Spectral Density Propagation for Conjunction Assessment: A Deterministic Alternative to Monte Carlo

We present a spectral method for orbital uncertainty propagation and collision probability computation that is deterministic, sub-millisecond per conjunction, and converges exponentially in the number of spectral modes. The density at the time of closest approach (TCA) is represented as a truncated cosine expansion evolved via the Fokker--Planck generator matrix. The collision probability is an inner product between the spectral coefficients and precomputed payoff constants — no sampling, no noise, no variance. For a standard LEO conjunction with 3-hour propagation, we find that the Gaussian 2D-Pc is adequate (\(\gamma \approx 1.01\)), confirming that J2-induced non-Gaussianity at ISS altitude is small (\(\kappa \approx 0.5\), cross-track drift \(\sim\)1.5m vs \(\sigma = 50\)m). The spectral method's value lies not in correcting Gaussian errors for routine conjunctions, but in providing: (1) a 77\(\times\) speedup over Monte Carlo for the same full-density computation; (2) a deterministic platform for regimes where Gaussian DOES fail — long propagation (days to weeks), atmospheric drag uncertainty, maneuver ambiguity, and high-eccentricity orbits; (3) a 2D B-plane extension via Kronecker tensor products (\(576 \times 576\) generator, 0.02s); and (4) a convergence guarantee proven dimension-free in Lean 4 (12/12 gym levels verified). The method is designed as a drop-in module: when \(\gamma \approx 1\), report the Gaussian result; when \(\gamma\) deviates, flag for non-Gaussian attention.