Three-Body Dynamics on the Shape Sphere: Tidal Constituent Prediction, Tribonacci Protection, and the Geometry of Gravitational Chaos

This monograph develops a geometric framework for the gravitational three-body problem based on the shape sphere — the S² manifold that exactly parameterises the triangle formed by three bodies at every instant, independent of size, position, and orientation. Three key advances are presented. 1. The tidal constituent catalog method. All dynamical frequencies of a hierarchical triple are predicted analytically from the known orbital periods as combination tones f (n, m) = n/Pᵢnner + m/Pₒuter, and only amplitudes are fitted via linear least squares. This reduces prediction error from 18% (blind FFT) to 1. 3% on simulated Galilean moon separations — a 14× improvement — and from 9. 8% to 4. 9% on real TESS photometry. Fourier decomposition of the shape sphere trajectory confirms that every oscillation matches a catalog frequency; no hidden modes exist. The method is the gravitational equivalent of ocean tidal harmonic analysis (Doodson, 1921). 2. The tribonacci equilibrium and τ-protection. The three independent frequencies of a hierarchical triple (inner orbital, outer orbital, apsidal precession) behave as three coupled oscillators on S². Their equilibrium frequency ratio satisfies the tribonacci equation r³ = r² + r + 1, yielding the tribonacci constant τ ≈ 1. 839. This ratio provides optimal KAM protection: a system initialised at frequency ratio τ maintains it with precision Δ < 10⁻⁴ across all eccentricity-damping strengths tested. Only orbital migration (da/dt ≠ 0) can break the protection and drive the system toward integer resonance. This establishes that resonance (e. g. , the Galilean 2: 1 lock) is the dissipative endpoint, not the natural state, of three-body systems. The bridge between the three oscillating edges of the gravitational triangle and the three great circles of the discrete S² geometry is established explicitly: the Diophantine condition assumed by KAM theory is derived from the topology of three coupled circles. 3. Two Poincaré regimes on S². The shape sphere reveals why chaos requires n ≥ 3 bodies (S² is the first shape space with ≥ 2 dimensions) and distinguishes two regimes: (A) hierarchical systems where the shape trajectory is a thin band and the catalog captures all dynamics (power-law divergence), and (B) compact systems where the trajectory fills S² and genuine topological chaos exists (exponential divergence). The boundary is the KAM critical η. The framework generalises to N bodies through hierarchical decomposition into (N−2) nested shape spheres. The equilibrium frequency ratio for n coupled oscillators (the n-nacci constant) converges from τ = 1. 839 (n = 3) toward 2. 000 (n → ∞), proving that the three-body problem has the strongest KAM protection of all n-body problems. A multi-system amplitude survey across 18 hierarchical triples gives A ∝ η⁰. 93 (R² = 0. 985), virtually independent of mass ratios. This enables one-dimensional prediction tables — the gravitational analog of Admiralty tide tables — requiring no numerical simulation. Validation spans Mercury's perihelion precession (42. 99"/century, 0. 02% agreement with observation), the Galilean moon Laplace resonance (libration period 306 orbits encoding the tribonacci partial quotient a₅ = 305 to 0. 4%), and six real TESS triply eclipsing triple star systems with independent test sectors achieving prediction errors of 1. 6–20%. The monograph presents 85 quantitative results organised in three confidence tiers. Tier 1 (60 results) stands on standard celestial mechanics alone. Tier 2 (17 results) documents suggestive numerical matches with the companion discrete S² publication. Tier 3 (8 results) provides alternative geometric interpretations requiring the (3+3) framework. Notes: Contains numerical code, simulation results, and real TESS light curve analysis Self-contained: all Tier 1 results are independently reproducible without the companion publication Graduate-level exposition with full derivations accessible to researchers in celestial mechanics, dynamical systems, and observational astronomy