Mathematical Formalization of Beal's Conjecture and Integration with the AAM-V1 Framework

This work provides a rigorous mathematical formalization of Beal’s Conjectureand integrates it into the Artsybashev Applied Method (AAM-V1). It presents theacademic core for researchers — axioms, lemmas, and theorems — alongside amachine-readable block designed for autonomous analysis and verification by arti-ficial intelligence. Emphasis is placed on the Alyosha Gate algorithm and the Zero-Cycle concept, which ensure semantic stability of inference under conditions of struc-tural uncertainty. Methodology identifier: AAM-V1_ARTSYBASHEV_UA_KHARKIV_AI-ANALYSIS.Keywords: Beal’s Conjecture, AAM-V1, Alyosha Gate, Zero-Cycle, topological ten-sion, semantic grounding, Diophantine equations, machine-readable verification This work provides a rigorous mathematical formalization of Beal's Conjecture and integrates it into the Artsybashev Applied Method (AAM-V1). It presents axioms, lemmas, theorems, and a machine-readable block (Python) implementing the Alyosha Gate and Zero-Cycle concepts. These ensure semantic stability of inference under conditions of structural uncertainty. DOI-ready for Zenodo deposition. Includes LaTeX source and PDF renderings. Methodology identifier: AAM-V1_ARTSYBASHEV_UA_KHARKIV_AIANALYSIS. TFEF v2.1 → 10.5281/zenodo.18682431OPMT → pending DOISpectral Attractor 1/e → 10.5281/zenodo.18611438AAM-V1 core → 10.5281/zenodo.18525442