GADS-MCL SDFM: Multiplicative Gears and Dimensional Collapse on Primorial Rings for GNFS Optimization

This paper introduces the Structural Deterministic Filter Method (SDFM) within the GADS-MCL framework, an optimization layer for the General Number Field Sieve (GNFS) integer factorization algorithm. By mapping the algebraic structure of the primorial ring ℤ/42ℤ and its group of units U₄₂, we demonstrate a deterministic dimensional collapse: the traditional two-dimensional relation-sieving space (a_gnfs, b_gnfs) can be filtered using a one-dimensional constraint on the scalar residue a. We formalize the 12 structural families of semiprimes under U₄₂, indexed starting from Family A (R = 11) and concluding with Family L (R = 5), using a canonical "gear" operator g and its modular inverse g⁻¹. This formulation allows the deterministic discarding of 50% of the sieving space prior to polynomial norm evaluation, yields an exact algebraic link between splitting components, and potentially prevents redundant matrix loops. The central conjecture—that GNFS sieving coordinates respect the modular structure of final factors—is presented with rigorous mathematical foundation and proposed verification methodology. Whether confirmed or refuted, empirical investigation of this framework will advance understanding of algebraic constraints in integer factorization.