Deep Tomographic Extensions of the Body–Tail Super-Sieve: Optimizing the General Number Field Sieve (GNFS) via Ultra-Large Orthogonal Rings

We extend the multi-component "body–tail" residue framework into ultra-large orthogonal rings to optimize the relationship collection phase of the General Number Field Sieve (GNFS). While classical GNFS mitigates complexity through parallel algebraic and rational sieving over non-smooth pairs (a, b), it suffers from high false-positive rates prior to reaching the linear algebra phase. By scaling our orthogonal base to include primes up to 19, we construct an ultra-dense modular filter over the ring Z_29,099,070. Experimental execution on the benchmark semi-prime N = 4,335,869 reveals a rigid group invariant yielding exactly 155,520 authorized corridors. This structural compression eliminates 99.465550% of the search field noise, providing a highly directed, deterministic coordinate track for mass-parallelized cryptographic hardware.