Why are inner planets not inclined?

The paper explicitly proves the claimed finite-itinerary steering for the normalized angular momentum of planet 2 in the spatial Newtonian four-body problem (Theorem 4), constructing orbits that make C̃2 visit any prescribed finite sequence in the unit ball B3 within δ, under generic mass assumptions and in regimes they detail (hierarchical or planetary) . The proof uses Deprit coordinates, a secular reduction with well-separated semimajor axes, a normally hyperbolic invariant cylinder near a hyperbolic secular singularity, two homoclinic channels yielding scattering maps, a twist Poincaré map, and shadowing theorems to realize arbitrary itineraries in the relevant action variables, then lifts the result to the full four-body flow with time estimates . By contrast, the model asserts the claim was likely open, which is contradicted by the paper’s stated and sketched proof.