==============================================================================
PART A. Example 1.4 cell (exact over Q, both methods)
case generic n 3 m 3
  d=0  S^1_d (BB, exact)=1  sum_k<=d H^1_k (cofactor, exact)=1  paper=1  OWR [LB,UB]=(None, None)
  d=1  S^1_d (BB, exact)=4  sum_k<=d H^1_k (cofactor, exact)=4  paper=4  OWR [LB,UB]=(None, None)
  d=2  S^1_d (BB, exact)=11  sum_k<=d H^1_k (cofactor, exact)=11  paper=11  OWR [LB,UB]=(10, 12)
  d=3  S^1_d (BB, exact)=24  sum_k<=d H^1_k (cofactor, exact)=24  paper=24  OWR [LB,UB]=(22, 24)
  d=4  S^1_d (BB, exact)=48  sum_k<=d H^1_k (cofactor, exact)=48  paper=48  OWR [LB,UB]=(46, 48)
  d=5  S^1_d (BB, exact)=90  sum_k<=d H^1_k (cofactor, exact)=90  paper=90  OWR [LB,UB]=(88, 90)
  d=6  S^1_d (BB, exact)=156  sum_k<=d H^1_k (cofactor, exact)=156  paper=156  OWR [LB,UB]=(154, 156)
  d=7  S^1_d (BB, exact)=252  sum_k<=d H^1_k (cofactor, exact)=252  paper=252  OWR [LB,UB]=(250, 252)
  Prop 1.3 on Ex1.4: (l_B)_+^2 satisfies all degree-2 cofactor equations: True ; dim H^1_2 = 7
==============================================================================
PART B. Random GENERIC cells, Method A (BB form) on affinely transformed cells,
        exact ranks over Q; Method B (homogeneous cofactors) exact; compare with the paper
number of generic cells: 62
  n=3 m=3 base=[(-4, -10), (3, -1), (-2, 4)] apices=(2, 2, 3),(-1, 2, -1)  S(d<=5)=[1, 4, 11, 24, 48, 90]  H(k<=8)=[1, 3, 7, 13, 24, 42, 66, 96, 132]  [0s]
  n=3 m=3 base=[(-6, -2), (10, 2), (-8, 2)] apices=(0, 2, 1),(1, 3, -3)  S(d<=5)=[1, 4, 11, 24, 48, 90]  H(k<=8)=[1, 3, 7, 13, 24, 42, 66, 96, 132]  [1s]
  n=3 m=3 base=[(-6, -8), (1, -5), (3, 18)] apices=(-1, -1, 1),(-3, 2, -2)  S(d<=5)=[1, 4, 11, 24, 48, 90]  H(k<=8)=[1, 3, 7, 13, 24, 42, 66, 96, 132]  [1s]
  n=4 m=2 base=[(-4, -2), (-10, -8), (6, 3), (10, 8)] apices=(-2, 3, 3),(-1, 2, -1)  S(d<=5)=[1, 4, 11, 28, 61, 117]  H(k<=8)=[1, 3, 7, 17, 33, 56, 88, 128, 176]  [1s]
  n=4 m=2 base=[(-8, -6), (-3, -5), (12, 9), (3, 5)] apices=(-1, -3, 4),(1, 1, -2)  S(d<=5)=[1, 4, 11, 28, 61, 117]  H(k<=8)=[1, 3, 7, 17, 33, 56, 88, 128, 176]  [1s]
  n=4 m=2 base=[(1, -6), (3, -4), (-1, 6), (-6, 8)] apices=(1, 3, 1),(0, 0, -2)  S(d<=5)=[1, 4, 11, 28, 61, 117]  H(k<=8)=[1, 3, 7, 17, 33, 56, 88, 128, 176]  [2s]
  n=4 m=3 base=[(-15, -3), (-4, -2), (5, 1), (4, 6)] apices=(3, 0, 3),(-2, 0, -4)  S(d<=5)=[1, 4, 11, 26, 56, 110]  H(k<=8)=[1, 3, 7, 15, 30, 54, 86, 126, 174]  [2s]
  n=4 m=3 base=[(-10, -4), (1, 0), (5, 2), (-6, 3)] apices=(3, -2, 1),(0, -3, -4)  S(d<=5)=[1, 4, 11, 26, 56, 110]  H(k<=8)=[1, 3, 7, 15, 30, 54, 86, 126, 174]  [2s]
  n=4 m=3 base=[(-2, -3), (-3, -12), (3, 2), (3, 12)] apices=(3, -1, 4),(2, 2, -3)  S(d<=5)=[1, 4, 11, 26, 56, 110]  H(k<=8)=[1, 3, 7, 15, 30, 54, 86, 126, 174]  [2s]
  n=4 m=4 base=[(-8, -10), (-2, -3), (4, -2), (-4, 3)] apices=(2, 0, 2),(-1, -2, -4)  S(d<=5)=[1, 4, 11, 25, 55, 109]  H(k<=8)=[1, 3, 7, 14, 30, 54, 86, 126, 174]  [3s]
  n=4 m=4 base=[(-6, -1), (-3, -4), (15, 18), (2, 10)] apices=(1, 0, 1),(3, -1, -3)  S(d<=5)=[1, 4, 11, 25, 55, 109]  H(k<=8)=[1, 3, 7, 14, 30, 54, 86, 126, 174]  [3s]
  n=4 m=4 base=[(-10, -4), (-12, -9), (-12, -10), (3, 2)] apices=(3, -1, 4),(1, 3, -2)  S(d<=5)=[1, 4, 11, 25, 55, 109]  H(k<=8)=[1, 3, 7, 14, 30, 54, 86, 126, 174]  [3s]
  n=5 m=3 base=[(-3, -5), (3, -4), (3, 5), (-6, 8), (-12, 2)] apices=(3, 0, 3),(-3, 1, -4)  S(d<=5)=[1, 4, 11, 28, 64, 130]  H(k<=8)=[1, 3, 7, 17, 36, 66, 106, 156, 216]  [4s]
  n=5 m=3 base=[(-6, -1), (8, -6), (9, -3), (-4, 3), (-9, 3)] apices=(2, 2, 1),(1, 1, -3)  S(d<=5)=[1, 4, 11, 28, 64, 130]  H(k<=8)=[1, 3, 7, 17, 36, 66, 106, 156, 216]  [4s]
