= Solution
Fit radiatively corrected visible <cross sections> across the <Z boson> <resonance>. The line shape determines $M_Z$, the total <decay width> $\Gamma_Z$, and pole normalizations
$$
\sigma_f^0=\frac{12\pi}{M_Z^2}\frac{\Gamma_e\Gamma_f}{\Gamma_Z^2}.
$$
The first extraction uses the fitted total and visible <partial decay widths>. Assuming <lepton universality>, $\Gamma_e=\Gamma_\mu=\Gamma_\tau=\Gamma_\ell$, an absolute leptonic pole <cross section> gives $\Gamma_\ell=\Gamma_Z\sqrt{M_Z^2\sigma_\ell^0/(12\pi)}$. The hadronic-to-leptonic ratio gives $R_\ell=\Gamma_{\rm had}/\Gamma_\ell$. Subtracting every visible channel then yields
$$
\boxed{\Gamma_{\rm inv}=\Gamma_Z-\Gamma_{\rm had}-3\Gamma_\ell.}
$$
Without <lepton universality>, separately fitted visible leptonic widths replace $3\Gamma_\ell$, with sufficient independent information to disentangle their products with $\Gamma_e$.
A second, useful peak-normalization extraction first determines the invisible-to-leptonic ratio. Because
$$
\sigma_{\rm had}^0=\frac{12\pi}{M_Z^2}R_\ell\left(\frac{\Gamma_\ell}{\Gamma_Z}\right)^2,
$$
one obtains
$$
\boxed{R_{\rm inv}\equiv\frac{\Gamma_{\rm inv}}{\Gamma_\ell}=\sqrt{\frac{12\pi R_\ell}{M_Z^2\sigma_{\rm had}^0}}-R_\ell-3.}
$$
This ratio extraction does not require the measured line width. Multiplication by an independently predicted leptonic <partial decay width>, or by $\Gamma_Z/(R_{\rm inv}+R_\ell+3)$ from the scan, converts it to $\Gamma_{\rm inv}$. Alternatively dividing $R_{\rm inv}$ by the predicted single-species ratio $\Gamma_\nu/\Gamma_\ell$ gives the number of light active <neutrinos>. These are two standard uses of line-shape information, with different choices of theoretical input; if both use the same fitted $\Gamma_\ell$, they are algebraically equivalent and statistically correlated, not independent determinations.
Statistical uncertainties in the reference-process counts give fluctuations in the <collider luminosity> at individual scan points, propagating into normalization and fitted line width. They decrease with additional reference events. A common multiplicative luminosity-scale systematic largely cancels in $R_\ell$ and the normalized line shape, but shifts every absolute peak <cross section>. It therefore does not cancel from either invisible-width extraction. For a fractional upward luminosity shift $\varepsilon$, $\sigma_{\rm had}^0$ shifts by $-\varepsilon$, and at fixed $R_\ell,M_Z,\Gamma_Z$,
$$
\delta R_{\rm inv}\simeq\frac\varepsilon2(R_{\rm inv}+R_\ell+3),\qquad
\delta\Gamma_{\rm inv}\simeq\frac\varepsilon2(\Gamma_{\rm had}+3\Gamma_\ell).
$$
Thus overestimating luminosity increases the inferred invisible contribution. Energy-dependent systematic errors can also distort the line shape and its total <decay width>. The covariance matrix must retain correlations between scan points and experiments, including common theoretical uncertainty in the <Bhabha scattering> calculation.
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