 in the labelling used in earlier computations) satisfies Conjecture 3
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STAR7 relabelled: 7 simplices
filtration : 1 2 3 0 03 02 01
P          : (0,03) (3,02) (2,01)
V_P        : (0,03)
non-incident: (3,02) (2,01)
essential  : ['1']
orders with pairing P: 1, with consecutive V-pairs: 1

ALL ASSERTIONS PASSED
