We introduce an exponent-globalized construction in Arithmetic Power Geometry by integrating an exact closure defect along the continuous exponent path from the Euclidean baseline n = 2 to a target exponent n > 2. For positive coordinates a and b, we define the Integrated Closure Defect Functional as the integral of the exact closure defect from 2 to n. We prove that the Integrated Closure Defect Functional is strictly positive for every n > 2 and derive its third-order local asymptotic expansion near the Euclidean baseline. The leading coefficient is determined by the Shannon entropy of the normalized squared-coordinate weights, while the cubic correction is determined by their quadratic logarithmic moment. Numerical quadrature for (a, b) = (3, 4) confirms the local accuracy of the expansion. We then formulate, without asserting its validity, a conditional research program asking whether a suitably normalized version of this continuous defect can be related to arithmetic invariants of Frey curves. No result concerning Fermat's Last Theorem, the abc conjecture, modularity, or a Frey-Faltings height obstruction is claimed.
Md. Amir Khusru Akhtar (Sun,) studied this question.