Arithmetic Power Geometry 0: Unified Foundations, Axioms, Core Invariants, Examples, and Open Problems - Toward a Unified Framework for Exponent Deformation Geometry and Arithmetic Defect Theory

Arithmetic Power Geometry (APG) is presented as a unified framework for studying algebraic closure relations under exponent deformation. The paper consolidates the APG I–VII program into a single foundational architecture, including the Power-Deformation Space, Euclidean Target, local closure defect, entropy-governed deformation, integrated defect energy, scale-corrected invariants, discrete prime-chain regularization, source invariants, discrepancy-source structures, and modular-efficiency quantities. The manuscript distinguishes established internal theorems from conditional projection and height-coupling programs, and it explicitly records examples, failure modes, and open problems. APG is not claimed to prove Fermat's Last Theorem, the Birch and Swinnerton-Dyer conjecture, the abc conjecture, or the Szpiro conjecture. Instead, it is developed as a coherent research framework for exponent-deformation geometry and arithmetic defect theory.