This preprint states and proves the Ananke (Gravitational Closure) Theorem, a foundational classification result for classical gravity. The theorem classifies the admissible structure of closed classical gravity and shows that, under quadratic closure and orthogonal response, gravitational dynamics admit a unique structural form up to equivalence. Vacuum exactness, inverse-square behaviour, and rigidity in isolated regimes emerge as necessary consequences rather than assumptions. The result is classificatory and independent of specific field equations or phenomenology.