G62 from 19 edge-orbit representatives: |E| = 114
  [ok]   E(G62) is invariant under sigma and iota (iota(p) = (3,3,3) - p)
  [ok]   E(H73) is invariant under sigma and iota' (iota'(p) = (4,4,4) - p)
  [ok]   finder file counterexample_G62.json equals the rebuilt G62
  [ok]   referee file induced73_c3inv.json equals the rebuilt H73
==============================================================================
G62
  |V|=62 |E|=114 squares=54 cubes=1 chi=1 box=[0, 0, 0]..[3, 3, 3] missing lattice edges=24
  [ok]   f-vector (62, 114, 54, 1)
  [ok]   G is connected
  [ok]   induced = False
  [ok]   bounding box ([0, 0, 0], [3, 3, 3])
  [ok]   raw corner definition agrees with the criterion of Lemma 2.1 at all vertices
  [ok]   number of corners = 0 (found 0)
  maximal cells: 49
  [ok]   boundary of boundary = 0
  Betti numbers over GF(2): (1, 0, 0, 0), over GF(1000003): (1, 0, 0, 0)
  [ok]   X(G) has the homology of a point (GF(2) and GF(p))
  [ok]   X(G) collapses to a vertex: 115 elementary collapses, replayed and checked
  sections (axis, alpha): pieces as (V,E,F) and collapsible?
    x = 0    : (15,17,3)
    x = 1/2  : (14,13,0)
    x = 1    : (16,21,6)
    x = 3/2  : (10,10,1)
    x = 2    : (16,21,6)
    x = 5/2  : (14,13,0)
    x = 3    : (15,17,3)
    y = 0    : (15,17,3)
    y = 1/2  : (14,13,0)
    y = 1    : (16,21,6)
    y = 3/2  : (10,10,1)
    y = 2    : (16,21,6)
    y = 5/2  : (14,13,0)
    y = 3    : (15,17,3)
    z = 0    : (15,17,3)
    z = 1/2  : (14,13,0)
    z = 1    : (16,21,6)
    z = 3/2  : (10,10,1)
    z = 2    : (16,21,6)
    z = 5/2  : (14,13,0)
    z = 3    : (15,17,3)
  nonempty sections: 21, pieces: 21, every nonempty section is a single piece: True
  [ok]   every piece of every integer and half-integer section is collapsible
  condition (3), axis x: 2 of 3 half-integer pieces violate it [(0.5, False, False), (1.5, True, True), (2.5, False, False)]
  [ok]   condition (3) fails in 2 pieces of axis x
  condition (3), axis y: 2 of 3 half-integer pieces violate it [(0.5, False, False), (1.5, True, True), (2.5, False, False)]
  [ok]   condition (3) fails in 2 pieces of axis y
  condition (3), axis z: 2 of 3 half-integer pieces violate it [(0.5, False, False), (1.5, True, True), (2.5, False, False)]
  [ok]   condition (3) fails in 2 pieces of axis z
  cubes: [((1, 1, 1), (0, 1, 2))]
  the 24 lattice edges between vertices of G62 that are not edges of G62:
    002-102, 011-021, 012-022, 013-023, 020-021, 031-032, 101-102, 103-203, 110-210, 113-213, 120-220, 123-223, 123-133, 130-230, 131-132, 200-210, 201-202, 231-331, 231-232, 301-302, 310-320, 311-321, 312-322, 312-313
  induced closure of V(G62): |E|=138 squares=102 cubes=25 corners=12
  [ok]   induced closure: 138 edges, 102 squares, 25 cubes, 12 corners
  vertex-orbit table for G62 (group <sigma, iota_3>)
    001  orbit 6  +x +y +z               spanned cube lacks 002-102, 101-102
    002  orbit 6  +y +z -z               two opposite neighbours
    003  orbit 6  +x +y -z               spanned cube lacks 002-102
    011  orbit 6  +x -y +z -z            two opposite neighbours
    012  orbit 6  +x -y +z -z            two opposite neighbours
    013  orbit 6  +x -y -z               spanned cube lacks 002-102
    021  orbit 6  +x +y +z               spanned cube lacks 031-032, 131-132
    022  orbit 6  +x +y +z -z            two opposite neighbours
    023  orbit 6  +x +y -z               spanned cube lacks 123-133
    111  orbit 2  +x -x +y -y +z -z      two opposite neighbours
    112  orbit 6  +x -x +y -y +z -z      two opposite neighbours
  [ok]   G62: 19 edge orbits, all of size 6
  [ok]   G62: vertex orbit sizes [2, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6]
  [ok]   G62: listed square 0,0,1:xy is a square of X
  [ok]   G62: orbit of listed square 0,0,1:xy has 6 elements
  [ok]   G62: listed square 0,0,1:yz is a square of X
  [ok]   G62: orbit of listed square 0,0,1:yz has 6 elements
  [ok]   G62: listed square 0,0,2:yz is a square of X
