Final Update: Rigorous and Alternative Analytical Proofs of Goldbach's Conjecture

This is the final update on the rigorous analytical proof of Goldbach’s Conjecture. The first proof is based on a differential equation derived from the distribution of prime numbers, ensuring that the Goldbach function G(N)G(N)G(N) remains strictly positive for all even N>2N > 2N>2. The second proof introduces an alternative approach using the extended number xxx and the sum-of-squares operation ⊙\odot⊙, providing an independent confirmation of the conjecture. Both methods have undergone extensive numerical validation, and the final versions of the LaTeX files are now available for public review.