= Solution
For the <static-emitter redshift in a de Sitter patch>, a stationary clock at radius $r_E$ satisfies $d\tau_E=\sqrt{\alpha_E}\,dt_E$. The geometry is stationary, so successive inward pulses emitted at the same radius have the same coordinate travel delay. Their coordinate arrival interval is therefore $\Delta t_O=\Delta t_E$. At the central receiver $\alpha_O=1$, giving
$$
\boxed{\Delta\tau_O=\frac{\Delta\tau_E}{\sqrt{1-r_E^2/r_S^2}}.}
$$
For $r_E=r_S/2$, this becomes
$$
\boxed{\Delta\tau_O=\frac2{\sqrt3}\Delta\tau_E,\qquad 1+z=\frac{\nu_E}{\nu_O}=\frac2{\sqrt3},\qquad z=\frac2{\sqrt3}-1.}
$$
The pulses are redshifted at the centre. At fixed proper emission period, taking a stationary emitter toward the horizon gives $\Delta\tau_O\to\infty$, $\nu_O\to0$ and $z\to\infty$. The inward coordinate travel delay also diverges. A stationary worldline at the horizon itself is not timelike, and the proper acceleration required to hold an emitter static diverges in this limit; the result concerns a sequence of idealized emitters with $r_E<r_S$, not a stationary physical lighthouse exactly on the horizon.
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