A Proof of Goldbach's Conjecture within the Ω-N Pure Primitive System and Fundamental Innovation of Mathematics

This preprint presents a rigorous proof of the Goldbach Conjecture within the Ω-N pure primitive system, an autonomous mathematical framework constructed entirely independent of ZFC set theory and Peano arithmetic. Background & Framework For nearly three centuries, the Goldbach Conjecture has remained unsolved within traditional mathematical paradigms, which treat numbers as a priori entities and arithmetic operations as artificial definitions. This work abandons all prior concepts of numbers, sets, membership relations, and formal operations, building the Ω-N system solely on two primitive notions: the generative order relation and primitive order Γ, anchored to three core axioms: symmetric generation, boundary expansion, and universal coverage. Core Contributions 1. Ontological Innovation: Establishes a relational ontology of mathematics, framing "primons" (the system's fundamental elements) not as static "prime numbers", but as indivisible nodes formed in a dynamic symmetric generative process. 2. Dual Independent Proofs: Provides two mutually independent rigorous derivations of the Ω-version Goldbach Theorem: the symmetric adjacent primon method and the generative density lower bound method, eliminating accidental reasoning deviations. 3. Large-Scale Empirical Verification: Validates the proof with a complete primon generation sequence up to the 100th primitive order, including 1275 verified prime partition cases, all perfectly consistent with theoretical predictions. 4. Homology Discovery: Reveals for the first time that the Goldbach Conjecture and the Riemann Hypothesis are not isolated problems, but homologous corollaries of the symmetric generation axiom, sharing the same core constraints of critical line distribution and bounded oscillation. Critical Boundary Statement The Ω-N system maintains absolute incommensurability with traditional mathematical frameworks. All conclusions in this work are valid only within the Ω-N pure primitive system, and no claim is made regarding their truth value within the ZFC/Peano arithmetic framework. Evaluating this work using external axiomatic systems constitutes a category error, analogous to judging non-Euclidean geometry by Euclidean axioms. This work opens a new path for foundational research in mathematics, demonstrating that rigorous, explanatory mathematical systems can be built on fundamentally different ontological assumptions, beyond the limits of traditional set-theoretic foundations.