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 = −Covw (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.
Lee Michael John Rich (Wed,) studied this question.