Toward the Inscribed Square Theorem via the Dalal Expansion Principle

We develop a geometric framework for the inscribed square problem (Toeplitz conjecture, 1911) and present a novel proof that every C² Jordan curve inscribes at least one square. The approach is based on the Dalal Expansion Principle: grow a Jordan curve continuously from a point and track the inscribed square from birth. The main result, the A-Orbit Parity Theorem (Theorem J), shows that the 90° rotational antisymmetry of the inscribed-rectangle parameter space creates an indestructible parity invariant — the number of geometric inscribed squares changes by exactly ±2 at each fold bifurcation, so an odd starting count (verified on convex curves) remains odd forever. Key ingredients include a Birth Collapse Lemma (no total collapse of rectangle vertices on C² curves), a Square Compactness Lemma (equal side lengths force compactness of the inscribed-square set), equivariant transversality for the free Z/4Z action, and the algebraic fact that the antisymmetry involution has no fixed points. The proof is elementary — it uses the implicit function theorem, compactness, and parity counting, without algebraic topology or configuration-space machinery. Keywords: inscribed square problem, Toeplitz conjecture, square peg problem, Jordan curve, inscribed rectangles, equivariant topology, parity invariant, geometric topology.