A Discrete Topological Resolution to Toeplitz's Conjecture for Non‑Differentiable Planar Curves

Toeplitz's Inscribed Square Conjecture (1911) postulates that every simple closed curve in the Euclidean plane contains the vertices of a square. While analytically proven for sufficiently smooth curves, non-differentiable (rough) planar curves present localized variations that render continuous calculus and standard tangent spaces ineffective. This paper bypasses traditional continuous methods by applying a discrete spatial constraint model. Extending Herbert Vaughan's 1977 higher-dimensional phase-space mappings, we explore how bounded topological limits necessitate discrete orthogonal symmetries. We demonstrate that extreme geometric tortuosity enforces a coordinate phase-lock, proving the existence of the inscribed square not as a geometric coincidence, but as a bounding tensor matrix required to stabilize the manifold. This provides a formalized, non-calculus pathway to resolving the conjecture for fractal boundaries. AI‑Assisted Development. The mathematical formulation, computational implementation, and algorithmic structure presented in this work were developed through an interactive workflow in which the author provided the conceptual framework, theoretical direction, and problem‑solving strategy, while AI‑based tools assisted with mathematical execution, code generation, and computational refinement. The AI did not originate the scientific ideas or conclusions; it operated as a technical assistant under the author’s guidance. All results, interpretations, and claims were reviewed and validated by the author.