Phenomenological Consequences of the Ananke Theorem

This preprint records the phenomenological consequences of the Ananke (Gravitational Closure) Theorem under explicit symmetry reduction. It is released to establish intellectual priority for the algebraic scaling relations that follow from gravitational closure once admissible symmetry classes are imposed. Starting from the unique covariant gravitational action fixed by the Ananke Theorem, the paper derives the complete algebraic consequences of closure in non-vacuum, symmetry-reduced regimes. No new gravitational model is proposed, no phenomenological modification is introduced, and no observational fitting is performed. All results follow deductively from closure, conservation, orthogonality, and symmetry. Two symmetry classes are analysed. In stationary, axisymmetric, rotationally supported extended systems, closure admits exactly one residual redistributive degree of freedom, yielding a unique algebraic balance relation governing circular motion. The baryonic Tully–Fisher relation, the radial-acceleration relation, and the mass-discrepancy–acceleration relation are shown to arise as distinct observational projections of this single closure relation, with slopes fixed algebraically and no independent freedom. In homogeneous (or homogeneous–isotropic) configurations, the same residual degree of freedom yields pressureless a−3 scaling as the unique evolution law compatible with closure. The scope of this document is intentionally restricted to symmetry-reduced regimes where functional freedom is exhausted. Outside these regimes, deviations are expected and are interpreted as consequences of symmetry breaking rather than failures of closure. Any additional universal scaling relations, slopes, or independent constants would violate gravitational closure within the stated domain of validity. This record is intended as a definitive, time-stamped statement of the deductive consequences of gravitational closure under symmetry reduction.