Meta-Primitive Natural Number Positioning System ——A Complete System of Natural Number Laws Generated from 0 and 2 𝑖3 𝑟 (6𝑚 ± 1)

This paper establishes a complete natural number positioning system, named the Core-Reductive Natural Number Positioning System (CRNNPS). The system is founded on only two primitive elements: the origin 0 and the generative expression N=2i3r(6m±1) where i,r,m∈Z and the sign is either +1 or −1. The theory is built upon six self-contained laws: Generation Law: Every nonzero integer n is uniquely generated as n=2i3r(6m±1) Dual-Track Positioning Law: Every odd integer N can be expressed as the common boundary of two 3-smooth number tracks:N=2i3rR-1=2j3pQ+1, with R=6m±1, Q=6n±1. Phase Constraint Law: The exponents must satisfy r⋅p=0 (i−1)(j−1)=0, and i≠jj, ensuring that only one side carries a factor of 3, and the powers of 2 are properly phased. Singularity Law: For N=±1, only i=1 with all other five coefficients zero; for N=0, all six coefficients vanish. Mirror Verification Law: Base-5 provides an independent coordinate system with the formN=2i′5r′R′−1=2j′5p′Q′+1, where R′=4m′±1, Q′=4n′±1 and 5∤R′Q′ , serving as a verification layer for the base-3 system. Coordinate Mapping Law: When a number is representable in both base-3 and base-5 systems, the coordinates are related by i=i′ ′ , j=j′ ′ , r′=p′=0 and R=5r′R′ , Q=5p′Q′ . Together, these six laws form a closed, self-consistent, and generative framework that transforms the natural numbers from a linear sequence into a six‑dimensional coordinate grid. The system does not aim to solve specific open problems but provides a precise “coordinate map” for any number‑theoretic question, revealing the internal structure of natural numbers through generation, positioning, constraint, boundary handling, verification, and cross‑kernel correspondence