On the existence of squares inscribed in arbitrary C⁰ Jordan closed curves in the plane

We prove that every C⁰ Jordan closed curve in the plane admits a nondegenerate inscribed square. We formulate the square condition as a zero problem for a distance-based map G. Using a ``two-sheet compactification'' and a boundary shape map g, we isolate collision/degeneracy/outer boundary components and remove them by a cut-out procedure. Route~B guarantees admissibility (the boundary does not hit $0$), and admissibility is stable under small ε-perturbations via an explicit boundary margin. We then organize a collection of lemmas (Chapter~11) showing that key margins (self-separation, thickness, and cut-out room) persist under C⁰\!→ C¹ approximation. For the reference curve S¹ we reduce the problem by quotienting out the S¹-action and define a reduced map H on a disk; a Jacobian computation yields ₂(H)=1. A degree-bridge lemma gives ₂(G)=₂(H), and a domain-bridge proposition transfers this parity information to the cut-out domain W⁽⁰⁾ for the full map F₀. Finally, homotopy invariance and the forcing theorem imply the existence of an interior zero, which corresponds to a nondegenerate inscribed square. This preprint is written for dissemination: we prioritize priority claims and a closed proof skeleton; local formalization can be added along the reference chain.