A Proof of Goldbach's Conjecture via the Golden Ratio Numeral System

While investigating the mathematical foundations of Qi theory in The Mathematical Essence of the Universe, the author systematically examined the relationship between the golden ratio $\Phi$ and recursive algorithms. Following the "$\Phi$-ary system + recursive algorithm" approach, a new proof of Fermat's Last Theorem was first completed. The core of that proof is the Sphere Area Golden Law: among all ellipsoids, only the sphere has an area that is $\Phi$-constructible in the $\Phi$-ary numeral system. Encouraged by this, the present paper applies the same methodology to Goldbach's conjecture. We continue to use the idea of dimensional normalization from physics: represent integers in the golden ratio numeral system (the $\Phi$-ary system), and then analyze them using recursive algorithms. In essence, the mathematical core of this method remains "Fibonacci numbers + infinite descent," merely expressed in the language of modern mathematics. The core of the proof is the Additive Uniformization Golden Law: in the $\Phi$-ary system, the addition map on pairs of ordinary primes realizes the most uniform covering of the space of legal strings. This extremality is jointly enforced by the optimal robustness and optimal coverage properties of $\Phi$. Goldbach's conjecture follows as a direct corollary. The entire proof avoids traditional analytic number-theoretic tools such as the circle method, the $\zeta$-function, and exponential sum estimates, relying instead on the intrinsic mathematical structure of the $\Phi$-ary numeral system. Note: The complete Chinese version of this paper is also provided as a supplementary file.