Lemma 3:
I claim that the permutation of the centers in each orbit on the even n x n x n cube is wholly independent of the parity of any other orbit on the cube.

Proof:
First note that if all 24 center pieces in each orbit were distinct, then indeed restrictions on the parity of that orbit would exist given that we've already permuted the corners and edges of the cube. Since there are groups of pieces that are indistinguishable from each other this will not be the case, as will be shown below.

We define a commutator as applied to the n x n x n cube to be two ordered sequences of slice turns A and B. Clearly the inverse of each exists, so the commutator ABA'B' exists as well. Below we will prove how to 3 cycle 3 centers in the same orbit.

Definition: Let the (m,n) 1<=m,n<=(n-2)/2 center orbit be defined as the orbit in which the center pieces lying on the intersection of the f[m] and l[n] inner slices are contained. To three cycle three centers in this orbit the following commutator may be used.
l[n] u[m] l[n]' U' l[n] u[m]' l[n]' U. This commutator 3-cycles the pieces in the Ru[m]f[n] , Ub[m]l[n] , Ur[m]b[n] locations.

Since any piece in the (m,n) edge orbit is allowed to occupy each of the 3 slots affected by this commutator, then we can choose any three pieces we want to place into these locations. If these pieces are spread across the cube in the incorrect locations, let us perform a setup algorithm, call it P, that will place them into the correct positions. Notice that if we form the conjugate of our commutator by P, that we again have a commutator that only affects those three pieces of the cube. Now choose two of the pieces we will 3-cycle to be the same color and the other piece to be a different color. By doing this we can appear to create a transposition within the (m,n) center orbit. Notice also that we are doing a commutator that performs and even number of inner slice and outer slice moves on the cube. Therefore we can appear to create a transposition within the (m,n) orbit while actually doing an even permutation. Therefore all possible arrangements of the pieces in that orbit, taking into account the groups of indistinguishable pieces are possible. This is because we can appear to perform a transposition within that orbit without affecting the parity of any other orbit on the cube, and even without moving any other pieces on the whole cube due to the fact that we have used a commutator that moves only three pieces on the cube. Therefore the lemma is proved.

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