Foundations of Arithmetic Power Geometry: Local Parameter Deformations and Entropy at the Euclidean Target

We present Arithmetic Power Geometry (APG). This axiomatic local deformation framework reframes the algebraic solvability of power-type Diophantine equations as a problem of localized functional deformation. Global infimum calculations over infinite discrete subsets are often susceptible to asymptotic collapse arising from approximation phenomena. To mitigate this, APG isolates algebraic deformation locally around a fixed coordinate baseline designated as the Euclidean Target. We formalize the set-theoretic structure of a localized parameter space, the Power-Deformation Space. This space treats the exponent as a continuous deformation parameter. We state and prove the First-Order Closure-Obstruction Asymptotic Expansion Theorem. This theorem demonstrates that the first-order growth of the normalized closure defect away from the classical baseline (n = 2) possesses a leading-order coefficient proportional to the Shannon entropy of the normalized base weights. Finally, we establish the formal scope and analytical boundaries of the framework to ensure mathematical and conceptual clarity.