"Extended Explanation and Mathematical Justification of the Goldbach's Conjecture Proof"

This document provides a detailed explanation and mathematical justification of the analytical proof of Goldbach’s Conjecture, initially presented in our previous work. The proof is based on two independent approaches:

1. The Extended Number xxx and the Sum-of-Squares Operation ⊙\odot⊙: This method introduces a novel numerical structure that preserves key arithmetic properties and leads to a governing equation that ensures Gx(N)>0G_x(N) > 0Gx(N)>0 for all even N>2N > 2N>2.

2. The Differential Equation Approach: This approach refines previous formulations by deriving an accurate differential equation that models the behavior of G(N)G(N)G(N) and eliminates prior inconsistencies.

The explanations in this document are intended to provide further clarity on the methodology and facilitate mathematical discussions. Feedback and opinions from the mathematical community are highly encouraged.

We invite mathematicians, researchers, and enthusiasts to review this extended explanation and share their perspectives on the presented proof. Constructive feedback is welcomed to ensure transparency and further refinement of the approach.

Keywords: Goldbach’s Conjecture, Extended Number xxx, Sum-of-Squares Operation, Differential Equations, Number Theory.