Relational Lysis and the Three-Body Problem: A Deterministic Derivation of Newtonian Gravity and General Relativity from Solved-State Geometry

We derive classical gravitational dynamics and exact relativistic geometry from first principles within the Relational Lysis (RL) framework, a deterministic operator construction in which solved-state geometry is primitive. For a translation-invariant relational encoding of a three-body system, every admissible configuration admits a unique decomposition X = D ⊕ S into symmetric alignment (Diamond) and antisymmetric transport (Shadow) components. From D, we construct a canonical covariant self-adjoint Laplacian and show that scaling uniquely determines the inverse-cubic kernel w(r) = Gmimj/r3, yielding the exact Newtonian equations of motion (cf. Theorem 8.9). The Shadow induces a transport operator whose curvature produces a correction with uniquely determined coefficient 3, recovering Mercury’s anomalous perihelion precession (∼ 43 arcseconds per century) without parameter fitting. Under refinement, the system converges to a self-adjoint elliptic continuum operator via Mosco and strong resolvent convergence, inducing a metric and Levi–Civita connection whose curvature satisfies Gμν = 0 by Lovelock uniqueness and elimination of the cosmological constant through RL scaling and bounded dynamics. Experiments of the RL operator chain L ⇒ Γ ⇒ g ⇒ ∇ ⇒ R ⇒ G confirms metric emergence, Einstein vacuum closure (max |Gμν| ∼ 10−7), and Lorentzian signature (−,+,+, +), with the time-like direction arising uniquely from the Shadow component. These results establish that Newtonian gravity and General Relativity arise as necessary operator consequences of solved-state geometry.