Three-Dimensional Phenomenological Correlations of Solar System Orbital Inclinations and Ellipsoidal Boundary Geometries

Standard celestial mechanics generally treats orbital parameters in isolation or within the framework of secular perturbations. In this paper, we present an alternative geometric and phenomenological model based on NASA JPL Horizons DE441 ephemeris data, describing the three-dimensional distribution of solar system planets using principles of effective media and analogue gravity. First, using an ellipsoidal Hill sphere model, we show that this geometric correlation (utilizing perihelion, aphelion, and semi-major orbital distances as axes, with precession correction) yields a universal density ratio of κ = ρplanet/ρref(r) = 3.005 ± 0.050 for the eight planets. This reduces Mercury's anomaly from approximately 100% in the standard spherical model to 4.3%, indicating a strong link between Hill boundaries and orbital eccentricity. Second, the maximum vertical deviation of each planet above the solar equatorial plane, Zmax = a(1 + e) · sin(ihelio), globally reveals a correlation of Zmax ∝ n^2.192 (R^2 = 0.993), where n is a scale-relative integer. The ideal two-dimensional Nottale scaling prediction is Z ∝ n^2.000; the observed index differs by Δβ = 0.192. Third, we propose that this discrete splitting of the l parameter may be associated with spatial inhomogeneity of the vacuum substrate (∇ρ ∝ r^-4), which determines local wave impedance and redistributes resonant angular modes. The model recovers the ideal radial n^2 law, yields a mean absolute error of 4.26%, and is validated by Monte Carlo simulations (p < 0.001). We note that the precessional coefficient α = 0.0134 rad^-1 exhibits a proportion to the global mass ratio of the solar system: α ≈ π^2 · (∑ mplanet/M⊙).