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

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):TraditionalMath(R¹) = Approx(TopoMath(Π²)) 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 Z₈ × Z₁₂₀ 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): NumTheory = UniTopo_Π 2. Dimension projection operator and approximation error operator are defined as: PΠ: Π² → R¹, Eₐₚₚᵣₒₓ = \|TopoMath-TraditionalMath\|₂ 2-1. The projection operator satisfies the irreversible loss constraint: (PΠ) ≠ ∅, Im(PΠ) R¹ 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.