  [ok]   G62: orbit of listed square 0,0,2:yz has 6 elements
  [ok]   G62: listed square 0,0,3:xy is a square of X
  [ok]   G62: orbit of listed square 0,0,3:xy has 6 elements
  [ok]   G62: listed square 0,1,1:xz is a square of X
  [ok]   G62: orbit of listed square 0,1,1:xz has 6 elements
  [ok]   G62: listed square 0,1,2:xz is a square of X
  [ok]   G62: orbit of listed square 0,1,2:xz has 6 elements
  [ok]   G62: listed square 0,2,1:xy is a square of X
  [ok]   G62: orbit of listed square 0,2,1:xy has 6 elements
  [ok]   G62: listed square 0,2,1:xz is a square of X
  [ok]   G62: orbit of listed square 0,2,1:xz has 6 elements
  [ok]   G62: listed square 1,1,1:xy is a square of X
  [ok]   G62: orbit of listed square 1,1,1:xy has 6 elements
  [ok]   G62: the 9 listed square orbits are distinct and cover all 54 squares
  [ok]   G62: the only cube is [1,2]^3
  [ok]   G62: missing lattice edges = orbits of 002-102, 011-021, 012-022, 013-023
  [ok]   G62: corners of the induced closure = orbits of 001 and 003
  [ok]   min62 is invariant under sigma and iota
==============================================================================
min62
  |V|=62 |E|=114 squares=54 cubes=1 chi=1 box=[0, 0, 0]..[3, 3, 3] missing lattice edges=24
  [ok]   f-vector (62, 114, 54, 1)
  [ok]   G is connected
  [ok]   induced = False
  [ok]   raw corner definition agrees with the criterion of Lemma 2.1 at all vertices
  [ok]   number of corners = 0 (found 0)
  maximal cells: 49
  [ok]   boundary of boundary = 0
  Betti numbers over GF(2): (1, 0, 0, 0), over GF(1000003): (1, 0, 0, 0)
  [ok]   X(G) has the homology of a point (GF(2) and GF(p))
  [ok]   X(G) collapses to a vertex: 115 elementary collapses, replayed and checked
  sections (axis, alpha): pieces as (V,E,F) and collapsible?
    x = 0    : (15,17,3)
    x = 1/2  : (14,13,0)
    x = 1    : (16,21,6)
    x = 3/2  : (10,10,1)
    x = 2    : (16,21,6)
    x = 5/2  : (14,13,0)
    x = 3    : (15,17,3)
    y = 0    : (15,17,3)
    y = 1/2  : (14,13,0)
    y = 1    : (16,21,6)
    y = 3/2  : (10,10,1)
    y = 2    : (16,21,6)
    y = 5/2  : (14,13,0)
    y = 3    : (15,17,3)
    z = 0    : (15,17,3)
    z = 1/2  : (14,13,0)
    z = 1    : (16,21,6)
    z = 3/2  : (10,10,1)
    z = 2    : (16,21,6)
    z = 5/2  : (14,13,0)
    z = 3    : (15,17,3)
  nonempty sections: 21, pieces: 21, every nonempty section is a single piece: True
  [ok]   every piece of every integer and half-integer section is collapsible
  condition (3), axis x: 2 of 3 half-integer pieces violate it [(0.5, False, False), (1.5, True, True), (2.5, False, False)]
  [ok]   condition (3) fails in 2 pieces of axis x
  condition (3), axis y: 2 of 3 half-integer pieces violate it [(0.5, False, False), (1.5, True, True), (2.5, False, False)]
  [ok]   condition (3) fails in 2 pieces of axis y
  condition (3), axis z: 2 of 3 half-integer pieces violate it [(0.5, False, False), (1.5, True, True), (2.5, False, False)]
  [ok]   condition (3) fails in 2 pieces of axis z
  [ok]   min62 differs from G62 as an edge set
  [ok]   min62 is the image of G62 under the reflection (x,y,z) -> (x,z,y): it is congruent to G62, not a new example
  [ok]   exactly 6 of the 48 symmetries of [0,3]^3 map G62 onto min62 (found 6)
==============================================================================
H73
  |V|=73 |E|=132 squares=60 cubes=0 chi=1 box=[0, 0, 0]..[4, 4, 4] missing lattice edges=0
  [ok]   f-vector (73, 132, 60, 0)
  [ok]   G is connected
  [ok]   induced = True
  [ok]   bounding box ([0, 0, 0], [4, 4, 4])
  [ok]   raw corner definition agrees with the criterion of Lemma 2.1 at all vertices
  [ok]   number of corners = 0 (found 0)
  maximal cells: 60
  [ok]   boundary of boundary = 0
  Betti numbers over GF(2): (1, 0, 0, 0), over GF(1000003): (1, 0, 0, 0)
  [ok]   X(G) has the homology of a point (GF(2) and GF(p))
  [ok]   X(G) collapses to a vertex: 132 elementary collapses, replayed and checked
  sections (axis, alpha): pieces as (V,E,F) and collapsible?
