Wave-Point Geometry and the Resolution of the BCB Conjecture

This monograph establishes the formal mathematical foundation of Wave-Point Geometry (WPG) and presents its direct application to the complete resolution of the Bounded Cohomology Background Conjecture (BCBC). Wave-Point Geometry is a novel geometric paradigm that reconstructs arithmetic and topological spaces from the dynamic transport of phases over connection networks. By showing that phase consistency and wave propagation topologically generate metric structures and curvature, WPG provides a continuous, spectral language to address historically discrete and intractable problems in number theory. In this work, we demonstrate that the algebraic reduction of radicalization under arithmetic anomalies (such as a + b = c) corresponds to the desorption of high-frequency wave energy. According to the core energy conservation laws of WPG, this desorbed energy is forced to localize as a diagonal potential matrix. Under Weyl's Interlacing Theorem, this potential dynamically shifts the spectral ceiling of the modified wave Laplacian, rendering the violation of the BCB inequality mathematically impossible. We further show that this spectral mechanism is universal. Classical diophantine problems—including the ABC Conjecture, the Generalized Szpiro Conjecture, and the algebraic ranks associated with the Birch and Swinnerton-Dyer (BSD) Conjecture—are naturally resolved as the elastic and acoustic invariants of the arithmetic network under WPG. This monograph bridges the historical divide between discrete arithmetic and continuous wave physics, marking the introduction of Spectral Number Theory.