A Complete Proof of the C⁰ Toeplitz Conjecture: An Approach via the Concept of Permanence and Degree Theory

Abstract This paper presents a complete proof of the unresolved Toeplitz Conjecture (Inscribed Square Problem) for the most general class: C⁰ Jordan closed curves. The proof is based on the existence theorem for C¹ curves using the Brouwer Degree and its extension to the C⁰ class via the Stability Theorem of Degree. The core argument revolves around guaranteeing the permanence of the zero set. This concept is formally defined as the existence of a zero set $Z(F)$ situated at a distance ε > 0 from the boundary ∂ T⁴ of the parameter space. This condition rigorously avoids the degeneration of the inscribed square during the limiting process, thereby affirmatively resolving the C⁰ Toeplitz Conjecture.