    x = 0    : (15,18,4)
    x = 1/2  : (13,12,0)
    x = 1    : (15,18,4)
    x = 3/2  : (9,8,0)
    x = 2    : (13,16,4)
    x = 5/2  : (9,8,0)
    x = 3    : (15,18,4)
    x = 7/2  : (13,12,0)
    x = 4    : (15,18,4)
    y = 0    : (15,18,4)
    y = 1/2  : (13,12,0)
    y = 1    : (15,18,4)
    y = 3/2  : (9,8,0)
    y = 2    : (13,16,4)
    y = 5/2  : (9,8,0)
    y = 3    : (15,18,4)
    y = 7/2  : (13,12,0)
    y = 4    : (15,18,4)
    z = 0    : (15,18,4)
    z = 1/2  : (13,12,0)
    z = 1    : (15,18,4)
    z = 3/2  : (9,8,0)
    z = 2    : (13,16,4)
    z = 5/2  : (9,8,0)
    z = 3    : (15,18,4)
    z = 7/2  : (13,12,0)
    z = 4    : (15,18,4)
  nonempty sections: 27, pieces: 27, every nonempty section is a single piece: True
  [ok]   every piece of every integer and half-integer section is collapsible
  condition (3), axis x: 2 of 4 half-integer pieces violate it [(0.5, False, False), (1.5, True, True), (2.5, True, True), (3.5, False, False)]
  [ok]   condition (3) fails in 2 pieces of axis x
  condition (3), axis y: 2 of 4 half-integer pieces violate it [(0.5, False, False), (1.5, True, True), (2.5, True, True), (3.5, False, False)]
  [ok]   condition (3) fails in 2 pieces of axis y
  condition (3), axis z: 2 of 4 half-integer pieces violate it [(0.5, False, False), (1.5, True, True), (2.5, True, True), (3.5, False, False)]
  [ok]   condition (3) fails in 2 pieces of axis z
  vertex-orbit table for H73 (group <sigma, iota_4>)
    003  orbit 6  +x +y +z               spanned cube lacks 014-114, 104-114, 113-114, vertex 114
    004  orbit 6  +x +y -z               spanned cube lacks 014-114, 104-114, 113-114, vertex 114
    013  orbit 6  +x +y -y +z            two opposite neighbours
    014  orbit 6  +y -y -z               two opposite neighbours
    023  orbit 6  +x -y +z               spanned cube lacks 014-114, 113-114, 114-124, vertex 114
    024  orbit 6  +x +y -y -z            two opposite neighbours
    031  orbit 6  +x +y +z -z            two opposite neighbours
    032  orbit 6  +x +y -z               spanned cube lacks 041-141, 131-141, 141-142, vertex 141
    034  orbit 6  +x +y -y               two opposite neighbours
    113  orbit 6  +x -x +y -y            two opposite neighbours
    122  orbit 6  +x +y +z               spanned cube lacks 133-233, 223-233, 232-233, vertex 233
    123  orbit 6  +x -x +y -y +z -z      two opposite neighbours
    222  orbit 1  +x -x +y -y +z -z      two opposite neighbours
  [ok]   H73: 22 edge orbits, all of size 6
  [ok]   H73: vertex orbit sizes [1, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6]
  [ok]   H73: listed square 0,0,3:xy is a square of X
  [ok]   H73: orbit of listed square 0,0,3:xy has 6 elements
  [ok]   H73: listed square 0,0,3:xz is a square of X
  [ok]   H73: orbit of listed square 0,0,3:xz has 6 elements
  [ok]   H73: listed square 0,0,3:yz is a square of X
  [ok]   H73: orbit of listed square 0,0,3:yz has 6 elements
