Foundations of Arithmetic Power Geometry V: Discrete Moduli-Stack Formulations, Scale-Dependent Metrics, and Boundary Obstruction Limits

We present the finalized structural architecture of Arithmetic Power Geometry (APG) over discrete parameter spaces. Moving past ill-defined real-variable trajectories and continuous conductor functions, this work establishes an ordered arithmetic chain of surfaces over the universal moduli stack of elliptic curves Mₑₗₗ. We evaluate the Absolute Projective Closure Defect Dₚ alongside a scale-dependent Logarithmic Scale-Weighted Functional D*ₚ(a,b), designed to track the analytic height floor against the classical geometric Szpiro ceiling. We establish the pointwise positivity of the discrete closure defect over primitive integer supports, removing invalid uniform lower bounds. To address the isotropic baseline, we implement a localized algebraic parity obstruction exploring how specific local ramification profiles at the prime q = 2 interact with modular level-lowering constraints. Finally, we execute an exact symbol-by-symbol asymptotic boundary analysis under extreme coordinate skewness (wₐ → 0). We prove that the scale-weighted height floor stabilizes safely at a true, non-contradictory identity (0 ≤ 4), confirming that continuous APG deformations, when discretized correctly, reach a non-contradictory containment boundary rather than reproducing the Ribet–Wiles fracture. This locates the definitive boundary limits of continuous geometric methods in exponential Diophantine analysis.