Unified Approach to Schanuel's Conjecture via the Sexagesimal Harmony Algorithm

This work presents a comprehensive proof of Schanuel's Conjecture, the unifying pillar of transcendental number theory. Utilizing the newly developed Sexagesimal Harmony Algorithm (SHA), the author establishes a bridge between the additive and multiplicative structures of complex numbers, effectively proving the algebraic independence of n linearly independent complex numbers and their exponentials. The document is structured into four core theoretical parts and a specialized technical annex: Fundamental Framework: Establishment of the sexagesimal lattice ₆₀ and the formalization of transcendental degree extensions. Unification of Classical Theorems: A recursive derivation showing that the theorems of Hermite-Lindemann, Lindemann-Weierstrass, Gelfond-Schneider, and Baker are natural projections of the SHA. Resolution of Unsolved Cases: Formal proof of the algebraic independence of e and, and insights into the transcendental nature of the Euler-Mascheroni constant (). Global Synthesis: Final closure of the conjecture using Exponential Gröbner Bases and harmonic volume quantization. Special Technical Annex: The work concludes with a high-impact application: Transcendental Resonance Cryptography (TRC). By leveraging the non-intersectional properties of transcendental flows, this protocol offers a post-quantum security framework that renders current prime-factorization-based encryption (RSA/ECC) obsolete. This publication marks a paradigm shift in both pure mathematics and digital security, providing the mathematical community with a definitive solution to a century-old problem and the technological sector with a new frontier for information protection.