Research on Quantum Topological Number Theory System Based on Π Two-Dimensional Vortex Domain——On the Underlying Defects of Traditional Number Theory and the Original Proofs of Century-Old Conjectures

Abstract The traditional decimal number theory system is constructed based on a one-dimensional linear number line, serving as an artificially defined low-dimensional approximate mathematical tool. It has inherent underlying defects including dimension deficiency, information loss, rule fragmentation, and absence of intrinsic symmetry, which leave numerous long-standing problems unsolved for thousands of years, such as prime distribution, large integer factorization, and classical number theory conjectures. Breaking the shackles of one-dimensional linear counting, this paper constructs a Π two-dimensional vortex domain quantum topological number theory system and redefines the ontology of numbers, number-theoretic operations, number system classification, and symmetric evolution rules. The core theoretical premise of this study is expressed as Equation (1):$$\text{TraditionalMath}(\mathbb{R}^1) = \text{Approx}\big(\text{TopoMath}(\Pi^2)\big) \tag{1}$$. This paper strictly proves thatall natural numbers, primes, and composite numbers are quantum steady-state occupations of the two-dimensional Π vortex domain, and all rules and problems in traditional number theory are superficial projections of high-dimensional topological structures onto one-dimensional space. Relying on the $\mathbb{Z}_8 \times \mathbb{Z}_{120}$ dual symmetric group control system, this paper traces the topological origin of fundamental number theory rules including parity, divisibility, remainder, and large-number evolution. From the perspective of geometric ontology, it provides rigorous intrinsic proofs for century-old number theory conjectures including the twin prime conjecture, Goldbach conjecture, and Riemann hypothesis, completely solving industrial pain points such as inefficient large-number factorization, computational errors, and irregular prime distribution. This system satisfies the unified universal Equation (2): $$\text{NumTheory}\cup\text{Geo}\cup\text{QPhys}\cup\text{CosmoTopo} = \text{UniTopo}_\Pi \tag{2}$$. Dimension projection operator and approximation error operator are defined as: $$\mathcal{P}_{\Pi\to\mathbb{R}}: \Pi^2 \to \mathbb{R}^1,\quad \mathcal{E}_{\text{approx}} = \big\|\text{TopoMath}-\text{TraditionalMath}\big\|_2 \tag{2-1}$$. The projection operator satisfies the irreversible loss constraint: $$\ker(\mathcal{P}_{\Pi\to\mathbb{R}}) \neq \emptyset,\quad \text{Im}(\mathcal{P}_{\Pi\to\mathbb{R}}) \subsetneq \mathbb{R}^1 \tag{2-2}$$. This research realizes the universal unification of four major disciplines, subverts the three-thousand-year artificial linear mathematical paradigm, and provides brand-new fundamental theoretical support for quantum computing, advanced cryptography, underlying artificial intelligence logic, and cosmic fundamental physics research.