Dynamical and Optimization Trade-offs of Levi--Civita Coordinates for Learned Close-Encounter Dynamics

Classical regularization removes the binary-collision singularity from the Kepler problem, but its value as a representation for learned Hamiltonian dynamics has not been systematically isolated.

We compare Cartesian and planar Levi--Civita formulations of a perturbed Kepler system with a smooth quadrupole potential.

With the perturbation supplied analytically, a Levi--Civita Hamiltonian splitting holds the maximum relative energy error near $2.1\times10^{-5}$ through eccentricity $e=0.99$, while the Cartesian splitting becomes unstable.

This advantage persists at matched physical horizon and force-evaluation budget, where the regularized baseline is $3\times10^{-5}$, about $4.7$--$8.3$ orders of magnitude below the Cartesian arm depending on eccentricity.

In held-out high-eccentricity tests with matched sampling, regularized models produce finite rollouts in $40/40$ runs versus $0/40$ for Cartesian.

However, the fixed-shell construction supplies the regularized model with the exact initial orbit energy, and survival still carries $\mathcal{O}(1)$ energy error.

Four neural residual objectives fail to approach the analytic result.

Exact-feature controls show that the regularized residual is a four-monomial degree-6 polynomial that a direct least-squares solve fits to the baseline.

The remaining exact-feature gap is due to severe raw-basis ill-conditioning: orthogonalization restores baseline fitting for L-BFGS in two iterations.

Small MLPs remain at $\mathcal{O}(1)$ rollout error even after gauge symmetrization.

Levi--Civita coordinates therefore improve dynamical conditioning while worsening raw-basis optimization conditioning; accurate neural residual learning remains unresolved.

This is a controlled falsification-plus-trade-off study, not a solution to learned close-encounter dynamics.