Left = latent space (a regular grid + a unit circle at the draggable probe). Right = the same grid pushed through a coupling-layer normalizing flow. Two things to see. (1) The warped grid never folds — lines don't cross — so the map is a bijection. (2) The probe's unit circle becomes an ellipse: its long axis is the local stretch σ_max (= L), its short axis is σ_min (= ℓ). The flow is bi-Lipschitz exactly when that ellipse never collapses to a line, i.e. σ_min stays above 0 — and the inverse map's Lipschitz constant is 1/σ_min, the very 1/ℓ from the inverse-projection bound. Each coupling layer clamps its log-scale to [−c, c], so each contributes a factor in [e^−c, e^c]; stacking layers just multiplies the rails.