  n=5 m=3 base=[(-6, -3), (-6, -10), (2, 1), (6, 10), (1, 3)] apices=(2, -2, 4),(0, -1, -1)  S(d<=5)=[1, 4, 11, 28, 64, 130]  H(k<=8)=[1, 3, 7, 17, 36, 66, 106, 156, 216]  [5s]
  n=5 m=4 base=[(-15, -12), (2, -3), (12, -2), (-2, 3), (-5, 6)] apices=(-1, 0, 4),(2, 1, -1)  S(d<=5)=[1, 4, 11, 27, 63, 129]  H(k<=8)=[1, 3, 7, 16, 36, 66, 106, 156, 216]  [5s]
  n=5 m=4 base=[(-15, -6), (-6, -4), (5, -2), (3, 4), (-15, 6)] apices=(1, 1, 4),(-3, 1, -1)  S(d<=5)=[1, 4, 11, 27, 63, 129]  H(k<=8)=[1, 3, 7, 16, 36, 66, 106, 156, 216]  [5s]
  n=5 m=4 base=[(-15, -9), (-5, -4), (-18, -15), (5, 4), (-6, 5)] apices=(-3, 3, 3),(-1, 0, -3)  S(d<=5)=[1, 4, 11, 27, 63, 129]  H(k<=8)=[1, 3, 7, 16, 36, 66, 106, 156, 216]  [6s]
  n=5 m=5 base=[(-8, -2), (5, 1), (9, 15), (0, 1), (-10, 6)] apices=(-3, 1, 4),(0, 2, -1)  S(d<=5)=[1, 4, 11, 27, 63, 129]  H(k<=8)=[1, 3, 7, 16, 36, 66, 106, 156, 216]  [6s]
  n=5 m=5 base=[(-12, -2), (12, -9), (1, 0), (3, 6), (-6, 2)] apices=(0, 3, 4),(1, 3, -4)  S(d<=5)=[1, 4, 11, 27, 63, 129]  H(k<=8)=[1, 3, 7, 16, 36, 66, 106, 156, 216]  [7s]
  n=5 m=5 base=[(-3, -9), (10, 4), (18, 15), (4, 5), (-15, 3)] apices=(-2, -1, 3),(-2, 0, -4)  S(d<=5)=[1, 4, 11, 27, 63, 129]  H(k<=8)=[1, 3, 7, 16, 36, 66, 106, 156, 216]  [7s]
  n=6 m=3 base=[(-4, -1), (-9, -15), (2, -12), (4, 1), (3, 5), (-3, 18)] apices=(2, 2, 2),(3, 2, -2)  S(d<=5)=[1, 4, 11, 30, 72, 150]  H(k<=8)=[1, 3, 7, 19, 42, 78, 126, 186, 258]  [8s]
  n=6 m=3 base=[(-15, -9), (3, -15), (5, -4), (10, 6), (-2, 10), (-15, 12)] apices=(-1, -3, 4),(-2, -1, -3)  S(d<=5)=[1, 4, 11, 30, 72, 150]  H(k<=8)=[1, 3, 7, 19, 42, 78, 126, 186, 258]  [9s]
  n=6 m=3 base=[(6, -8), (3, -1), (15, -3), (-6, 8), (-3, 1), (-15, 3)] apices=(-1, -3, 1),(-2, -2, -4)  S(d<=5)=[1, 4, 11, 30, 72, 150]  H(k<=8)=[1, 3, 7, 19, 42, 78, 126, 186, 258]  [9s]
  n=6 m=4 base=[(-4, -2), (-3, -9), (-1, -6), (6, 3), (3, 6), (1, 6)] apices=(3, -1, 2),(-1, 3, -4)  S(d<=5)=[1, 4, 11, 29, 71, 149]  H(k<=8)=[1, 3, 7, 18, 42, 78, 126, 186, 258]  [10s]
  n=6 m=4 base=[(-9, -15), (10, -8), (6, -2), (8, 10), (3, 5), (-3, 1)] apices=(0, 2, 1),(2, 3, -1)  S(d<=5)=[1, 4, 11, 29, 71, 149]  H(k<=8)=[1, 3, 7, 18, 42, 78, 126, 186, 258]  [10s]
  n=6 m=4 base=[(-3, -2), (-4, -5), (3, -6), (15, -12), (9, 6), (-2, 4)] apices=(-2, 3, 1),(-2, 2, -4)  S(d<=5)=[1, 4, 11, 29, 71, 149]  H(k<=8)=[1, 3, 7, 18, 42, 78, 126, 186, 258]  [11s]
  n=6 m=5 base=[(-6, -9), (1, -4), (2, -6), (6, 9), (3, 6), (3, 15)] apices=(1, -2, 4),(1, 3, -3)  S(d<=5)=[1, 4, 11, 29, 71, 149]  H(k<=8)=[1, 3, 7, 18, 42, 78, 126, 186, 258]  [12s]
  n=6 m=5 base=[(-10, -4), (2, -5), (15, 6), (10, 12), (4, 6), (0, 2)] apices=(-2, -2, 2),(-1, -1, -4)  S(d<=5)=[1, 4, 11, 29, 71, 149]  H(k<=8)=[1, 3, 7, 18, 42, 78, 126, 186, 258]  [12s]
  n=6 m=5 base=[(2, -12), (15, 12), (1, 4), (-1, 6), (-3, 6), (-6, 8)] apices=(2, 1, 4),(2, 3, -2)  S(d<=5)=[1, 4, 11, 29, 71, 149]  H(k<=8)=[1, 3, 7, 18, 42, 78, 126, 186, 258]  [13s]
  n=6 m=6 base=[(4, -6), (3, -4), (4, 2), (2, 12), (-1, 5), (-12, 2)] apices=(-3, 0, 1),(2, 1, -3)  S(d<=5)=[1, 4, 11, 29, 71, 149]  H(k<=8)=[1, 3, 7, 18, 42, 78, 126, 186, 258]  [13s]
  n=6 m=6 base=[(1, -5), (3, -5), (10, -4), (6, 4), (-3, 3), (-12, 9)] apices=(2, 3, 2),(-2, 3, -2)  S(d<=5)=[1, 4, 11, 29, 71, 149]  H(k<=8)=[1, 3, 7, 18, 42, 78, 126, 186, 258]  [14s]
  n=6 m=6 base=[(-15, -3), (-2, -4), (6, -15), (6, 3), (6, 10), (-8, 6)] apices=(-2, 2, 3),(-3, 1, -3)  S(d<=5)=[1, 4, 11, 29, 71, 149]  H(k<=8)=[1, 3, 7, 18, 42, 78, 126, 186, 258]  [15s]
