This paper presents an analytical mathematical framework for investigating the gravitational constraints imposed by the coupled giant‑planet system on the rotational evolution of inner bodies in the outer Solar System. Within this framework, the Saturn–Neptune 2: 5 orbital resonance is modelled as a composite boundary condition that regulates the transfer of angular momentum between orbital planes and planetary spin axes. By constructing a closed‑form solution for the solar torque as a function of Uranus's obliquity (), the framework rigorously demonstrates the existence of a strict geometric singularity: at = 90^, the net torque vanishes identically ( (2) 0). This forces a hard limit on the effective domain of pure gravitational resonance in altering planetary attitudes. Crucially, we identify the early solar mass-loss during the T Tauri phase (∼5–10%) as the physical mechanism that permits adiabatic passage through the 90° deadlock, eliminating the need for external collision events. The formalism defines a characteristic response time T₁ₑ 10⁶ to 10⁷ years as an inherent timescale derived from the mathematical structure, and derives the necessary condition for resonance crossing: M_/M_ 10^-3 (0. 1%) as the theoretical lower bound, consistent with observational constraints on pre-main-sequence stellar mass-loss. No new physical laws are introduced; rather, the framework provides a rigorous solution space for the classical three‑body + gyroscopic coupling problem, offering analytically pre‑screened initial‑condition constraints and dynamical boundaries for future numerical simulations. The mathematical architecture reveals that Uranus's 98° obliquity is not a probabilistic collision outcome but an inevitable consequence of stellar evolution coupled with planetary resonance. The framework further yields a falsifiable prediction for the Solar System's terminal epoch: a potential resonance reactivation during the red giant phase, contingent upon the survival of the Saturn–Neptune resonance before the 0. 5% solar mass-loss threshold. (Note on AI-Assisted Computation Certain mathematical derivations and physical calculations in this paper were performed by an AI tool (large language model) based on the theoretical framework and postulate system provided by the author. Specifically, the AI tool contributed to: formula derivation, equation solving, integral evaluation, series summation, and recalculation verification of established quantum mechanical results. All physical insights, core assumptions, logical premises, and the theoretical framework itself were independently developed by the author. The AI tool served solely as an auxiliary instrument for mathematical derivation and computational verification, comparable in role to symbolic computation software or numerical tools routinely employed by researchers. The author has reviewed every derived result for physical plausibility, consistency with known experimental data, and logical coherence, and assumes full responsibility for all conclusions. This statement is provided in the interest of academic transparency, while clearly distinguishing between the originality of ideas and the auxiliary role of computation. )
Yanlei Liu (Sat,) studied this question.