=== d = 2, word ABBB: all binary tori with at most 25 cells ===
number of shapes (m <= n, mn <= 25): 46; total colourings examined: 160098126
maximum concentration over all shapes: 8/5
   1x1  N= 1 max count  0  concentration      0 = 0.0000  maximisers 2
   1x2  N= 2 max count  0  concentration      0 = 0.0000  maximisers 4
   1x3  N= 3 max count  0  concentration      0 = 0.0000  maximisers 8
   1x4  N= 4 max count  6  concentration    3/2 = 1.5000  maximisers 4
   1x5  N= 5 max count  6  concentration    6/5 = 1.2000  maximisers 10
   1x6  N= 6 max count  6  concentration      1 = 1.0000  maximisers 24
   1x7  N= 7 max count  6  concentration    6/7 = 0.8571  maximisers 56
   1x8  N= 8 max count 12  concentration    3/2 = 1.5000  maximisers 4
   1x9  N= 9 max count 12  concentration    4/3 = 1.3333  maximisers 18
   1x10 N=10 max count 12  concentration    6/5 = 1.2000  maximisers 60
   1x11 N=11 max count 12  concentration  12/11 = 1.0909  maximisers 176
   1x12 N=12 max count 18  concentration    3/2 = 1.5000  maximisers 4
   1x13 N=13 max count 18  concentration  18/13 = 1.3846  maximisers 26
   1x14 N=14 max count 18  concentration    9/7 = 1.2857  maximisers 112
   1x15 N=15 max count 18  concentration    6/5 = 1.2000  maximisers 400
   1x16 N=16 max count 24  concentration    3/2 = 1.5000  maximisers 4
   1x17 N=17 max count 24  concentration  24/17 = 1.4118  maximisers 34
   1x18 N=18 max count 24  concentration    4/3 = 1.3333  maximisers 180
   1x19 N=19 max count 24  concentration  24/19 = 1.2632  maximisers 760
   1x20 N=20 max count 30  concentration    3/2 = 1.5000  maximisers 4
   1x21 N=21 max count 30  concentration   10/7 = 1.4286  maximisers 42
   1x22 N=22 max count 30  concentration  15/11 = 1.3636  maximisers 264
   1x23 N=23 max count 30  concentration  30/23 = 1.3043  maximisers 1288
   1x24 N=24 max count 36  concentration    3/2 = 1.5000  maximisers 4
   1x25 N=25 max count 36  concentration  36/25 = 1.4400  maximisers 50
   2x2  N= 4 max count  0  concentration      0 = 0.0000  maximisers 16
   2x3  N= 6 max count  4  concentration    2/3 = 0.6667  maximisers 24
   2x4  N= 8 max count 12  concentration    3/2 = 1.5000  maximisers 8
   2x5  N=10 max count 12  concentration    6/5 = 1.2000  maximisers 30
   2x6  N=12 max count 12  concentration      1 = 1.0000  maximisers 240
   2x7  N=14 max count 16  concentration    8/7 = 1.1429  maximisers 28
   2x8  N=16 max count 24  concentration    3/2 = 1.5000  maximisers 8
   2x9  N=18 max count 24  concentration    4/3 = 1.3333  maximisers 54
   2x10 N=20 max count 24  concentration    6/5 = 1.2000  maximisers 660
   2x11 N=22 max count 28  concentration  14/11 = 1.2727  maximisers 44
   2x12 N=24 max count 36  concentration    3/2 = 1.5000  maximisers 8
   3x3  N= 9 max count  0  concentration      0 = 0.0000  maximisers 512
   3x4  N=12 max count 18  concentration    3/2 = 1.5000  maximisers 4
   3x5  N=15 max count 18  concentration    6/5 = 1.2000  maximisers 160
   3x6  N=18 max count 18  concentration      1 = 1.0000  maximisers 6540
   3x7  N=21 max count 22  concentration  22/21 = 1.0476  maximisers 609
   3x8  N=24 max count 36  concentration    3/2 = 1.5000  maximisers 4
   4x4  N=16 max count 24  concentration    3/2 = 1.5000  maximisers 96
   4x5  N=20 max count 30  concentration    3/2 = 1.5000  maximisers 4
   4x6  N=24 max count 36  concentration    3/2 = 1.5000  maximisers 32
   5x5  N=25 max count 40  concentration    8/5 = 1.6000  maximisers 10  <== maximum
maximisers on (5, 5):
    BBBAB/BABBB/BBBBA/BBABB/ABBBB
    BBABB/BBBBA/BABBB/BBBAB/ABBBB
    BBBBA/BBABB/ABBBB/BBBAB/BABBB
    BBBAB/ABBBB/BBABB/BBBBA/BABBB
    BBBBA/BABBB/BBBAB/ABBBB/BBABB
    ABBBB/BBBAB/BABBB/BBBBA/BBABB
    BABBB/BBBBA/BBABB/ABBBB/BBBAB
    ABBBB/BBABB/BBBBA/BABBB/BBBAB
    BBABB/ABBBB/BBBAB/BABBB/BBBBA
    BABBB/BBBAB/ABBBB/BBABB/BBBBA
  equal to the ten lattice grids A = {(i, s i + t mod 5)}, s in {2,3}: True

=== d = 3, word ABB: all binary tori with at most 24 cells ===
number of shapes (a <= b <= c, abc <= 24): 51; total colourings examined: 127924302
maximum concentration over all shapes: 6
  (1, 1, 1) N= 1 max   0 conc     0  maximisers     2   Theorem D predicts: < 6  ok
  (1, 1, 2) N= 2 max   0 conc     0  maximisers     4   Theorem D predicts: < 6  ok
  (1, 1, 3) N= 3 max  18 conc     6  maximisers     3   Theorem D predicts: 6 with 3 grids  ok
  (1, 1, 4) N= 4 max  18 conc   9/2  maximisers     8   Theorem D predicts: < 6  ok
  (1, 1, 5) N= 5 max  18 conc  18/5  maximisers    20   Theorem D predicts: < 6  ok
  (1, 1, 6) N= 6 max  36 conc     6  maximisers     3   Theorem D predicts: 6 with 3 grids  ok
  (1, 1, 7) N= 7 max  36 conc  36/7  maximisers    14   Theorem D predicts: < 6  ok
  (1, 1, 8) N= 8 max  36 conc   9/2  maximisers    48   Theorem D predicts: < 6  ok
  (1, 1, 9) N= 9 max  54 conc     6  maximisers     3   Theorem D predicts: 6 with 3 grids  ok
  (1, 1, 10) N=10 max  54 conc  27/5  maximisers    20   Theorem D predicts: < 6  ok
  (1, 1, 11) N=11 max  54 conc 54/11  maximisers    88   Theorem D predicts: < 6  ok
  (1, 1, 12) N=12 max  72 conc     6  maximisers     3   Theorem D predicts: 6 with 3 grids  ok
  (1, 1, 13) N=13 max  72 conc 72/13  maximisers    26   Theorem D predicts: < 6  ok
  (1, 1, 14) N=14 max  72 conc  36/7  maximisers   140   Theorem D predicts: < 6  ok
  (1, 1, 15) N=15 max  90 conc     6  maximisers     3   Theorem D predicts: 6 with 3 grids  ok
  (1, 1, 16) N=16 max  90 conc  45/8  maximisers    32   Theorem D predicts: < 6  ok
  (1, 1, 17) N=17 max  90 conc 90/17  maximisers   204   Theorem D predicts: < 6  ok
  (1, 1, 18) N=18 max 108 conc     6  maximisers     3   Theorem D predicts: 6 with 3 grids  ok
  (1, 1, 19) N=19 max 108 conc 108/19  maximisers    38   Theorem D predicts: < 6  ok
  (1, 1, 20) N=20 max 108 conc  27/5  maximisers   280   Theorem D predicts: < 6  ok
  (1, 1, 21) N=21 max 126 conc     6  maximisers     3   Theorem D predicts: 6 with 3 grids  ok
  (1, 1, 22) N=22 max 126 conc 63/11  maximisers    44   Theorem D predicts: < 6  ok
  (1, 1, 23) N=23 max 126 conc 126/23  maximisers   368   Theorem D predicts: < 6  ok
  (1, 1, 24) N=24 max 144 conc     6  maximisers     3   Theorem D predicts: 6 with 3 grids  ok
  (1, 2, 2) N= 4 max   0 conc     0  maximisers    16   Theorem D predicts: < 6  ok
  (1, 2, 3) N= 6 max  36 conc     6  maximisers     3   Theorem D predicts: 6 with 3 grids  ok
