Beyond the Plane: Abstracting Away Geometry from AlphaGeometry

While AlphaGeometry has achieved gold-medalist performance on International Mathematical Olympiad geometry problems, its success remains largely confined to a specialized domain. Developing a cross-domain successor requires that we first determine the underlying logical substrate of AlphaGeometry’s domain-specific language. In this paper, we identify and propose the logic of observation as a mathematically principled language designed to serve as AlphaGeometry's foundation. We present a formal implementation of this logic within the Lean 4 programming language. By explicitly embedding Euclidean geometry into this logic and providing a machine-checked proof of its soundness, we demonstrate how the reasoning patterns of specialized solvers can be generalized across a wide class of mathematical theories.