Prime Number Regularity and the Canonical Triple-Graph

This paper challenges the long-held view that prime numbers are distributed randomly among the integers. By examining primes through the structure of the Canonical Triple-Graph (CTG), we show that their distribution is entirely deterministic and governed by two interlocking periodicities. The first periodicity is horizontal: within any block of the CTG, consecutive elements follow the affine rule x → 4x + 1, which produces complete permutations of residue classes modulo powers of 3. Every 9-tuple contains each residue 0 through 8 exactly once. The second periodicity is vertical: across depth levels, the structure repeats with period 18. This period arises because 18 is the smallest exponent for which 2^18 ≡ 1 (mod 27). A lookup table based on residues modulo 27 completely determines the mod-9 behavior of descendants. The key insight is that these two periodicities intersect: every position in the CTG is root-affine in both dimensions, meaning it is determined by affine transformations from the root both horizontally and vertically. Primality at any position is simply the intersection of periodic sieves—one for each prime—operating at different frequencies. The apparent randomness of primes arises not from any intrinsic disorder but from two sources: the incommensurability of sieve periods for different primes, and the computational difficulty of intersecting infinitely many periodic conditions. The primes themselves follow perfectly regular laws. We use a ray metaphor to describe the structure: near the root, primes are plentiful; as one moves away horizontally or vertically, they become rare. But the walking distance is always affine—the path is straight, only the light fades. This work is part of a series on the Canonical Triple-Graph and its applications to structural number theory.