Overview

Introduction to《Metastructural Unification ( Mathematics Volume ): Unifying Algebra, Geometry, and Probability》 This book is not a collection of papers addressing isolated mathematical problems, but a systematic work that seeks to unify algebra, geometry, and probability at the structural level. The author proposes a unified structural grammar centered on the triad Dot - - Line 1 - Circle Ο, revealing the intrinsic isomorphic relationships among the three foundational mathematical languages across different levels. Guided by a ring-structured framework, the book develops mathematical problems through multiple hierarchical layers. At the Third-Ring level, it systematically examines represent-tative problems such as the generalized Goldbach conjecture, Fermat's Last Theorem, the abc conjecture, as well as the Poincaré problem, geodesics, prime number distributions, and spectral statistics, demonstrating their unity in structural roles. At the Fourth-Ring level, the discussion advances to selected topics including the Langlands program, higher-order L-functions, noncommutative geometry, Ricci flow, many-body random systems, and high-dimensional spectral statistics, revealing a unified mechanism underlying existence, stability, and generation. The central thesis of this book is that algebra, geometry, and probability are not parallel disciplines, but manifestations of the same structure expressed in different languages. Through unified formulations and structural closure analysis, the author presents a clear path from local conditions to global structures, offering a new perspective on the deep unifying principles of modern mathematics. This book is intended for readers interested in foundational mathematical structures, cross-domain unification theories, and advanced mathematical thought.

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