  n=7 m=4 base=[(-10, -4), (-6, -4), (-12, -15), (-2, -3), (10, 4), (8, 10), (6, 9)] apices=(-3, 0, 4),(-3, -3, -1)  S(d<=4)=[1, 4, 11, 31, 79]  H(k<=6)=[1, 3, 7, 20, 48, 90, 146]  [15s]
  n=7 m=4 base=[(-4, -1), (-6, -8), (9, -3), (8, 2), (3, 4), (-4, 2), (-9, 3)] apices=(2, -3, 1),(-3, 0, -1)  S(d<=4)=[1, 4, 11, 31, 79]  H(k<=6)=[1, 3, 7, 20, 48, 90, 146]  [15s]
  n=7 m=5 base=[(-8, -10), (4, -5), (18, -15), (12, 15), (-9, 12), (-8, 10), (-10, 4)] apices=(3, -1, 3),(0, -3, -1)  S(d<=4)=[1, 4, 11, 31, 79]  H(k<=6)=[1, 3, 7, 20, 48, 90, 146]  [15s]
  n=7 m=5 base=[(-4, -10), (-1, -5), (3, -18), (1, -5), (10, -6), (4, 10), (-10, 6)] apices=(3, 1, 4),(1, -1, -1)  S(d<=4)=[1, 4, 11, 31, 79]  H(k<=6)=[1, 3, 7, 20, 48, 90, 146]  [16s]
  n=7 m=6 base=[(-12, -15), (-6, -10), (-2, -5), (2, -3), (3, -3), (5, -2), (6, 15)] apices=(3, 1, 3),(0, 1, -4)  S(d<=4)=[1, 4, 11, 31, 79]  H(k<=6)=[1, 3, 7, 20, 48, 90, 146]  [16s]
  n=7 m=6 base=[(-6, -8), (-9, -15), (3, -12), (6, -2), (9, 15), (-3, 6), (-18, 3)] apices=(0, -2, 1),(1, 3, -4)  S(d<=4)=[1, 4, 11, 31, 79]  H(k<=6)=[1, 3, 7, 20, 48, 90, 146]  [16s]
  n=7 m=7 base=[(-9, -12), (-1, -6), (15, -6), (18, 3), (2, 2), (6, 15), (3, 12)] apices=(3, -1, 1),(-1, 1, -3)  S(d<=4)=[1, 4, 11, 31, 79]  H(k<=6)=[1, 3, 7, 20, 48, 90, 146]  [17s]
  n=7 m=7 base=[(-3, -3), (0, -2), (3, -15), (4, -1), (-9, 12), (-8, 10), (-1, 0)] apices=(-3, 3, 1),(-1, 0, -3)  S(d<=4)=[1, 4, 11, 31, 79]  H(k<=6)=[1, 3, 7, 20, 48, 90, 146]  [17s]
  n=8 m=4 base=[(-15, -12), (-10, -12), (0, -1), (12, -3), (15, 12), (15, 18), (0, 1), (-4, 1)] apices=(-2, -1, 2),(1, 2, -4)  S(d<=4)=[1, 4, 11, 33, 87]  H(k<=6)=[1, 3, 7, 22, 54, 102, 166]  [17s]
  n=8 m=4 base=[(-8, -6), (-5, -6), (3, -2), (10, -4), (12, 9), (5, 6), (-9, 6), (-15, 6)] apices=(-2, -3, 4),(3, 1, -3)  S(d<=4)=[1, 4, 11, 33, 87]  H(k<=6)=[1, 3, 7, 22, 54, 102, 166]  [18s]
  n=8 m=5 base=[(-6, -4), (3, -5), (12, -15), (3, 0), (3, 2), (-9, 15), (-6, 2), (-2, 0)] apices=(-3, 2, 4),(-1, 0, -3)  S(d<=4)=[1, 4, 11, 33, 87]  H(k<=6)=[1, 3, 7, 22, 54, 102, 166]  [18s]
  n=8 m=5 base=[(-12, -3), (-1, -3), (6, -2), (8, 2), (1, 3), (3, 18), (-18, 15), (-9, 3)] apices=(0, -1, 2),(0, -3, -1)  S(d<=4)=[1, 4, 11, 33, 87]  H(k<=6)=[1, 3, 7, 22, 54, 102, 166]  [18s]
  n=8 m=6 base=[(-6, -2), (6, -10), (6, -8), (2, -1), (10, 6), (4, 3), (-3, 5), (-6, 8)] apices=(-3, 0, 4),(2, -3, -4)  S(d<=4)=[1, 4, 11, 33, 87]  H(k<=6)=[1, 3, 7, 22, 54, 102, 166]  [18s]
  n=8 m=6 base=[(-3, -2), (-9, -15), (5, -1), (3, 2), (4, 5), (3, 5), (-3, 12), (-1, 2)] apices=(-2, 3, 4),(-3, -1, -1)  S(d<=4)=[1, 4, 11, 33, 87]  H(k<=6)=[1, 3, 7, 22, 54, 102, 166]  [19s]
  n=8 m=7 base=[(-9, -6), (-12, -15), (1, -1), (15, -12), (10, -6), (5, -1), (5, 6), (8, 10)] apices=(-1, -1, 4),(0, 0, -2)  S(d<=4)=[1, 4, 11, 33, 87]  H(k<=6)=[1, 3, 7, 22, 54, 102, 166]  [19s]
  n=8 m=7 base=[(-2, -2), (-3, -4), (12, -15), (10, -12), (12, 3), (6, 10), (-8, 10), (-15, 12)] apices=(2, -2, 1),(-1, 2, -2)  S(d<=4)=[1, 4, 11, 33, 87]  H(k<=6)=[1, 3, 7, 22, 54, 102, 166]  [20s]
  n=8 m=8 base=[(-3, -1), (-15, -18), (18, -15), (9, -6), (6, -2), (4, 2), (8, 6), (5, 4)] apices=(3, 0, 4),(2, -1, -4)  S(d<=4)=[1, 4, 11, 33, 87]  H(k<=6)=[1, 3, 7, 22, 54, 102, 166]  [20s]
  n=8 m=8 base=[(-12, -15), (-6, -10), (-1, -4), (10, -12), (6, -5), (15, 6), (-3, 6), (-4, 3)] apices=(0, -1, 3),(-2, 3, -1)  S(d<=4)=[1, 4, 11, 33, 87]  H(k<=6)=[1, 3, 7, 22, 54, 102, 166]  [21s]
  n=9 m=5 base=[(10, -8), (6, -3), (10, -4), (4, -1), (12, -2), (-15, 12), (-5, 2), (-4, 1), (-12, 2)] apices=(-1, -1, 4),(3, 2, -4)  S(d<=4)=[1, 4, 11, 35, 95]  H(k<=6)=[1, 3, 7, 24, 60, 114, 186]  [21s]
