= Solution
The <tau-neutrino charged-current threshold> requires specifying the stationary target and the lightest allowed final state. For a free <neutron>, the lowest <charged current> channel is $\nu_\tau n\to\tau^-p$. At threshold both final particles are at rest relative to one another in the <centre-of-momentum frame>, so
$$
s=m_n^2+2m_nE_\nu=(m_p+m_\tau)^2,\qquad
E_\nu^{\rm thr}=\frac{(m_p+m_\tau)^2-m_n^2}{2m_n}\simeq3.45\,\mathrm{GeV}.
$$
The equal-<nucleon>-mass approximation gives $m_\tau+m_\tau^2/(2m_N)\simeq3.46\,\mathrm{GeV}$. More generally $E_\nu^{\rm thr}=[(m_\tau+M_X)^2-M_T^2]/(2M_T)$ for a target of <mass> $M_T$ and a recoil system with minimum allowed <invariant mass> $M_X$. A stationary <electron> in $\nu_\tau e^-\to\tau^-\nu_e$ instead gives $(m_\tau^2-m_e^2)/(2m_e)\simeq3.09\,\mathrm{TeV}$. The customary $3.5\,\mathrm{GeV}$ answer therefore presumes a <nucleon> target; nuclear thresholds and practical <deep inelastic scattering> thresholds differ.
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