= Solution
An exterior misaligned planet exerts an orbit-averaged <secular torque> on each belt orbit. In the hierarchical limit $a_d\ll a$, its leading <quadrupole approximation> drives <nodal precession> about the planet's orbital angular-momentum axis at a rate of order
$$
|\dot\Omega_d|\sim n_d\frac{M_p}{M_*}
\left(\frac{a_d}{a}\right)^3|\cos I_m|,
$$
up to a coefficient of order unity. Because this rate varies across a radially extended belt, <differential nodal precession> winds an initially flat belt into a warp and eventually a vertically thick, phase-mixed structure. Collisions can damp the free inclinations and make material approach a forced or <Laplace plane>; at sufficiently large <mutual inclination>, coupled eccentricity-inclination evolution such as the <Kozai–Lidov mechanism> may also become important.
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