  n=9 m=6 base=[(-12, -2), (-10, -8), (-12, -10), (10, -12), (5, -3), (18, 3), (5, 4), (-1, 1), (-5, 3)] apices=(-3, -1, 1),(1, 3, -2)  S(d<=4)=[1, 4, 11, 35, 95]  H(k<=6)=[1, 3, 7, 24, 60, 114, 186]  [22s]
  n=9 m=7 base=[(-3, -5), (6, -15), (12, -9), (4, -1), (4, 1), (8, 10), (9, 15), (-10, 12), (-8, 2)] apices=(3, -2, 2),(-3, -2, -1)  S(d<=4)=[1, 4, 11, 35, 95]  H(k<=6)=[1, 3, 7, 24, 60, 114, 186]  [22s]
  n=9 m=8 base=[(-6, -1), (-4, -2), (4, -10), (6, -10), (8, -10), (2, -2), (12, 2), (6, 9), (-10, 2)] apices=(-2, -1, 2),(-2, 1, -2)  S(d<=4)=[1, 4, 11, 35, 95]  H(k<=6)=[1, 3, 7, 24, 60, 114, 186]  [22s]
  n=9 m=9 base=[(1, -5), (6, -4), (15, -9), (8, 10), (1, 4), (-12, 15), (-15, 18), (-5, 2), (-18, 3)] apices=(-2, 1, 3),(3, 0, -1)  S(d<=4)=[1, 4, 11, 35, 95]  H(k<=6)=[1, 3, 7, 24, 60, 114, 186]  [23s]
  n=10 m=5 base=[(-8, -2), (-9, -6), (-2, -8), (2, -5), (4, -2), (4, 1), (6, 4), (3, 12), (-4, 10), (-2, 1)] apices=(-2, -1, 1),(3, 1, -4)  S(d<=4)=[1, 4, 11, 37, 103]  H(k<=6)=[1, 3, 7, 26, 66, 126, 206]  [24s]
  n=10 m=6 base=[(-6, -2), (-5, -2), (-15, -18), (5, -6), (9, 3), (5, 2), (5, 4), (15, 18), (-1, 5), (-15, 18)] apices=(2, 1, 1),(1, -1, -1)  S(d<=4)=[1, 4, 11, 37, 103]  H(k<=6)=[1, 3, 7, 26, 66, 126, 206]  [24s]
  n=10 m=7 base=[(-8, -10), (1, -2), (6, -8), (1, -1), (1, 0), (2, 3), (-2, 4), (-9, 12), (-6, 1), (-1, 0)] apices=(1, 1, 2),(1, 0, -2)  S(d<=4)=[1, 4, 11, 37, 103]  H(k<=6)=[1, 3, 7, 26, 66, 126, 206]  [25s]
  n=10 m=8 base=[(-2, -1), (-1, -2), (2, -12), (2, -4), (9, -6), (2, 0), (-2, 8), (-3, 6), (-6, 2), (-2, 0)] apices=(0, 1, 2),(3, -2, -1)  S(d<=4)=[1, 4, 11, 37, 103]  H(k<=6)=[1, 3, 7, 26, 66, 126, 206]  [25s]
  n=10 m=9 base=[(-15, -3), (-6, -10), (8, -6), (6, -1), (10, 2), (4, 2), (10, 12), (1, 2), (-1, 4), (-3, 9)] apices=(-3, -2, 2),(-3, 0, -3)  S(d<=4)=[1, 4, 11, 37, 103]  H(k<=6)=[1, 3, 7, 26, 66, 126, 206]  [26s]
  n=10 m=10 base=[(-8, -6), (-15, -18), (-12, -15), (4, -5), (10, -6), (6, 5), (2, 3), (-3, 18), (-3, 12), (-18, 3)] apices=(-3, -1, 3),(3, 2, -4)  S(d<=4)=[1, 4, 11, 37, 103]  H(k<=6)=[1, 3, 7, 26, 66, 126, 206]  [26s]
PART B comparisons: S^1_d: 343  H^1_k: 500  mismatches so far: 0
==============================================================================
PART B2. Higher degree: S^1_6 (BB, exact) for n<=6 and S^1_7 mod two primes for n<=5
  n=3 m=3 S^1_6={'Q': 156, 2147483647: 156} paper=156  S^1_7(mod p)={2147483647: 252, 2147483629: 252} paper=252
  n=4 m=2 S^1_6={'Q': 205, 2147483647: 205} paper=205  S^1_7(mod p)={2147483647: 333, 2147483629: 333} paper=333
  n=4 m=3 S^1_6={'Q': 196, 2147483647: 196} paper=196  S^1_7(mod p)={2147483647: 322, 2147483629: 322} paper=322
  n=4 m=4 S^1_6={'Q': 195, 2147483647: 195} paper=195  S^1_7(mod p)={2147483647: 321, 2147483629: 321} paper=321
  n=5 m=3 S^1_6={'Q': 236, 2147483647: 236} paper=236  S^1_7(mod p)={2147483647: 392, 2147483629: 392} paper=392
  n=5 m=4 S^1_6={'Q': 235, 2147483647: 235} paper=235  S^1_7(mod p)={2147483647: 391, 2147483629: 391} paper=391
  n=5 m=5 S^1_6={'Q': 235, 2147483647: 235} paper=235  S^1_7(mod p)={2147483647: 391, 2147483629: 391} paper=391
  n=6 m=3 S^1_6={'Q': 276, 2147483647: 276} paper=276
  n=6 m=4 S^1_6={'Q': 275, 2147483647: 275} paper=275
  n=6 m=5 S^1_6={'Q': 275, 2147483647: 275} paper=275
  n=6 m=6 S^1_6={'Q': 275, 2147483647: 275} paper=275
==============================================================================
PART C. Symmetric / special-position GENERIC cells (exact, BB form)
