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—, which also defines 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 its structure. The first factors into the 120-cell—which contains examples of every relationship among all the convex regular polytopes found in the first 4 dimensions. The coherent divisors of that structure ( 44, 35, 18, 32, and 8 ) define the quantized powers of the Planck constants.

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.

The hyperbolic figure eight knot's twisted zeros.