  (1, 2, 4) N= 8 max  36 conc   9/2  maximisers    56   Theorem D predicts: < 6  ok
  (1, 2, 5) N=10 max  48 conc  24/5  maximisers    20   Theorem D predicts: < 6  ok
  (1, 2, 6) N=12 max  72 conc     6  maximisers     3   Theorem D predicts: 6 with 3 grids  ok
  (1, 2, 7) N=14 max  72 conc  36/7  maximisers    56   Theorem D predicts: < 6  ok
  (1, 2, 8) N=16 max  72 conc   9/2  maximisers  1000   Theorem D predicts: < 6  ok
  (1, 2, 9) N=18 max 108 conc     6  maximisers     3   Theorem D predicts: 6 with 3 grids  ok
  (1, 2, 10) N=20 max 108 conc  27/5  maximisers    80   Theorem D predicts: < 6  ok
  (1, 2, 11) N=22 max 108 conc 54/11  maximisers  1496   Theorem D predicts: < 6  ok
  (1, 2, 12) N=24 max 144 conc     6  maximisers     3   Theorem D predicts: 6 with 3 grids  ok
  (1, 3, 3) N= 9 max  54 conc     6  maximisers    12   Theorem D predicts: 6 with 12 grids  ok
  (1, 3, 4) N=12 max  72 conc     6  maximisers     3   Theorem D predicts: 6 with 3 grids  ok
  (1, 3, 5) N=15 max  90 conc     6  maximisers     3   Theorem D predicts: 6 with 3 grids  ok
  (1, 3, 6) N=18 max 108 conc     6  maximisers    12   Theorem D predicts: 6 with 12 grids  ok
  (1, 3, 7) N=21 max 126 conc     6  maximisers     3   Theorem D predicts: 6 with 3 grids  ok
  (1, 3, 8) N=24 max 144 conc     6  maximisers     3   Theorem D predicts: 6 with 3 grids  ok
  (1, 4, 4) N=16 max  84 conc  21/4  maximisers   432   Theorem D predicts: < 6  ok
  (1, 4, 5) N=20 max 108 conc  27/5  maximisers   380   Theorem D predicts: < 6  ok
  (1, 4, 6) N=24 max 144 conc     6  maximisers     3   Theorem D predicts: 6 with 3 grids  ok
  (2, 2, 2) N= 8 max   0 conc     0  maximisers   256   Theorem D predicts: < 6  ok
  (2, 2, 3) N=12 max  72 conc     6  maximisers     3   Theorem D predicts: 6 with 3 grids  ok
  (2, 2, 4) N=16 max  72 conc   9/2  maximisers  2336   Theorem D predicts: < 6  ok
  (2, 2, 5) N=20 max  96 conc  24/5  maximisers   160   Theorem D predicts: < 6  ok
  (2, 2, 6) N=24 max 144 conc     6  maximisers     3   Theorem D predicts: 6 with 3 grids  ok
  (2, 3, 3) N=18 max 108 conc     6  maximisers    12   Theorem D predicts: 6 with 12 grids  ok
  (2, 3, 4) N=24 max 144 conc     6  maximisers     3   Theorem D predicts: 6 with 3 grids  ok
agreement with the classification of Theorem D (value 6 iff some side divisible by 3; number of extremal grids 3*(3^k-1)/2): True
maximisers with value 6 that are not hyperplane grids (or count mismatch): 0

=== examples of the claim, counted from the definition ===
5x5 lattice grid A at (i, 2i mod 5): ct = 40, cells 25, concentration 8/5, density of A 1/5
   same grid, word BABBB: concentration 8/5
   same grid, word ABBBB: concentration 8/5
4x4 grid AABA/BBAB/ABBB/ABBB: ct = 24, cells 16, concentration 3/2, density of A 3/8; c + (4/7) alpha = 12/7
   delta = 1 - c/(8/5) = 1/16; total variation distance to (1/5, 4/5) = 7/40; ratio TV/delta = 14/5
3x3x3 stacked grid (A on the plane y1 = 0): ct = 162, cells 27, concentration 6
3x3x3 hyperplane grid a = (1, 1, 0): concentration 6
3x3x3 hyperplane grid a = (1, 1, 1): concentration 6
3x3x3 hyperplane grid a = (1, -1, 1): concentration 6
