Left = latent space; drag z₂. Right = output f(z). The bound ℓ‖Δz‖ ≤ ‖Δf‖ ≤ L‖Δz‖ says f(z₂) must land inside the ring between the red inner circle (radius ℓ‖Δz‖) and the teal outer circle (radius L‖Δz‖) — a fair currency exchange that never shrinks a distance below ℓ× or grows it above L×. Drag ℓ toward 0 and the inner ring collapses: now two very different latents can map to almost the same output — the map is no longer invertible, and reading a latent gap off an output gap (the inverse projection we need) becomes impossible.