Orientation Reversal Under Antipodal Inversion A Geometric Constraint on Continuous Antipodal Paths

A short mathematical note proving that any continuous path connecting a vector xxx to its antipode −x-x−x in Rn\mathbb{R}^nRn must intersect the hyperplane orthogonal to a fixed reference direction. The result follows from a sign-reversal property of inner products combined with the Intermediate Value Theorem and holds in arbitrary dimension.