= Solution
Tag an energetic <initial-state radiation> <photon> in $e^+e^-\to\gamma Z^{(*)}\to\gamma+\text{invisible}$, observing a single-photon signature with <missing energy> and <missing momentum>. At known squared <center-of-mass energy> $s$, the leading single-<photon> recoil <invariant mass> is
$$
\boxed{m_{\rm recoil}^2=(p_{e^+}+p_{e^-}-p_\gamma)^2=s-2\sqrt s\,E_\gamma.}
$$
Above the <Z boson> <mass>, radiative-return events cluster at $m_{\rm recoil}\simeq M_Z$. At or near the pole the full off-shell recoil spectrum is fitted instead. Folding the <Breit-Wigner resonance cross section> with the calculable <QED> radiation kernel gives a predicted tagged rate proportional to $\Gamma_e\Gamma_{\rm inv}$ for fixed total <decay width>. A fit with the known luminosity and visible line-shape parameters therefore determines $\Gamma_{\rm inv}$ directly from invisible events, rather than subtracting visible widths. Comparing identically selected radiative-return events with visible <Z boson> decays can instead constrain $\Gamma_{\rm inv}/\Gamma_{\rm vis}$ while cancelling much of the luminosity normalization.
The detector needs efficient single-<photon> triggering, good <photon energy> and angular resolution, reliable timing, and nearly hermetic charged-particle and <particle calorimeter> coverage. Forward vetoes reject radiative <Bhabha scattering> with escaped <electrons>; additional-photon vetoes reject $\gamma\gamma$ events with a lost <photon>. Require sufficient transverse <photon momentum> that a recoiling visible system could not all disappear down a beam hole. Timing rejects cosmic rays and beam backgrounds. Detector inefficiencies and beam holes must be measured, since visible final states with lost particles imitate the signal. The irreducible $e^+e^-\to\nu_e\bar\nu_e\gamma$ contribution from $t$-channel <W boson> exchange and its interference must be included when isolating the <Z boson> invisible <decay width>.
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