Foundations of Arithmetic Power Geometry IV: The Coordinate-Dependent Defect Complex and the Modularity Barrier Obstruction Program

This paper establishes a rigorous analytical foundation for Arithmetic Power Geometry (APG) by formalising a scale-corrected geometric framework. By coupling the continuous path-integrated closure defect functional A₀(a,b;p) to the logarithmic minimal discriminant ln|ΔE| of a semi-stable Frey curve Eₐ,ᵦ,ₚ associated with a hypothetical primitive Fermat triple, we construct the Scale-Corrected Global APG Invariant AΔ. We prove an explicit global upper height bound mapping continuous deformation strain to absolute logarithmic Weil projective heights. Furthermore, we eliminate a major definitional vulnerability by proving the Local Coordinate-Dependent Damping Obstruction Theorem. We demonstrate that a universal rational damping constant K for the instantaneous defect field does not exist near the baseline. Instead, the local boundary condition is governed by a weight-dependent damping field that diverges to infinity under extreme coordinate skews. Finally, we formalise the modularity barrier roadmap, positioning the continuous analytic machinery as explicit structural motivation for a future obstruction program under the Ribet–Wiles regime.