PFUSRC-2026 New Prime Number Theory From Unary System and 0-Interface to Prime Nodes: A First-Principles Narrative of Cosmic Self-Generation

Classical analytic number theory, grounded in the artificial sequence of natural numbers and relying on complex analysis to prove the Prime Number Theorem and properties of the Riemann zeta function, is a phenomenological inductive theory with fundamental limitations: it cannot explain, from an ontological level, the generative mechanism of primes, local distribution fluctuations, hierarchical differentiation, and lacks a topological-logical foundation for the Riemann Hypothesis. Based on the core framework of the PFUSRC system, this paper establishes a first-principles New Prime Number Theory. It abandons all presuppositions of natural numbers, classical prime empirical formulas, complex analysis tools, and external references, relying solely on two native axioms: the Unary Topological Unit Axiom and the 0-Interface Structured Coupling Axiom. Together with the cosmic self-generating topological narrative of "no first cause in the global cosmos, first cause in the local cosmos," it fully derives the emergence process from the chaotic continuum to prime nodes. This paper rigorously completes a chain of logical derivations: topological truncation of the chaotic continuum → generation of structured 0-interface → growth of the 1D open chain → topological closure of the closed waist ring → emergence of rigid prime nodes → construction of the topological accumulation law → three-tier classification of primes (Anchor Primes, Drift Primes, Hidden Primes) → asymptotic law of prime distribution density → topological correspondence between hidden primes and the non-trivial zeros of the Riemann zeta function. This paper restores the essence of primes from a topological-ontological level: primes are not an indecomposable subset of natural numbers, but rigid structural anchor points within a closed cosmic topology that cannot be split or compounded. The classical prime sequence, the Prime Number Theorem, and the symmetry of Riemann zeros are all projections of topological prime nodes onto the linear dimension of natural numbers. This theory constructs a new foundational paradigm of isomorphism between number theory and cosmic topology, providing a self-consistent ontological framework for prime fluctuations, the underlying logic of the Riemann Hypothesis, and the growth of cosmic spacetime structure.