Straight grid lines in ℝ³ morph linearly into their images under F. The Jacobian determinant is the nonzero constant −2, so F is a local diffeomorphism everywhere — but it is not injective. It is generically three‑to‑one.
The three amber points (0,0,−¼), (1,−3⁄2,13⁄2), (−1,3⁄2,13⁄2) all collapse onto the single image (−¼,0,0) — a direct witness that F is not globally invertible, disproving the conjecture for n ≥ 3.