= Solution
Approximate the atmosphere by concentric spherical layers of <refractive index>. At every boundary, <Snell's law> conserves $nr\sin\psi$, where $\psi$ is the angle between the <light ray> and the radius. Straight propagation between boundaries conserves the same quantity because $r\sin\psi$ is the distance of that line from the origin. In the smooth limit this is the <spherical ray invariant>
$$
p=n(r)r\sin\psi.
$$
At the surface, a <light ray> apparently on the horizon is tangent, so $p=n_sR$. In vacuum far from the Earth, $n=1$, and the straight incident line has distance $p$ from the centre. Removing the atmosphere while retaining that incident line gives minimum altitude
$$
\boxed{H=p-R=(n_s-1)R=0.0003(6400\,\mathrm{km})=1.92\,\mathrm{km}.}
$$
The detailed variation of <refractive index> with altitude changes the local curvature of the <light ray>, but not the conserved $p$. This conclusion assumes a spherically stratified atmosphere and a <light ray> which actually connects the surface to space; nonspherical structure or a trapping layer would require separate analysis.
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