Numerical Hessian Rank and Keplerian Eigenvalue Structure as Diagnostics for Periodic Orbits in the Equal-Mass Three-Body Problem

We compute the Hessian of the gravitational three-body Hamiltonian in Jacobi coordinates across 3.24 million configurations and analyze its numerical rank, eigenvalue spectrum, and relationship to orbital periodicity. At configurations where the numerical rank of the potential-energy block V_qq drops to~3 (using a singular-value threshold = 10^-6), the three nonzero eigenvalues exhibit an exact 2:1:1 magnitude ratio---the curvature signature of a dominant Keplerian binary with a decoupled third body. This ratio is proven analytically from the Hessian of the 1/r potential and confirmed computationally to < 10^-12 relative error. We integrate 100,000 orbits on GPU and find that configurations at effective rank~9 (potential rank~3) show a periodicity rate of 5.5\ the highest among all rank levels. A gradient-boosted classifier trained on Hessian-derived features achieves cross-validated F_1 = 0.86 for periodicity prediction on a 51,000-configuration dataset, and an A*-discovered closed-form formula ^2 + _ ^2 > achieves F_1 = 0.48. The rank classification is threshold-dependent (vanishing at = 10^-10) and the Keplerian structure is asymptotic (exact only in the decoupled limit), which we characterize through threshold sensitivity sweeps. We compare against separation ratio as a baseline and find that Hessian features provide complementary but not superior predictive power for periodicity. Author preprint deposited for archival and citation. Draft — pending author review.