Unifying the Hyperbolic and Spherical 2-Body Problem with Biquaternions

The paper proves that all semisimple relative equilibria (RE) in the hyperbolic two-body problem are coaxial, satisfy the hyperbolic lever rule m1 sinh(2χ1) = m2 sinh(2χ2), admit |η|^2 = f(cosh ψ) sinh ψ/(2ζ), and that no parabolic RE exist for strictly attractive potentials; it then shows reduced Lyapunov stability for Γ(θ, ψ) < 0 and linear instability for Γ(θ, ψ) > 0 (Theorem 3.2, Proposition 4.3, Theorem 4.4) , with the gravitational potential V(z) = −m1 m2/(z√(z^2 − 1)) and the right-reduced equations (3.5) under the SL2(C) conjugation action (3.6) . The candidate solution reaches the same classification and the same stability discriminator Γ(θ, ψ) (including the same Z and cos 2θ dependence) but via a different route: the symmetric-space/energy–momentum method with an augmented potential and a direct slice-Hessian calculation. The only substantive discrepancy is the model’s general claim that the Γ-based stability criterion is potential-independent; the paper states and justifies that criterion for the gravitational potential (using continuity from H2), not for arbitrary strictly attractive potentials. Aside from that overreach, the two accounts agree on all structural facts and formulas.