  square, apex above v0, w=(1,1) diagonal            n=4 m=2  S^1_d, d=0..5: [1, 4, 11, 28, 61, 117]  paper: [1, 4, 11, 28, 61, 117]
  square, w=(1,2)                                    n=4 m=2  S^1_d, d=0..5: [1, 4, 11, 28, 61, 117]  paper: [1, 4, 11, 28, 61, 117]
  square, mirror-symmetric apices (1,1,+-1)          n=4 m=2  S^1_d, d=0..5: [1, 4, 11, 28, 61, 117]  paper: [1, 4, 11, 28, 61, 117]
  (configuration 'rhombus, antipodal apices (collinear; skipped here)' is collinear, not generic -- checked in part D instead)
  hexagon m=3, w=(1,-1)                              n=6 m=3  S^1_d, d=0..5: [1, 4, 11, 30, 72, 150]  paper: [1, 4, 11, 30, 72, 150]
  hexagon m=3, w=(2,1)                               n=6 m=3  S^1_d, d=0..5: [1, 4, 11, 30, 72, 150]  paper: [1, 4, 11, 30, 72, 150]
  hexagon m=3, mirror apices                         n=6 m=3  S^1_d, d=0..5: [1, 4, 11, 30, 72, 150]  paper: [1, 4, 11, 30, 72, 150]
  octagon m=4, w=(1,2)                               n=8 m=4  S^1_d, d=0..5: [1, 4, 11, 33, 87, 189]  paper: [1, 4, 11, 33, 87, 189]
  octagon m=4, w=(2,-1)                              n=8 m=4  S^1_d, d=0..5: [1, 4, 11, 33, 87, 189]  paper: [1, 4, 11, 33, 87, 189]
  triangle m=3, w=(1,1)                              n=3 m=3  S^1_d, d=0..5: [1, 4, 11, 24, 48, 90]  paper: [1, 4, 11, 24, 48, 90]
  equilateral-like triangle, w=(1,-1)                n=3 m=3  S^1_d, d=0..5: [1, 4, 11, 24, 48, 90]  paper: [1, 4, 11, 24, 48, 90]
  pentagon m=5, w=(1,2)                              n=5 m=5  S^1_d, d=0..5: [1, 4, 11, 27, 63, 129]  paper: [1, 4, 11, 27, 63, 129]
  pentagon n=5 m=3 (two opposite pairs + 1)          n=5 m=3  S^1_d, d=0..5: [1, 4, 11, 28, 64, 130]  paper: [1, 4, 11, 28, 64, 130]
  pentagon n=5 m=3                                   n=5 m=3  S^1_d, d=0..5: [1, 4, 11, 28, 64, 130]  paper: [1, 4, 11, 28, 64, 130]
==============================================================================
PART D. Collinear / coplanar I / coplanar II cells vs OWR Theorems 1-3 (exact, BB form)
  collinear  n=3 m=3 collinear n=3 m=3                      S^1_d (d=2..5)=[11, 26, 53, 98] OWR=[11, 26, 53, 98]
  collinear  n=3 m=3 collinear n=3 m=3                      S^1_d (d=2..5)=[11, 26, 53, 98] OWR=[11, 26, 53, 98]
  coplanarI  n=3 m=3 coplanarI n=3 m=3                      S^1_d (d=2..5)=[11, 24, 49, 92] OWR=[11, 24, 49, 92]
  coplanarI  n=3 m=3 coplanarI n=3 m=3                      S^1_d (d=2..5)=[11, 24, 49, 92] OWR=[11, 24, 49, 92]
  collinear  n=4 m=2 collinear n=4 m=2                      S^1_d (d=2..5)=[13, 32, 68, 128] OWR=[13, 32, 68, 128]
  collinear  n=4 m=2 collinear n=4 m=2                      S^1_d (d=2..5)=[13, 32, 68, 128] OWR=[13, 32, 68, 128]
  coplanarII n=4 m=2 coplanarII n=4 m=2                     S^1_d (d=2..5)=[12, 30, 64, 122] OWR=[12, 30, 64, 122]
  coplanarII n=4 m=2 coplanarII n=4 m=2                     S^1_d (d=2..5)=[12, 30, 64, 122] OWR=[12, 30, 64, 122]
  collinear  n=4 m=3 collinear n=4 m=3                      S^1_d (d=2..5)=[12, 30, 64, 122] OWR=[12, 30, 64, 122]
  collinear  n=4 m=3 collinear n=4 m=3                      S^1_d (d=2..5)=[12, 30, 64, 122] OWR=[12, 30, 64, 122]
  coplanarI  n=4 m=3 coplanarI n=4 m=3                      S^1_d (d=2..5)=[11, 26, 57, 112] OWR=[11, 26, 57, 112]
  coplanarI  n=4 m=3 coplanarI n=4 m=3                      S^1_d (d=2..5)=[11, 26, 57, 112] OWR=[11, 26, 57, 112]
  coplanarII n=4 m=3 coplanarII n=4 m=3                     S^1_d (d=2..5)=[12, 28, 60, 116] OWR=[12, 28, 60, 116]
  coplanarII n=4 m=3 coplanarII n=4 m=3                     S^1_d (d=2..5)=[12, 28, 60, 116] OWR=[12, 28, 60, 116]