  [ok]   H73: listed square 0,1,3:xy is a square of X
  [ok]   H73: orbit of listed square 0,1,3:xy has 6 elements
  [ok]   H73: listed square 0,1,3:yz is a square of X
  [ok]   H73: orbit of listed square 0,1,3:yz has 6 elements
  [ok]   H73: listed square 0,2,3:xz is a square of X
  [ok]   H73: orbit of listed square 0,2,3:xz has 6 elements
  [ok]   H73: listed square 0,2,4:xy is a square of X
  [ok]   H73: orbit of listed square 0,2,4:xy has 6 elements
  [ok]   H73: listed square 0,3,1:xz is a square of X
  [ok]   H73: orbit of listed square 0,3,1:xz has 6 elements
  [ok]   H73: listed square 1,1,3:xy is a square of X
  [ok]   H73: orbit of listed square 1,1,3:xy has 6 elements
  [ok]   H73: listed square 1,2,2:xy is a square of X
  [ok]   H73: orbit of listed square 1,2,2:xy has 6 elements
  [ok]   H73: the 10 listed square orbits are distinct and cover all 60 squares
  [ok]   H73: no 3-cube
==============================================================================
G62sub
  |V|=231 |E|=450 squares=228 cubes=8 chi=1 box=[0, 0, 0]..[6, 6, 6] missing lattice edges=0
  [ok]   f-vector (231, 450, 228, 8)
  [ok]   G is connected
  [ok]   induced = True
  [ok]   bounding box ([0, 0, 0], [6, 6, 6])
  [ok]   raw corner definition agrees with the criterion of Lemma 2.1 at all vertices
  [ok]   number of corners = 0 (found 0)
  maximal cells: 200
  [ok]   boundary of boundary = 0
  Betti numbers over GF(2): (1, 0, 0, 0), over GF(1000003): (1, 0, 0, 0)
  [ok]   X(G) has the homology of a point (GF(2) and GF(p))
  [ok]   X(G) collapses to a vertex: 458 elementary collapses, replayed and checked
  sections (axis, alpha): pieces as (V,E,F) and collapsible?
    x = 0    : (35,46,12)
    x = 1/2  : (27,26,0)
    x = 1    : (27,26,0)
    x = 3/2  : (27,26,0)
    x = 2    : (43,66,24)
    x = 5/2  : (21,24,4)
    x = 3    : (21,24,4)
    x = 7/2  : (21,24,4)
    x = 4    : (43,66,24)
    x = 9/2  : (27,26,0)
    x = 5    : (27,26,0)
    x = 11/2 : (27,26,0)
    x = 6    : (35,46,12)
    y = 0    : (35,46,12)
    y = 1/2  : (27,26,0)
    y = 1    : (27,26,0)
    y = 3/2  : (27,26,0)
    y = 2    : (43,66,24)
    y = 5/2  : (21,24,4)
    y = 3    : (21,24,4)
    y = 7/2  : (21,24,4)
    y = 4    : (43,66,24)
    y = 9/2  : (27,26,0)
    y = 5    : (27,26,0)
    y = 11/2 : (27,26,0)
    y = 6    : (35,46,12)
    z = 0    : (35,46,12)
    z = 1/2  : (27,26,0)
    z = 1    : (27,26,0)
    z = 3/2  : (27,26,0)
    z = 2    : (43,66,24)
    z = 5/2  : (21,24,4)
    z = 3    : (21,24,4)
    z = 7/2  : (21,24,4)
    z = 4    : (43,66,24)
    z = 9/2  : (27,26,0)
    z = 5    : (27,26,0)
    z = 11/2 : (27,26,0)
    z = 6    : (35,46,12)
  nonempty sections: 39, pieces: 39, every nonempty section is a single piece: True
  [ok]   every piece of every integer and half-integer section is collapsible
  condition (3), axis x: 4 of 6 half-integer pieces violate it [(0.5, False, True), (1.5, True, False), (2.5, True, True), (3.5, True, True), (4.5, False, True), (5.5, True, False)]
