The Spectral Generator of the N-Body Latent: Connecting Padé Poles, Koopman Eigenvalues, and Dynamical Classification

The N-body Latent [Nagy 2026g, 2026i] provides an exact, finite representation of gravitational trajectories via the generating function \(G(z; \mathbf{v}_0)\). The Universal Spectral Representation Theorem [Nagy 2026b] shows that for stochastic systems, the Fokker–Planck spectral generator \(M\) — a grade-2 Latent — is a sufficient statistic for the entire process. But what plays the role of \(M\) for the deterministic N-body problem?

We answer this by connecting the Latent framework to the Koopman operator of Hamiltonian dynamics. The Koopman generator \(\mathcal{L}\) is the deterministic counterpart of the Fokker–Planck generator: it advances observables forward in time and its spectrum encodes the full dynamical structure. We prove four results that establish the connection:

1. Padé–Koopman correspondence (Theorem 1). The poles of the Padé approximant to \(G(z; \mathbf{v}_0)\) converge to the singularities of the Koopman resolvent. For regular orbits, these approximate the discrete Koopman eigenvalues; for chaotic orbits, they approximate the continuous spectrum.

2. Spectral complexity theorem (Theorem 2). The spectral entropy of the Padé pole distribution characterizes the topological entropy of the orbit. Low entropy ↔ regular (quasi-periodic); high entropy ↔ chaotic.

3. Quantitative Sundman theorem (Theorem 3). The Padé convergence rate \(\rho\) equals \(\exp(2\pi \tau_{\min}/T)\), where \(\tau_{\min}\) is the distance from the Koopman spectrum to the real axis in complex time — determined by the nearest collision singularity.

4. Classification theorem (Theorem 4). The orbit classification (bounded/ejection/collision) is encoded in the topology of the Koopman spectrum, which is approximated by the Padé pole landscape.

The deterministic spectral generator \(\mathcal{L}\) is a grade-2 Latent in the hierarchy of [Nagy 2026e], completing the parallel:

| | Stochastic (USRT) | Deterministic (this paper) | |---|---|---| | Grade-1 Latent | Density \(p(x)\) → spectral coefficients | Trajectory \(\mathbf{q}(t)\) → Fourier modes \(\Lambda_n\) | | Grade-2 Latent | Fokker–Planck generator \(M\) | Koopman generator \(\mathcal{L}\) | | Grade-3 Latent | Time-varying generator \(M(t)\) | Meta-Latent over IC space |

All four results are supported by numerical evidence from 30,000+ three-body orbit computations. The spectral generator framework unifies the N-body Latent, the USRT, and the Latent of Latents into a single hierarchy.