  collinear  n=4 m=4 collinear n=4 m=4                      S^1_d (d=2..5)=[12, 30, 64, 122] OWR=[12, 30, 64, 122]
  collinear  n=4 m=4 collinear n=4 m=4                      S^1_d (d=2..5)=[12, 30, 64, 122] OWR=[12, 30, 64, 122]
  coplanarI  n=4 m=4 coplanarI n=4 m=4                      S^1_d (d=2..5)=[11, 26, 57, 112] OWR=[11, 26, 57, 112]
  coplanarI  n=4 m=4 coplanarI n=4 m=4                      S^1_d (d=2..5)=[11, 26, 57, 112] OWR=[11, 26, 57, 112]
  collinear  n=5 m=3 collinear n=5 m=3                      S^1_d (d=2..5)=[13, 34, 75, 146] OWR=[13, 34, 75, 146]
  collinear  n=5 m=3 collinear n=5 m=3                      S^1_d (d=2..5)=[13, 34, 75, 146] OWR=[13, 34, 75, 146]
  coplanarI  n=5 m=3 coplanarI n=5 m=3                      S^1_d (d=2..5)=[11, 28, 65, 132] OWR=[11, 28, 65, 132]
  coplanarI  n=5 m=3 coplanarI n=5 m=3                      S^1_d (d=2..5)=[11, 28, 65, 132] OWR=[11, 28, 65, 132]
  coplanarII n=5 m=3 coplanarII n=5 m=3                     S^1_d (d=2..5)=[12, 30, 68, 136] OWR=[12, 30, 68, 136]
  coplanarII n=5 m=3 coplanarII n=5 m=3                     S^1_d (d=2..5)=[12, 30, 68, 136] OWR=[12, 30, 68, 136]
  collinear  n=5 m=4 collinear n=5 m=4                      S^1_d (d=2..5)=[13, 34, 75, 146] OWR=[13, 34, 75, 146]
  collinear  n=5 m=4 collinear n=5 m=4                      S^1_d (d=2..5)=[13, 34, 75, 146] OWR=[13, 34, 75, 146]
  coplanarI  n=5 m=4 coplanarI n=5 m=4                      S^1_d (d=2..5)=[11, 28, 65, 132] OWR=[11, 28, 65, 132]
  coplanarI  n=5 m=4 coplanarI n=5 m=4                      S^1_d (d=2..5)=[11, 28, 65, 132] OWR=[11, 28, 65, 132]
  coplanarII n=5 m=4 coplanarII n=5 m=4                     S^1_d (d=2..5)=[12, 30, 68, 136] OWR=[12, 30, 68, 136]
  coplanarII n=5 m=4 coplanarII n=5 m=4                     S^1_d (d=2..5)=[12, 30, 68, 136] OWR=[12, 30, 68, 136]
  collinear  n=5 m=5 collinear n=5 m=5                      S^1_d (d=2..5)=[13, 34, 75, 146] OWR=[13, 34, 75, 146]
  collinear  n=5 m=5 collinear n=5 m=5                      S^1_d (d=2..5)=[13, 34, 75, 146] OWR=[13, 34, 75, 146]
  coplanarI  n=5 m=5 coplanarI n=5 m=5                      S^1_d (d=2..5)=[11, 28, 65, 132] OWR=[11, 28, 65, 132]
  coplanarI  n=5 m=5 coplanarI n=5 m=5                      S^1_d (d=2..5)=[11, 28, 65, 132] OWR=[11, 28, 65, 132]
  collinear  n=6 m=3 collinear n=6 m=3                      S^1_d (d=2..5)=[14, 38, 86, 170] OWR=[14, 38, 86, 170]
  collinear  n=6 m=3 collinear n=6 m=3                      S^1_d (d=2..5)=[14, 38, 86, 170] OWR=[14, 38, 86, 170]
  coplanarII n=6 m=3 coplanarII n=6 m=3                     S^1_d (d=2..5)=[12, 32, 76, 156] OWR=[12, 32, 76, 156]
  coplanarII n=6 m=3 coplanarII n=6 m=3                     S^1_d (d=2..5)=[12, 32, 76, 156] OWR=[12, 32, 76, 156]
  collinear  n=6 m=4 collinear n=6 m=4                      S^1_d (d=2..5)=[14, 38, 86, 170] OWR=[14, 38, 86, 170]
  collinear  n=6 m=4 collinear n=6 m=4                      S^1_d (d=2..5)=[14, 38, 86, 170] OWR=[14, 38, 86, 170]
  coplanarI  n=6 m=4 coplanarI n=6 m=4                      S^1_d (d=2..5)=[11, 30, 73, 152] OWR=[11, 30, 73, 152]
  coplanarI  n=6 m=4 coplanarI n=6 m=4                      S^1_d (d=2..5)=[11, 30, 73, 152] OWR=[11, 30, 73, 152]
  coplanarII n=6 m=4 coplanarII n=6 m=4                     S^1_d (d=2..5)=[12, 32, 76, 156] OWR=[12, 32, 76, 156]
  coplanarII n=6 m=4 coplanarII n=6 m=4                     S^1_d (d=2..5)=[12, 32, 76, 156] OWR=[12, 32, 76, 156]
  collinear  n=6 m=5 collinear n=6 m=5                      S^1_d (d=2..5)=[14, 38, 86, 170] OWR=[14, 38, 86, 170]
