Arithmetic Power Geometry III: The Integrated Closure Defect Functional and the Conditional Frey-Curve Height Obstruction Program

We establish an exponent-globalized analytical framework for Arithmetic Power Geometry (APG) by constructing a path-integrated analytical system over the continuous exponent deformation space. Moving beyond localized static evaluations and artificially regularized tensor fields, this paper introduces the Integrated Closure Defect Functional (A0), defined as the unregularized path integral of the exact continuous closure defect from the classical Euclidean baseline (n=2) to an arbitrary deformed exponent state (n>2). We state and prove two foundational internal theorems: the strict positivity of the integrated closure defect functional for arbitrary positive base coordinates, and its third-order asymptotic expansion near the baseline, governed explicitly by the normalized coordinate Shannon entropy (H(W)) and a newly isolated quadratic logarithmic weight dispersion parameter (Q(a,b)). Finally, we formalize a rigorous, conditional Frey-Curve Height Obstruction Program. We detail the proposed cross-disciplinary criteria under which this continuous exponent-globalized path integral can be coupled to the discrete arithmetic attributes of semi-stable elliptic schemes, defining the precise target bridge theorems required to interface continuous geometric defect functionals with the modular representation landscape.