PFUSRC-123: Topological Ontology Ultimate Settlement The Goldbach Conjecture: Finite Closure Validity and the Breakdown of the Infinite Paradigm

The Goldbach Conjecture (every even integer greater than 2 can be expressed as the sum of two primes) has remained unsettled within classical number theory for nearly three centuries since its proposal in 1742. Massive numerical verifications have confirmed it in all finite cases, and the logical intuition is highly self-consistent, yet a rigorous complete proof has never been achieved—constituting the most typical "true but unprovable" paradoxical proposition in modern mathematics. This paper departs from traditional paths of analytic number theory, sieve methods, and almost-prime theory. Relying on the PFUSRC 11-dimensional triple coaxial 45° biconical topological ontological system, it carries out a five-layer foundational reconstruction of the conjecture’s underlying elements: the ontology of numbers, the essence of infinity, the boundaries of arithmetic axioms, the topological origin of primes, and the legitimacy of universal quantification. This paper establishes the ultimate core judgment: all finite even-number verifications are entirely valid, but can never be extrapolated to the infinite domain. The unprovability of Goldbach’s Conjecture is not a defect of arithmetic laws, but a topological boundary transgression and a natural breakdown of infinite universal paradigm. Four ultimate ontological judgments are put forward: 1. Numbers are topological boundary markers, not pure counting symbols. All valid numbers are strictly constrained within the 55-closed topological steady-state domain. 2. Infinity is merely a structural dissipation tendency, not an operable existential object. Classical mathematical infinite universal propositions are empty symbols operating outside the domain. 3. 1+1=2 is a conditional conclusion of boundary conditions, not an a priori axiom—it holds only under same-level steady-state topological coupling. 4. Primes are topological rigid anchors; even numbers are closed topological structures. In the finite domain, anchor decomposition is fully self-consistent; in the infinite domain, there is no corresponding topological structure. Through topological projection verification of typical even numbers including 4, 6 and 54, this paper realizes the ultimate settlement of the Goldbach Conjecture: the conjecture holds valid within finite topological closure while fails in the infinite domain, and its classical formulation has inherent positioning defects, which permanently resolves the 300-year academic deadlock.