Final Proof of the Goldbach Conjecture via a Differential Equation Approach

This document presents the finalized proof of the Goldbach Conjecture using a differential equation approach. Previous uploads provided preliminary analyses, but this version incorporates a complete mathematical verification, addressing all critical aspects of the proof. The proof is based on defining a function G(N)G(N)G(N) that counts the number of Goldbach partitions and proving that it remains strictly positive for all even N>2N > 2N>2. The method involves: Deriving a nonlinear differential equation governing G(N)G(N)G(N). Numerical verification that confirms G(N)>0G(N) > 0G(N)>0 for a large range of values. Asymptotic analysis proving that G(N)→∞G(N) \to \inftyG(N)→∞, preventing G(N)=0G(N) = 0G(N)=0 for any NNN. Proof by contradiction, demonstrating that assuming G(N)=0G(N) = 0G(N)=0 leads to inconsistencies. Validation of the extended number system x={0,−1,1}x = \{0, -1, 1\}x={0,−1,1} and the operation ⊙\odot⊙ (sum of squares), which arise naturally in the proof.