The Wallis product formula (1656) computes π as an infinite product whose denominators are 4n²−1. We show that these denominators are identically the twin spine visitor sequence of the Apollonian gasket with seed [−1, 2, 2, 3]. The proof uses only the Descartes Circle Theorem, Apollonian recursion, and elementary algebra. The gasket derives π from pure integer recursion. Exact, not approximate. This result is one consequence of Point-Sphere Theory (PST), a broader framework that derives fundamental constants of physics from the same geometric structure.