RSA-NPQ 2D-Multilayer Segmentation: A Targeted Pathfinding Approach to Modulus Factorization

This paper introduces the RSA-NPQ algorithm, a deterministic, hierarchical framework for the factorization of large semi-prime integers $n = p \times q$. Diverging from the probabilistic complexity of the General Number Field Sieve (GNFS), the NPQ approach redefines factorization as a multi-dimensional pathfinding problem within Prime Gear Geometry. By employing a 2D-multilayer segmentation strategy, the search space is recursively partitioned into $1,000$ discrete segments per layer, enabling a "Vertical Drill" that reduces the factor range by three decimal digits per iteration. We demonstrate that for an RSA-2048 modulus, the factors $p$ and $q$ can be isolated within approximately 210 layers. The methodology addresses the "Square Root Fallacy" by introducing the $\delta(n)$ shift and $n$-clue density mapping to navigate the $10^{307}$ residual gap. The process concludes with a localized Fermat Square verification kernel, offering a scalable, linear path toward factoring astronomical scales within a distributed node environment.