Tidal Harmonic Prediction for Triply Eclipsing Triple Stars: A Catalog-Based Approach from Shape Sphere Dynamics

We present a tidal constituent catalog method for predicting the dynamics of hierarchical triple star systems. All significant frequencies are specified analytically from the known orbital periods — f (n, m) = n/Pᵢnner + m/Pₒuter — and only amplitudes and phases are fitted via linear least squares. On simulated Galilean moon separations, the catalog reduces prediction error from 18. 0% (blind FFT, 25 constituents) to 1. 3% (catalog order 4, 21 constituents), a 14× improvement. On real TESS photometry of six triply eclipsing triples with independent test epochs separated by 130–710 days from training data, all systems achieve < 2% NRMSE; TIC 14839347 shows 5× catalog superiority at 683-day extrapolation. The method is grounded in the shape sphere — the S² manifold on which the triangle formed by three bodies evolves (u² + v² + w² = 1 exactly). Fourier decomposition of the shape trajectory confirms that every oscillation matches a catalog frequency; no hidden modes exist. The dominant perturbation amplitude scales as A ∝ η⁰. 93 (R² = 0. 985) with virtual independence from mass ratios, enabling prediction tables without simulation. The equilibrium frequency ratio for three coupled oscillators satisfies the tribonacci equation r³ = r² + r + 1, yielding τ ≈ 1. 839. This ratio provides optimal KAM protection: systems at τ maintain their frequency ratio under all non-migratory perturbations tested, with shape band width 25× narrower than equivalent 2: 1 resonant systems. The framework generalises to N bodies through hierarchical decomposition into nested shape spheres, with the n-nacci equilibrium constant converging from τ = 1. 839 (n=3) toward 2. 000 (n→∞).