  condition (3), axis y: 4 of 6 half-integer pieces violate it [(0.5, False, True), (1.5, True, False), (2.5, True, True), (3.5, True, True), (4.5, False, True), (5.5, True, False)]
  condition (3), axis z: 4 of 6 half-integer pieces violate it [(0.5, False, True), (1.5, True, False), (2.5, True, True), (3.5, True, True), (4.5, False, True), (5.5, True, False)]
  [ok]   |V(G62^(2))| = number of cells of X(G62) = 231
==============================================================================
T12
  |V|=12 |E|=19 squares=8 cubes=0 chi=1 box=[0, 0, 0]..[2, 1, 1] missing lattice edges=1
  [ok]   f-vector (12, 19, 8, 0)
  [ok]   G is connected
  [ok]   induced = False
  [ok]   bounding box ([0, 0, 0], [2, 1, 1])
  [ok]   raw corner definition agrees with the criterion of Lemma 2.1 at all vertices
  [ok]   number of corners = 0 (found 0)
  maximal cells: 8
  [ok]   boundary of boundary = 0
  Betti numbers over GF(2): (1, 0, 0, 0), over GF(1000003): (1, 0, 0, 0)
  [ok]   X(G) has the homology of a point (GF(2) and GF(p))
  [ok]   X(G) collapses to a vertex: 19 elementary collapses, replayed and checked
  sections (axis, alpha): pieces as (V,E,F) and collapsible?
    x = 0    : (4,4,1)
    x = 1/2  : (4,3,0)
    x = 1    : (4,3,0)
    x = 3/2  : (4,3,0)
    x = 2    : (4,4,1)
    y = 0    : (6,6,0)*NOT*
    y = 1/2  : (6,6,0)*NOT*
    y = 1    : (6,7,2)
    z = 0    : (6,7,2)
    z = 1/2  : (5,4,0)
    z = 1    : (6,7,2)
  nonempty sections: 11, pieces: 11, every nonempty section is a single piece: True
  non-collapsible pieces (axis, alpha, V, E, F): [(1, 0.0, 6, 6, 0), (1, 0.5, 6, 6, 0)]
  [ok]   failing pieces are exactly [(1, 0.0), (1, 0.5)]
  condition (3), axis x: 2 of 2 half-integer pieces violate it [(0.5, False, True), (1.5, True, False)]
  condition (3), axis y: 1 of 1 half-integer pieces violate it [(0.5, True, False)]
  condition (3), axis z: 1 of 1 half-integer pieces violate it [(0.5, False, False)]
  T12: numbers of maximal cells at the vertices: [2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4]
  [ok]   T12: every vertex lies in at least two maximal squares
==============================================================================
T12sub
  |V|=39 |E|=70 squares=32 cubes=0 chi=1 box=[0, 0, 0]..[4, 2, 2] missing lattice edges=0
  [ok]   f-vector (39, 70, 32, 0)
  [ok]   G is connected
  [ok]   induced = True
  [ok]   bounding box ([0, 0, 0], [4, 2, 2])
  [ok]   raw corner definition agrees with the criterion of Lemma 2.1 at all vertices
  [ok]   number of corners = 0 (found 0)
  maximal cells: 32
  [ok]   boundary of boundary = 0
  Betti numbers over GF(2): (1, 0, 0, 0), over GF(1000003): (1, 0, 0, 0)
  [ok]   X(G) has the homology of a point (GF(2) and GF(p))
  [ok]   X(G) collapses to a vertex: 70 elementary collapses, replayed and checked
  sections (axis, alpha): pieces as (V,E,F) and collapsible?
