Huh? Obviously, you cannot even begin to justify your dubious "Diamond Theorem"! The questions asked in the original posting have not even been addressed. Instead you are attacking other people again. Someone is going to sue you one of these days. I quote you again:
"The Diamond Theorem: Inscribe a white diamond in a black square. Split the resulting figure along its vertical and horizontal midlines into four quadrants so that each quadrant is a square divided by one of its diagonals into a black half and a white half. Call the resulting figure D.
Let G be the group of 24 transformations of D obtained by randomly permuting (without rotating) the four quadrants of D. Let S4 denote the
symmetric group acting on four elements. Then
(1) Every G-image of D has some ordinary or color-interchange symmetry,
(2) G is an affine group generated by S4 actions on parts of D, and
(3) Results (1) and (2) generalize, through intermediate stages, to symmetry invariance under a group of approximately 1.3 trillion transformations generated by S4 actions on parts of a 4x4x4 cube."
What does (2) mean? By Cullinane's definition of G, G is isomorphic to S4, the symmetric group on 4 letters, with 24 elements. What does he mean by saying G is generated by S4 actions? Every group generates itself as its own subgroup! This is a theorem?
What does (3) mean? Generalize how? What is "approximately 1.3 trillion"? What "parts" of a 4x4x4 cube? Who is Cullinane trying to fool?
See the "4x4 Case" following the "theorem". Cullinane states that "G is
a group of 322,560 permutations". Where does this number come from? Going by his own definition, if 4x4=16 objects are to be permuted (this
is the only way to "generalize" his "theorem") then G should be isomorphic to S16, the symmetric group on 16 letters, with 16!=20922789888000 elements. Why 322560? Huh?