Prime Walls of Primes: A New Structure and Modular Constraint Hierarchy in Prime Number Theory

We introduce the theory of prime walls for prime numbers, a novel structurethat reveals deep modular constraints governing symmetric prime pairs. For an oddprime p > 3, we define its prime wall distance k(p) as the smallest positive integer ksuch that both p − k and p + k are prime. We establish four fundamental theorems:(1) the Evenness Theorem—k(p) must be even; (2) the Mod-3 Theorem—k(p) ≡ 0(mod 3) for p > 7; (3) the Mod-6 Theorem—k(p) ≡ 0 (mod 6) for p > 7; (4)the Mod-q Constraint Theorem—k(p) ̸≡ ±p (mod q) for all primes q < p (with anatural exceptional case). The first three theorems have strict elementary proofsrequiring no unproven assumptions. We discover a modular constraint hierarchy bydefining and analyzing the hierarchy function h(q), prove its existence and finiteness,and provide extensive numerical verification for all odd primes p ≤ 500,000 with zerocounterexamples. Using Brun’s sieve, we prove the Universality Theorem: for anyfixed prime q ≥ 5, the proportion of primes p with k(p) ≡ 0 (mod q) tends to 1.We present a heuristic argument for the asymptotic bound k(p) = O((log p)2+ε ),supported by strong numerical evidence, and identify several open problems.