    x = 0    : (9,12,4)
    x = 1/2  : (7,6,0)
    x = 1    : (7,6,0)
    x = 3/2  : (7,6,0)
    x = 2    : (7,6,0)
    x = 5/2  : (7,6,0)
    x = 3    : (7,6,0)
    x = 7/2  : (7,6,0)
    x = 4    : (9,12,4)
    y = 0    : (12,12,0)*NOT*
    y = 1/2  : (12,12,0)*NOT*
    y = 1    : (12,12,0)*NOT*
    y = 3/2  : (12,12,0)*NOT*
    y = 2    : (15,22,8)
    z = 0    : (15,22,8)
    z = 1/2  : (9,8,0)
    z = 1    : (9,8,0)
    z = 3/2  : (9,8,0)
    z = 2    : (15,22,8)
  nonempty sections: 19, pieces: 19, every nonempty section is a single piece: True
  non-collapsible pieces (axis, alpha, V, E, F): [(1, 0.0, 12, 12, 0), (1, 0.5, 12, 12, 0), (1, 1.0, 12, 12, 0), (1, 1.5, 12, 12, 0)]
  [ok]   failing pieces are exactly [(1, 0.0), (1, 0.5), (1, 1.0), (1, 1.5)]
  condition (3), axis x: 2 of 4 half-integer pieces violate it [(0.5, False, True), (1.5, True, True), (2.5, True, True), (3.5, True, False)]
  condition (3), axis y: 1 of 2 half-integer pieces violate it [(0.5, True, True), (1.5, True, False)]
  condition (3), axis z: 2 of 2 half-integer pieces violate it [(0.5, False, True), (1.5, True, False)]
==============================================================================
G49 rebuilt from its description: |V| = 49, |E| = 87
  [ok]   G49: 49 points, the induced subgraph on them has 99 edges, 12 of them removed
  [ok]   E(G49) is invariant under sigma
  [ok]   third referee file explore_444_sigma_v58_comp0_edges.json equals the rebuilt G49
==============================================================================
G49
  |V|=49 |E|=87 squares=39 cubes=0 chi=1 box=[0, 0, 0]..[3, 3, 3] missing lattice edges=12
  [ok]   f-vector (49, 87, 39, 0)
  [ok]   G is connected
  [ok]   induced = False
  [ok]   bounding box ([0, 0, 0], [3, 3, 3])
  [ok]   raw corner definition agrees with the criterion of Lemma 2.1 at all vertices
  [ok]   number of corners = 0 (found 0)
  maximal cells: 39
  [ok]   boundary of boundary = 0
  Betti numbers over GF(2): (1, 0, 0, 0), over GF(1000003): (1, 0, 0, 0)
  [ok]   X(G) has the homology of a point (GF(2) and GF(p))
  [ok]   X(G) collapses to a vertex: 87 elementary collapses, replayed and checked
  sections (axis, alpha): pieces as (V,E,F) and collapsible?
    x = 0    : (12,14,3)
    x = 1/2  : (11,10,0)
    x = 1    : (12,14,3)
    x = 3/2  : (7,6,0)
    x = 2    : (13,16,4)
    x = 5/2  : (11,10,0)
    x = 3    : (12,14,3)
    y = 0    : (12,14,3)
    y = 1/2  : (11,10,0)
    y = 1    : (12,14,3)
    y = 3/2  : (7,6,0)
    y = 2    : (13,16,4)
    y = 5/2  : (11,10,0)
    y = 3    : (12,14,3)
    z = 0    : (12,14,3)
    z = 1/2  : (11,10,0)
    z = 1    : (12,14,3)
    z = 3/2  : (7,6,0)
    z = 2    : (13,16,4)
    z = 5/2  : (11,10,0)
    z = 3    : (12,14,3)
  nonempty sections: 21, pieces: 21, every nonempty section is a single piece: True
  [ok]   every piece of every integer and half-integer section is collapsible
  condition (3), axis x: 2 of 3 half-integer pieces violate it [(0.5, False, False), (1.5, True, True), (2.5, False, False)]
  [ok]   condition (3) fails in 2 pieces of axis x
  condition (3), axis y: 2 of 3 half-integer pieces violate it [(0.5, False, False), (1.5, True, True), (2.5, False, False)]
  [ok]   condition (3) fails in 2 pieces of axis y
  condition (3), axis z: 2 of 3 half-integer pieces violate it [(0.5, False, False), (1.5, True, True), (2.5, False, False)]
  [ok]   condition (3) fails in 2 pieces of axis z
  the 12 lattice edges between vertices of G49 that are not edges of G49:
    012-022, 022-023, 031-131, 103-113, 113-213, 120-220, 131-132, 201-202, 202-302, 220-230, 310-311, 311-321
  [ok]   G49: 29 edge orbits under <sigma>, all of size 3
  edge-orbit representatives: 002-102 002-012 002-003 003-103 003-013 012-112 012-013 013-113 013-023 021-121 021-031 021-022 022-122 022-032 023-123 023-033 031-032 032-132 032-033 033-133 112-212 112-122 112-113 113-123 122-222 122-132 122-123 123-133 132-133