  collinear  n=6 m=5 collinear n=6 m=5                      S^1_d (d=2..5)=[14, 38, 86, 170] OWR=[14, 38, 86, 170]
  coplanarI  n=6 m=5 coplanarI n=6 m=5                      S^1_d (d=2..5)=[11, 30, 73, 152] OWR=[11, 30, 73, 152]
  coplanarI  n=6 m=5 coplanarI n=6 m=5                      S^1_d (d=2..5)=[11, 30, 73, 152] OWR=[11, 30, 73, 152]
  coplanarII n=6 m=5 coplanarII n=6 m=5                     S^1_d (d=2..5)=[12, 32, 76, 156] OWR=[12, 32, 76, 156]
  coplanarII n=6 m=5 coplanarII n=6 m=5                     S^1_d (d=2..5)=[12, 32, 76, 156] OWR=[12, 32, 76, 156]
  collinear  n=6 m=6 collinear n=6 m=6                      S^1_d (d=2..5)=[14, 38, 86, 170] OWR=[14, 38, 86, 170]
  collinear  n=6 m=6 collinear n=6 m=6                      S^1_d (d=2..5)=[14, 38, 86, 170] OWR=[14, 38, 86, 170]
  coplanarI  n=6 m=6 coplanarI n=6 m=6                      S^1_d (d=2..5)=[11, 30, 73, 152] OWR=[11, 30, 73, 152]
  coplanarI  n=6 m=6 coplanarI n=6 m=6                      S^1_d (d=2..5)=[11, 30, 73, 152] OWR=[11, 30, 73, 152]
  collinear  n=7 m=4 collinear n=7 m=4                      S^1_d (d=2..4)=[15, 42, 97] OWR=[15, 42, 97]
  collinear  n=7 m=4 collinear n=7 m=4                      S^1_d (d=2..4)=[15, 42, 97] OWR=[15, 42, 97]
  coplanarI  n=7 m=4 coplanarI n=7 m=4                      S^1_d (d=2..4)=[11, 32, 81] OWR=[11, 32, 81]
  coplanarI  n=7 m=4 coplanarI n=7 m=4                      S^1_d (d=2..4)=[11, 32, 81] OWR=[11, 32, 81]
  coplanarII n=7 m=4 coplanarII n=7 m=4                     S^1_d (d=2..4)=[12, 34, 84] OWR=[12, 34, 84]
  coplanarII n=7 m=4 coplanarII n=7 m=4                     S^1_d (d=2..4)=[12, 34, 84] OWR=[12, 34, 84]
  collinear  n=7 m=5 collinear n=7 m=5                      S^1_d (d=2..4)=[15, 42, 97] OWR=[15, 42, 97]
  collinear  n=7 m=5 collinear n=7 m=5                      S^1_d (d=2..4)=[15, 42, 97] OWR=[15, 42, 97]
  coplanarI  n=7 m=5 coplanarI n=7 m=5                      S^1_d (d=2..4)=[11, 32, 81] OWR=[11, 32, 81]
  coplanarI  n=7 m=5 coplanarI n=7 m=5                      S^1_d (d=2..4)=[11, 32, 81] OWR=[11, 32, 81]
  coplanarII n=7 m=5 coplanarII n=7 m=5                     S^1_d (d=2..4)=[12, 34, 84] OWR=[12, 34, 84]
  coplanarII n=7 m=5 coplanarII n=7 m=5                     S^1_d (d=2..4)=[12, 34, 84] OWR=[12, 34, 84]
  collinear  n=7 m=6 collinear n=7 m=6                      S^1_d (d=2..4)=[15, 42, 97] OWR=[15, 42, 97]
  collinear  n=7 m=6 collinear n=7 m=6                      S^1_d (d=2..4)=[15, 42, 97] OWR=[15, 42, 97]
  coplanarI  n=7 m=6 coplanarI n=7 m=6                      S^1_d (d=2..4)=[11, 32, 81] OWR=[11, 32, 81]
  coplanarI  n=7 m=6 coplanarI n=7 m=6                      S^1_d (d=2..4)=[11, 32, 81] OWR=[11, 32, 81]
  coplanarII n=7 m=6 coplanarII n=7 m=6                     S^1_d (d=2..4)=[12, 34, 84] OWR=[12, 34, 84]
  coplanarII n=7 m=6 coplanarII n=7 m=6                     S^1_d (d=2..4)=[12, 34, 84] OWR=[12, 34, 84]
  collinear  n=7 m=7 collinear n=7 m=7                      S^1_d (d=2..4)=[15, 42, 97] OWR=[15, 42, 97]
  collinear  n=7 m=7 collinear n=7 m=7                      S^1_d (d=2..4)=[15, 42, 97] OWR=[15, 42, 97]
  coplanarI  n=7 m=7 coplanarI n=7 m=7                      S^1_d (d=2..4)=[11, 32, 81] OWR=[11, 32, 81]
  coplanarI  n=7 m=7 coplanarI n=7 m=7                      S^1_d (d=2..4)=[11, 32, 81] OWR=[11, 32, 81]
  collinear  n=8 m=4 collinear n=8 m=4                      S^1_d (d=2..4)=[16, 46, 108] OWR=[16, 46, 108]
  collinear  n=8 m=4 collinear n=8 m=4                      S^1_d (d=2..4)=[16, 46, 108] OWR=[16, 46, 108]
