Inscribed Square Proof

This proof of the Inscribed Square Problem demonstrates that every simple closed loop in R^2 contains four points that form the vertices of a square. Grounded in topology and geometry, the proof relies on recursive approximations of loops using smooth methods like mollifiers, Fourier series, and wavelets, ensuring that square-forming conditions persist under uniform convergence. It employs Brouwer’s Fixed-Point Theorem to guarantee solutions within a compact parameter space and uses topological invariants to show that the square property is preserved under deformation. By addressing edge cases, including loops with sharp corners or fractal-like irregularities, the proof establishes that removing infinite complexities does not compromise the existence of an inscribed square. Inspired by natural forms like sine waves and cycles, the proof reflects Nature’s inherent symmetry and balance, emphasizing the universal persistence of squares in simple closed loops.