  We study the contribution of $\nu_e \to \nu_\tau \to \tau \to \mu$
transitions to the wrong-sign muon sample of the {\em golden} channel of the
Neutrino Factory. Muons from tau decays are not really a background, since they
contain information from the oscillation signal, and represent a small fraction
of the sample. However, if not properly handled they introduce serious
systematic error, in particular if the detector/analysis are sensitive to muons
of low energy. This systematic effect is particularly troublesome for large
$\theta_{13} \geq 1^\circ$ and prevents the use of the Neutrino Factory as a
precision facility for large $\theta_{13}$. Such systematic error disappears if
the tau contribution to the golden muon sample is taken into account. The fact
that the fluxes of the Neutrino Factory are exactly calculable permits the
knowledge of the tau sample due to the $\nu_e \to \nu_\tau$ oscillation. We
then compute the contribution to the muon sample arising from this sample {\em
in terms of the apparent muon energy}. This requires the computation of a
migration matrix $M_{ij}$ which describes the contributions of the tau
neutrinos of a given energy $E_i$, to the muon neutrinos of an apparent energy
$E_j$. We demonstrate that applying $M_{ij}$ to the data permits the full
correction of the otherwise intolerable systematic error.
