严格凸曲线内接正方形的存在性:拓扑度方法的证明 The Existence of Inscribed Squares on Strictly Convex Curves via Degree Theory

This work studies the classical inscribed square problem on strictly convex Jordan curves. On a four-point configuration space denoted by U_delta, we construct a continuous map H: U_delta → R^4 whose zeros correspond exactly to inscribed squares on the curve. Using a second-order geometric expansion of this map, we establish a uniform lower bound for the size of H on the boundary of U_delta, which excludes the possibility that zeros occur on the boundary and yields a well-defined degree modulo 2. By constructing a homotopy between the circle and an arbitrary strictly convex curve, and proving that this mod 2 degree remains invariant along the homotopy, we show that every C^3 strictly convex Jordan curve contains at least one inscribed square. This preprint presents a geometric–topological proof framework based on configuration spaces, boundary estimates, and degree theory, offering a structurally stable and geometrically transparent approach to the inscribed square problem in the strictly convex setting.