The external transform space defines the set of coherent transformations available to the minimal arena—the hyperbolic figure eight knot. It is equal to the volume of the 24-dimensional unit hypersphere ( ), and the density of the Leech lattice ( ). Where   = 120 ( the factorial of   ), and   = 44 ( the derangements of   ).

The divisors of that space ( ,  , 35, 18, 32, and 8 ) define how it is coherently structured. The first divisor introduces the structure of the 120–cell ( a compound of 120 regular –cells ), which contains examples of every relationship among all the convex regular polytopes found in the first 4 dimensions. The remaining divisors ( 44, 35, 18, 32, and 8 ) define the quantized powers of the Planck constants—the division boundaries connecting that space.

The controlling root of that structure ( its highest power component ) is equal to the product of the hyperbolic figure eight knot's twisted zeros.

The hyperbolic figure eight knot's constructive zeros are:

And the hyperbolic figure eight knot's twisted zeros are: