The Toeplitz Conjecture has remained a focal point in geometry for over a century, yet traditional analytical approaches often overlook the fundamental logical constraints of geometric entities. This study addresses the "Inscribed Square Problem" by shifting the focus from empirical measurement to a first-principles logical audit. By revising the logical definition of "inscription" and treating a square as a rigid four-node structure with inherent 90-degree angular equilibrium constraints, we demonstrate that the existence of such a structure is not a probabilistic occurrence but a topological necessity of curve closure. Our results introduce the concept of "270-degree Space Compensation", showing that any simple closed curve that returns exactly to its starting point must generate sufficient spatial tension to accommodate the vertices of an inscribed square; curves failing to meet this threshold are logically invalid closed entities. We conclude that within a valid geometric domain, a closed curve and its inscribed square are logically inseparable. This finding reframes the conjecture from an existence proof to a deterministic outcome of topological cost. Logically, this necessitates any geometrically valid closed curve to sweep through topological coordinates sufficient to accommodate the four vertices of a square; conversely, a curve that cannot sweep through these coordinates fails to meet the 270-degree threshold.