Axiomatic Structure and Closure of the Geometric Field Theory

This paper proposes a framework for a unified Axiomatic Field Theory, establishing the logical closure of the geometric information system based on Information Geometry. By postulating the axiom of Maximum Information Efficiency, we derive the Ideal Planck Constant and demonstrate that physical reality emerges from Saturated Excitation within a constrained Phase Space Topology. Applying the Shannon Entropy Limit and Channel Capacity, we prove that the Fine Structure Constant is a geometric projection of the Vacuum Polarization Background. The framework utilizes the Paley-Wiener Theorem and Orthogonal Decomposition to identify the Deviation Field—manifesting as a Evanescent wave and radiating as Topological Radiation. We derive the Gravitational Constant from the Residue caused by the decay of Geometric Fidelity, explicitly defining gravity as a Recoil Force. Furthermore, the model introduces Field-Cavity Duality and Vacuum Breathing modes. Through Geometric Screening rooted in Measure Theory, we explain Momentum Asymmetry. The system's structural closure is secured via Quantum Phase Locking and Generalized Rabi Oscillation, confirming the G Efficiency Structure aligns closely with the CODATA 1986/1998 historical baseline, while discussing potential theoretical implications for the deviation observed in recent high-precision measurements. Furthermore, the theory identifies a synchronized ~0.025% vacuum polarization shift across both G and a, suggesting a distinction between derived ‘Geometric Naked Values’ and experimentally screened effective values.