= Solution
Estimate characteristic bulk terms. Put $\epsilon=H/r\ll1$ and assume smooth radial variation on scale $r$, secular evolution, and $\alpha\lesssim1$. Vertical support gives $c_s^2\sim p/\rho\sim\Omega^2H^2$, while the <alpha disk> prescription and slow accretion imply
$$
\nu\sim\alpha\Omega H^2,\qquad u_r\sim\frac\nu r,\qquad u_z\sim\frac Hr u_r.
$$
These velocity estimates follow from the viscous radial evolution scale and <mass conservation>. They are estimates for a slowly evolving thin disc, not a claim that geometric thinness excludes independently forced rapid vertical motion.
For vertical balance, the exact central force expands as
$$
\frac{GMz}{(r^2+z^2)^{3/2}}=\Omega_K^2z\left[1-\frac32\frac{z^2}{r^2}+O\!\left(\frac{z^4}{r^4}\right)\right].
$$
At $z\sim H$, the neglected correction has magnitude $\rho\Omega^2H\epsilon^2$, compared with the retained force $\rho\Omega^2H$. Radial <pressure> support similarly changes the rotation from its Keplerian value by
$$
\frac{\delta\Omega^2}{\Omega_K^2}\sim\frac{c_s^2}{r^2\Omega_K^2}=O(\epsilon^2).
$$
Vertical meridional inertia scales as $u_r u_z/r$ or $u_z^2/H$, both $O(\alpha^2\epsilon^4\Omega^2H)$. Meridional viscous accelerations can scale as $\nu u_z/H^2$ or $\nu u_r/(rH)$, both $O(\alpha^2\epsilon^2\Omega^2H)$. Thus the gravity correction and possible viscous correction give
$$
\boxed{\frac{\text{neglected vertical-force terms}}{\rho\Omega^2H}=O(\epsilon^2).}
$$
Some omitted terms are smaller than this leading order.
For the heating equation, the retained dissipation is $q_0^+\sim\rho\nu\Omega^2\sim\alpha p\Omega$. Radial <pressure> support and the finite-height correction change the dominant shear by a relative $O(\epsilon^2)$. Smooth vertical variation of the rotation gives $\partial_z(r\Omega)=O(\epsilon\Omega)$, whose additional squared shear contributes $O(\epsilon^2q_0^+)$. Meridional shear of order $u_r/H$ gives an additional relative $O(\alpha^2\epsilon^2)$ contribution.
Thermal <advection> and compressional work are of order $\rho u_r c_s^2/r$ and $p u_r/r$, respectively. Relative to $\alpha p\Omega$ they are $O(\epsilon^2)$. Thermal storage on the viscous evolution timescale $t_\nu\sim r^2/\nu$ has the same order, since $p/t_\nu\sim\alpha p\Omega\epsilon^2$. For <radiative diffusion> with the same local diffusivity in different directions, smooth gradients give $F_r/F_z\sim H/r$, so
$$
\frac{r^{-1}\partial_r(rF_r)}{\partial_zF_z}\sim\frac{F_r/r}{F_z/H}=O(\epsilon^2).
$$
Consequently
$$
\boxed{\frac{\text{neglected local-energy terms}}{(9/4)\alpha\Omega p}=O\!\left((H/r)^2\right).}
$$
This is the <thin-accretion-disk truncation error> for the secular, radially smooth approximation. Neglect of disc <self-gravity> and <radiation pressure>, optical thickness, the opacity model and the idealized zero surface boundary have their own validity conditions; their errors do not become small solely because $H/r$ does. Near a sharp radial <boundary layer> or during thermal evolution on the heating timescale, the above estimates need not apply.
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