The Three-Body Problem as a Replicator Dynamical System: A Complete Geometric Reformulation via Kelvin-Transform Quaternions

We present a complete geometric reformulation of the Newtonian three-body problem in five state variables derived from the gravitational potential of each body. Mapping each body to a unit quaternion on S³ via its normalised Kelvin transform - the fraction of total gravitational potential it contributes - we derive a closed coordinate system of five equations that fully encodes the three-body dynamics. The scalar parts of the quaternions satisfy the replicator equation of evolutionary game theory, dwᵢ/dt = wᵢ(⟨vr⟩ − vrᵢ), establishing an exact equivalence between gravitational dynamics and population selection. A missing-harmonics theorem follows directly from the normalisation constraint. The replicator structure further induces an exact entropy evolution law, dH/dt = −Cov_w(log(w/w*), vr). Linearising near a fixed point yields H ∝ exp(2λt) for unstable modes, connecting the entropy growth rate, in the linear regime, to the Lyapunov exponent of the dominant mode. The closed system is verified to exact precision on non-singular orbits, and the ejection-detection property of the Kelvin fractions offers a practical diagnostic for N-body simulation. Version 2 (uploaded 25/03/2026) This version substantially strengthens the paper relative to the initial submission. Key additions and corrections: Invertibility remark (Section 3.1): For physically admissible, non-singular states in the centre-of-mass frame, the full Kelvin variables determine the physical state uniquely, and conversely. Section 5.5 -- Near-collision analysis (new): Introduces an explicit dual-formulation framework separating CoM-based Kelvin fractions (the paper's primary definition, satisfying the replicator equation) from pair-potential fractions (an auxiliary formulation). Proposition 3 is now correctly scoped to the pair-potential variant, where the clean analytic zero-limit result holds. The CoM fractions show qualitative suppression without a universal zero limit -- this distinction is stated honestly. Pythagorean three-body benchmark (Section 5.2, Figure 6): Numerical integration of the Burrau (1913) three-body problem (masses 3, 4, 5 at rest at vertices of a 3-4-5 right triangle) confirms 9 close-approach events and validates the suppression behaviour under both definitions. Figure 6 shows the CoM vs. pair-potential comparison directly. Section 7.4 -- Scope and representation: Explicit statement that the replicator framework is not Hamiltonian (simplex structure trades for symplectic structure), and that singular orbits (collision, near-collision) lie outside the current scope. The physics is preserved in the full reconstructed state. Abstract and conclusions: Updated to scope the near-collision result ("in the pair-potential formulation") consistently throughout. Reference [8]: Burrau, C. (1913). Über die gegenseitige Anziehung zweier Körper. Astronomische Nachrichten, 136, 281.