The Prime Arithmetic Theorem, The Twin Prime Theorem, The Cousin Prime Theorem, and The Sexy Prime Theorem in the Prime Field Architecture of the Arithmetic Modulation Field

The proof is a quality argument. It proceeds beyond counting prime pairs. It establishesthat the arithmetic structure of M = 30 makes permanent complete coverage ofall K-gap positions structurally precluded. The K-gap positions are permanentdiscrete features of the ℤ/30ℤ lattice. The binary field dictates where primesMUST BY LAW land. The geometric lane structure remains invariant under any N.