Hierarchical Spirals in Equal-Scale Lattice Spaces and Geometric Rejection of Higher-Dimensional Diophantine Equations

This paper establishes a definitive geometric framework for analyzing prime distribution and primality testing within a2D L-shaped coordinate system. We define four quadratic sieve polynomials (the “Composite Nets”) that exhaustivelycapture composite numbers based on the dimensional step y and the lateral positioning x. By analyzing the structuralmisalignment between the expansion rate of square intervals (2y + 1) and the resetting cycle of the sum of integers(Σn), we prove the Lattice Reset-Driven Radical Inversion Limit, restricting total resets within any single interval toa maximum of two. Furthermore, we demonstrate the absolute inequality Remainder Count > Tier Number, geometricallyvalidating the permanence of prime pockets to resolve Legendre’s conjecture. We introduce the primal complement sievebased on the n−0.5 half-step axis, showcasing its algebraic symmetry with Riemann’s critical line (1/2). Finally, wedelineate a geometric subtraction-based factorization scheme against RSA-moduli, computing its exact complexity boundsto prove its computational supremacy over conventional sieves in the 10-digit domain while preserving macro-scalecryptographic safety due to the physical barriers of massive digit spaces.