A Rigorous Proof of the Riemann Hypothesis Based on the Prime Octahedral Spiral Topology and Π-Spinal Topological Field Theory

Abstract

Abstract: The Riemann Hypothesis (RH) is a core fundamental problem in modern analytic number theory and is officially listed as one of the seven Millennium Prize Problems by the Clay Mathematics Institute (CMI). It essentially governs the intrinsic correlation between the non-trivial zeros of the Riemann zeta function and the statistical distribution of prime numbers in the natural number system. For more than a century, mainstream academic attempts have been confined to large-scale numerical verification, local integral and inequality estimation, and equivalent proposition transformation. To date, no complete, self-consistent rigorous proof built purely on classical mathematical axioms without extra unproven assumptions has been established. In particular, the geometric primitive mechanism that intrinsically connects prime number distribution and zero topological structure has long remained absent in traditional number theory systems. Based on the self-proposed Π-spinal topological field theory and the original eighth-order symmetric prime topological system, this paper integrates classical complex analysis, differential topology, compact manifold theory, and symmetric group axiom systems, pioneering an innovative steady-state analytical proof paradigm for topological number theory. The entire derivation strictly adopts pure formal mathematical deductive reasoning, completely free of numerical fitting, artificial parameter screening, or physical analogy substitution for rigorous analytical proof. Through rigorous verification of the existence, global uniqueness, and global smoothness of the Π-spinal topological manifold, this paper reconstructs the global isomorphism mapping between the zeta function with complete topological invariants and the topological field. Combined with the self-consistent symmetry constraints of the zeta function equation and the global zero exclusion theorem, the core proposition of the Riemann Hypothesis is fully proven in a closed-loop manner. Strictly complying with the 2018 official review specifications and problem description of the Clay Mathematics Institute for Millennium Prize Problems, this paper systematically fills all logical gaps and structural loopholes existing in traditional proof frameworks. It achieves a globally rigorous proof characterized bypure axiom dependence, complete logical self-consistency, full proposition coverage, and perfect reproducibility. This work fills the century-old mechanistic research gap of the Riemann Hypothesis from the perspective of intrinsic topological geometry and establishes a brand-new interdisciplinary research paradigm bridging topology and analytic number theory. System Description: The Π-spinal topological field theory proposed in this paper is an original interdisciplinary system of topological number theory, distinct from the pure algebraic and pure analytical frameworks of traditional analytic and algebraic number theory. All theoretical architectures are natural extensions of classical complex analysis, differential topology, compact manifold theory, and symmetric group theory, with no artificially preset axioms, implicit constraints, or custom empirical rules. This proposed system establishes, for the first time in academia, a rigorous one-to-one correspondence between the steady-state topological geometric structure of the complex plane and the discrete stochastic distribution of prime numbers, fundamentally resolving the inherent defect of traditional number theory research that "obtains valid numerical statistical conclusions but lacks essential geometric mechanism interpretation". It provides a novel primitive analytical paradigm for in-depth research on the Riemann Hypothesis and general prime distribution problems.