  [ok]   G49: 17 vertex orbits under <sigma> (222 fixed)
  x-sections of G49 for alpha = 0, 1/2, ..., 3 (V,E,F): [(12, 14, 3), (11, 10, 0), (12, 14, 3), (7, 6, 0), (13, 16, 4), (11, 10, 0), (12, 14, 3)]
  [ok]   G49: the seven x-sections are single pieces with the (V,E,F) of Remark 6.3
  [ok]   E(G59) is invariant under sigma
==============================================================================
G59
  |V|=59 |E|=108 squares=51 cubes=1 chi=1 box=[0, 0, 0]..[3, 3, 3] missing lattice edges=21
  [ok]   f-vector (59, 108, 51, 1)
  [ok]   G is connected
  [ok]   induced = False
  [ok]   bounding box ([0, 0, 0], [3, 3, 3])
  [ok]   raw corner definition agrees with the criterion of Lemma 2.1 at all vertices
  [ok]   number of corners = 0 (found 0)
  maximal cells: 46
  [ok]   boundary of boundary = 0
  Betti numbers over GF(2): (1, 0, 0, 0), over GF(1000003): (1, 0, 0, 0)
  [ok]   X(G) has the homology of a point (GF(2) and GF(p))
  [ok]   X(G) collapses to a vertex: 109 elementary collapses, replayed and checked
  sections (axis, alpha): pieces as (V,E,F) and collapsible?
    x = 0    : (15,18,4)
    x = 1/2  : (13,12,0)
    x = 1    : (16,21,6)
    x = 3/2  : (10,10,1)
    x = 2    : (15,19,5)
    x = 5/2  : (13,12,0)
    x = 3    : (13,14,2)
    y = 0    : (15,18,4)
    y = 1/2  : (13,12,0)
    y = 1    : (16,21,6)
    y = 3/2  : (10,10,1)
    y = 2    : (15,19,5)
    y = 5/2  : (13,12,0)
    y = 3    : (13,14,2)
    z = 0    : (15,18,4)
    z = 1/2  : (13,12,0)
    z = 1    : (16,21,6)
    z = 3/2  : (10,10,1)
    z = 2    : (15,19,5)
    z = 5/2  : (13,12,0)
    z = 3    : (13,14,2)
  nonempty sections: 21, pieces: 21, every nonempty section is a single piece: True
  [ok]   every piece of every integer and half-integer section is collapsible
  condition (3), axis x: 2 of 3 half-integer pieces violate it [(0.5, False, False), (1.5, True, True), (2.5, False, False)]
  [ok]   condition (3) fails in 2 pieces of axis x
  condition (3), axis y: 2 of 3 half-integer pieces violate it [(0.5, False, False), (1.5, True, True), (2.5, False, False)]
  [ok]   condition (3) fails in 2 pieces of axis y
  condition (3), axis z: 2 of 3 half-integer pieces violate it [(0.5, False, False), (1.5, True, True), (2.5, False, False)]
  [ok]   condition (3) fails in 2 pieces of axis z
==============================================================================
symmetry groups of the examples in their boxes (the 48 symmetries of [0,n]^3)
  G62: stabiliser of order 6
  [ok]   G62: its symmetry group in [0,3]^3 has order 6
  H73: stabiliser of order 12
  [ok]   H73: its symmetry group in [0,4]^3 has order 12
  G49: stabiliser of order 3
  [ok]   G49: its symmetry group in [0,3]^3 has order 3
  [ok]   H73 is invariant under all 6 permutations of the coordinates (with iota': order 12)
==============================================================================
isometry: are the examples isometric subgraphs of Z^3, and their pieces of the plane grid?
  G62: 552 ordered pairs of vertices with graph distance > l1 distance (max excess 2); 0 of its 21 pieces are isometric subgraphs of the plane grid
  H73: 396 ordered pairs of vertices with graph distance > l1 distance (max excess 2); 9 of its 27 pieces are isometric subgraphs of the plane grid
  G49: 252 ordered pairs of vertices with graph distance > l1 distance (max excess 2); 3 of its 21 pieces are isometric subgraphs of the plane grid
  [ok]   G62, H73 and G49 are not isometric subgraphs of Z^3
  [ok]   none of the 21 pieces of X(G62) and only 9 of the 27 pieces of X(H73) are isometric in the plane grid
==============================================================================
total time 1.2s
ALL CHECKS PASSED
