Expanding Sophie Germain Primes Concept to Ultimate Numbers

This work expands the classical concept of Sophie Germain primes and safe primes into a broader category termed ultimate numbers. The underlying mathematical mechanism is extended to the entire set of whole numbers ℕ0, which are now classified as either Ultimate or non-Ultimate. This comprehensive investigation of ℕ0 offers a universal generalization of the Sophie Germain framework, enabling the formulation of a genetics of numbers—a structural analogy to biological genetics. According to these newly defined numerical genetic criteria, the initial topological organization and geometric distribution of whole numbers within various closed matrices exhibit exact 3:2 or 1:1 ratios, revealing a deep arithmetic harmony.