The Ananke (Gravitational Closure) Theorem

This preprint states and proves the Ananke (Gravitational Closure) Theorem, a foundational classification result for classical gravity. Rather than proposing a new model or fitting data, the theorem asks a prior structural question: which gravitational field structures are admissible once gravity is required to close as a classical field under covariance, quadratic action structure with finite conserved energy, and exhaustion of functional freedom under symmetry. From three minimal axioms—classical covariance, quadratic closure, and orthogonal modes of response—the theorem derives a complete classification. In isolated vacuum regimes, closure is exact and admits no residual degrees of freedom, enforcing rigidity and inverse-square scaling. In non-vacuum symmetry-reduced regimes, closure admits exactly one residual redistributive degree of freedom and no more. The result fixes the admissible covariant gravitational action uniquely up to equivalence and excludes additional vacuum degrees of freedom, screening mechanisms, or phenomenological supplements. This manuscript is released as a preprint to establish intellectual priority; regime-specific consequences are developed in subsequent work.