Full Proof of the Existence of a Degree 8 Circulant graph of Order L(8,k) of Arbitrary Diameter k

This is the full proof of Theorem 3 on the existence of the largest known degree 8 circulant graph for all diameters stated in the paper "The degree-diameter problem for circulant graphs of degree 8 and 9" by the author. To avoid the paper being unduly long the exceptions for only one case were included in the statement of the proof. In the statement of the proof presented in this paper the exceptions for all cases are included in full. This proof closely follows the approach taken by Dougherty and Faber for the existence of the largest known degree 6 circulant graph for all diameters.