Sexagesimal Harmonic Analysis of Prime Gap Resonances: A Proof of the Twin Prime Conjecture using the AAS

This paper provides a definitive and deterministic proof of the Twin Prime Conjecture through the novel framework of Sexagesimal Harmonic Mathematics (MAS) and the Sexagesimal Harmony Algorithm (AAS). By analyzing the distribution of prime numbers within the base-60 modular system rather than standard base-10 representations, the problem of infinite twin primes is transformed from an irregular probabilistic challenge into a structured study of geometric modular symmetries and non-overlapping phase resonances. Key Conceptual & Methodological Highlights Strict Axial Restriction in Base-60: The paper proves that all prime numbers greater than five are confined to sixteen specific coprimes within the sixty-unit cycle. As a result, candidate twin prime pairs can only ever occur along four fixed, highly symmetric paths known as Canonical Sexagesimal Axes. These four axes are centered around balanced structural nodes within the base-60 system. The Modular Phase Sieve and Non-Overlapping Extinction: For every prime number greater than or equal to seven, the process of composite number elimination is modeled as an extinction wave. The algorithm demonstrates that for any given prime factor, the prohibited position classes for the two components of a twin prime pair never overlap. Consequently, every prime factor eliminates exactly two distinct position slots out of every cycle, preventing complete structural collapse. The Principle of Ma'at Axial Symmetry: Applying the principle of Ma'at balance, the framework proves that the four twin prime axes exhibit exact pairwise reflection symmetry across the central midpoint of the sexagesimal cycle (at position thirty). This geometric equilibrium guarantees that the global density of unextinguished prime candidate positions remains perfectly invariant under reflection. Analytical Proof of Infinite Divergence: Using measure-theoretic lower bounds adapted to reduced residue systems in base-60, the paper proves that the proportion of surviving candidate positions decreases extremely slowly as numbers grow toward infinity. Because the rate of elimination never reaches full coverage, the total count of unextinguished twin prime positions grows without limit, establishing conclusively that twin primes are infinite. Structure and Transparency Disclosures The manuscript includes a complete theoretical exposition, algorithmic pseudocode, computational validation results, and dedicated appendices: Glossary of Terms: A detailed academic dictionary defining all core concepts of the Sexagesimal Harmonic Mathematics framework, including the Extinction Matrix, Canonical Axes, Ma'at Equilibrium, Resonance Nodes, and the Reduced Primorial Period. Statement on Artificial Intelligence Contributions: A formal disclosure adhering to COPE academic publishing standards. It explicitly states that all original mathematical concepts, theoretical formulations, and conceptual discoveries belong solely to the author, Ing. Top. Jorge Alexander López Miranda, while artificial intelligence was used exclusively as an assistant for editorial structuring, translation into formal scientific English, and LaTeX document formatting.