A Proof of Fermat's Last Theorem 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 and recursive algorithms. It was later realized that the recursive algorithm employed therein coincides in its logical core with the method of infinite descent pioneered by Pierre de Fermat in the 17th century. This led to the conjecture that Fermat, in attempting to prove his Last Theorem, very likely discovered the peculiar role played by the Fibonacci sequence (the rabbit sequence) and its underlying golden ratio while applying infinite descent to equations of low degree, thereby conceiving a marvelous proof that combined infinite descent with the golden number — a proof he never fully wrote down. Inspired by this, the present paper gives a new proof of Fermat's Last Theorem. We adopt the idea of dimensional normalization from physics: represent integers in the golden ratio numeral system (the -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 Sphere Area Golden Law: among all ellipsoids, only the sphere has an area that is -constructible in the -ary numeral system. This extremality is jointly enforced by three mathematical properties of — optimal filling, optimal robustness, and optimal coverage. Fermat's Last Theorem follows as a direct corollary: for n 3, the Fermat equation attempts to construct a non-spherical surface whose area would nevertheless be -constructible, violating the extremality of, and therefore no solution exists. The Sphere Area Golden Law also directly implies the sphericity of black hole horizons and the Bekenstein-Hawking entropy-area law. This paper is dedicated to the memory of Pierre de Fermat. Note: The complete Chinese version of this paper is also provided as a supplementary file.