Three-Factor Tomographic Sieve: Achieving 65% Prime Density Channels via Orthogonal Modulo-7 Grid Projections

We introduce a three-factor tomographic sieve architecture that establishes ultra-high density prime number channels by completely suppressing composite coordinate vectors. While classical single-layer wheels eliminate small primorial factors, they leave dense distributions of square-free composite numbers. By multi-layering a modulo-30 cyclic progression matrix with the author's 24 dual "body–tail" structural codes, and intersecting the output space with an orthogonal prime field modulo 7, we fragment the natural number stream into 168 discrete sub-channels. Experimental validation up to 15,000 demonstrates that our framework isolates exactly 24 absolute dead zones (0.00% prime density) alongside 69 hyper-focused prime corridors with densities surging beyond 50%, with the top two channels achieving 66.67% prime density. This transforms chaotic interval scanning into deterministic hardware vector paths with 3.8 times improvement over classical sieving density.