Description of the system of patterning involved in where prime numbers occur within the infinite set of natural numbers

This article describes the system of patterning involved in the distribution of prime numbers. The description of the system is based on the idea that two seemingly independent process are interacting in a non-dimensional state. And since the language of mathematical formulation is syntactically dimensional, it cannot describe the system as a whole. If we accept advance predictability as proof of the system in connection with viewing a model of the regularity of its relationships, rather than relying on a traditionally formulated mathematical proof, the rigorous system of prime distribution is clearly evident. The philosophical implications are significant, since it points to a weakness in mathematical language and the formulation of mathematical proofs. *This draft is based on ideas published in the author's memoir "The Image of Language" 2021:Artists Books Editions. It is planned to be published in a book by the author tentatively titled "Archetivity."