Gabriel's Horn the solid of revolution generated by rotating about the x-axisfor is the classical example of a geometric object with finite volume (π cubic units) butinfinite surface area. We prove that a Schwarzschild black hole, viewed as a 4D spacetimemanifold, is the exact dual of this object: it possesses a finite event horizon area but aninfinite interior spacetime volume, in the precise sense established by Christodoulou andRovelli (2015). The mechanism that inverts the relationship is the spacetime signature flip atthe event horizon, whereby the radial coordinate becomes time like and the time coordinatebecomes spacelike inside the horizon a feature entirely absent from Gabriel's Horn andfrom all prior geometric analogies in the literature, including Bousder (2023). We formalizethis as the Spacetime Horn Duality Theorem and derive its consequences for the informationparadox.
