Abstract: Traditional number theory has had fundamental paradigm defects for two thousand years. It defaults that all prime numbers are homogeneous, disordered and equivalent in status, resulting in long-term fragmentation and lack of unified ontological interpretation of core problems such as prime distribution, Riemann hypothesis, infinitude of twin primes, essence of pi, and large number factorization. This paper pioneers the axiom system of prime double-layer topological shell, strictly divides all primes into inner primitive compact closed shell prime sets and outer high-order harmonic prime sets, constructs three-level arithmetic topological compact manifold skeletons of 30, 210 and 30030, and establishes the core laws of topological duality and Fourier phase duality for inner and outer shells. This paper proves that the natural number arithmetic space is a steady-state closed manifold clamped by double-layer shells. The inner six-prime base locks the global topological symmetry and geometric ontology, and outer high-order primes only provide high-frequency harmonic perturbations without adding new topological degrees of freedom. Based on this axiom system, the real part of non-trivial zeros of the Riemann ζ-function is strictly locked as σ=1/2 by innate spatial geometry, completing the ontological proof of the Riemann hypothesis fundamentally; the circular constant π is no longer a random irrational decimal, but an inherent topological perimeter of the ultimate primitive shell, and its infinite decimals are interference ripples of high-order primes; the infinitude of twin primes is an inevitable result of local harmonic resonance. Meanwhile, relying on the double-layer topological stratification characteristics, this paper reconstructs the large number operation algorithm, reduces the traditional O (N^1/2) computational complexity to O (log N), and realizes a dual revolution of number theory theory and engineering computing power. This paper completely terminates the empirical induction paradigm of classical number theory, establishes the first principle of geometric number theory, and completes the unified axiomatic shaping of the arithmetic system.
xiaogang shui (Sat,) studied this question.