  coplanarII n=8 m=4 coplanarII n=8 m=4                     S^1_d (d=2..4)=[12, 36, 92] OWR=[12, 36, 92]
  coplanarII n=8 m=4 coplanarII n=8 m=4                     S^1_d (d=2..4)=[12, 36, 92] OWR=[12, 36, 92]
  collinear  n=8 m=5 collinear n=8 m=5                      S^1_d (d=2..4)=[16, 46, 108] OWR=[16, 46, 108]
  collinear  n=8 m=5 collinear n=8 m=5                      S^1_d (d=2..4)=[16, 46, 108] OWR=[16, 46, 108]
  coplanarI  n=8 m=5 coplanarI n=8 m=5                      S^1_d (d=2..4)=[11, 34, 89] OWR=[11, 34, 89]
  coplanarI  n=8 m=5 coplanarI n=8 m=5                      S^1_d (d=2..4)=[11, 34, 89] OWR=[11, 34, 89]
  coplanarII n=8 m=5 coplanarII n=8 m=5                     S^1_d (d=2..4)=[12, 36, 92] OWR=[12, 36, 92]
  coplanarII n=8 m=5 coplanarII n=8 m=5                     S^1_d (d=2..4)=[12, 36, 92] OWR=[12, 36, 92]
  collinear  n=8 m=6 collinear n=8 m=6                      S^1_d (d=2..4)=[16, 46, 108] OWR=[16, 46, 108]
  collinear  n=8 m=6 collinear n=8 m=6                      S^1_d (d=2..4)=[16, 46, 108] OWR=[16, 46, 108]
  coplanarI  n=8 m=6 coplanarI n=8 m=6                      S^1_d (d=2..4)=[11, 34, 89] OWR=[11, 34, 89]
  coplanarI  n=8 m=6 coplanarI n=8 m=6                      S^1_d (d=2..4)=[11, 34, 89] OWR=[11, 34, 89]
  coplanarII n=8 m=6 coplanarII n=8 m=6                     S^1_d (d=2..4)=[12, 36, 92] OWR=[12, 36, 92]
  coplanarII n=8 m=6 coplanarII n=8 m=6                     S^1_d (d=2..4)=[12, 36, 92] OWR=[12, 36, 92]
  collinear  n=8 m=7 collinear n=8 m=7                      S^1_d (d=2..4)=[16, 46, 108] OWR=[16, 46, 108]
  collinear  n=8 m=7 collinear n=8 m=7                      S^1_d (d=2..4)=[16, 46, 108] OWR=[16, 46, 108]
  coplanarI  n=8 m=7 coplanarI n=8 m=7                      S^1_d (d=2..4)=[11, 34, 89] OWR=[11, 34, 89]
  coplanarI  n=8 m=7 coplanarI n=8 m=7                      S^1_d (d=2..4)=[11, 34, 89] OWR=[11, 34, 89]
  coplanarII n=8 m=7 coplanarII n=8 m=7                     S^1_d (d=2..4)=[12, 36, 92] OWR=[12, 36, 92]
  coplanarII n=8 m=7 coplanarII n=8 m=7                     S^1_d (d=2..4)=[12, 36, 92] OWR=[12, 36, 92]
  collinear  n=8 m=8 collinear n=8 m=8                      S^1_d (d=2..4)=[16, 46, 108] OWR=[16, 46, 108]
  collinear  n=8 m=8 collinear n=8 m=8                      S^1_d (d=2..4)=[16, 46, 108] OWR=[16, 46, 108]
  coplanarI  n=8 m=8 coplanarI n=8 m=8                      S^1_d (d=2..4)=[11, 34, 89] OWR=[11, 34, 89]
  coplanarI  n=8 m=8 coplanarI n=8 m=8                      S^1_d (d=2..4)=[11, 34, 89] OWR=[11, 34, 89]
  collinear  n=4 m=2 square collinear symmetric             S^1_d (d=2..5)=[13, 32, 68, 128] OWR=[13, 32, 68, 128]
  coplanarII n=4 m=2 square coplanar II (w along axis)      S^1_d (d=2..5)=[12, 30, 64, 122] OWR=[12, 30, 64, 122]
  collinear  n=6 m=3 hexagon collinear                      S^1_d (d=2..5)=[14, 38, 86, 170] OWR=[14, 38, 86, 170]
  coplanarII n=6 m=3 hexagon coplanar II (w=(1,1))          S^1_d (d=2..5)=[12, 32, 76, 156] OWR=[12, 32, 76, 156]
  coplanarI  n=3 m=3 triangle coplanar I                    S^1_d (d=2..5)=[11, 24, 49, 92] OWR=[11, 24, 49, 92]
  collinear  n=4 m=2 rhombus collinear through v0           S^1_d (d=2..5)=[13, 32, 68, 128] OWR=[13, 32, 68, 128]
PART D comparisons: 384
==============================================================================
PART E. Prop 1.3 basis certificate on all generic cells of part B (exact)
  certified (l_B)_+^2 in H^1_2 and dim H^1_2 = 7 on 62 generic cells; mismatches so far: 0
==============================================================================
TOTAL mismatches: 0
elapsed 34.6s
