Wave-Point Geometric Interpretation and Evolution of Additive and Multiplicative Structures

This paper introduces a novel geometric framework termed Wave-Point Geometry (WPG) to bridge the divide between the additive and multiplicative structures of integers, offering a formalization of the Bounded Cohomology Background (BCB) conjecture. Deviating from traditional algebraic machinery, the proposed framework maps a+b=c triplets onto a continuous potential field governed strictly by metric-based curvature and logarithmic potentials. We define a scale-invariant wave distance metric and a dynamic, non-positive curvature penalty (η = min(0, 1 - log c / log rad(abc))) that acts as a contractive geometric force under high number-theoretic tension. Through this rectified curvature function, we show that negative spatial distortions naturally expand the upper bound of the radical field via the definitive structural boundary: c <= rad(abc)^(1 - η). Finally, we demonstrate the strict empirical verifiability and algorithmic computability of the framework by providing exact numerical tracking for historical exceptions, such as the (1, 8, 9) system, yielding mathematically consistent and precise boundary containment without free parameters.