  This article presents a model independent search for an additional, mostly
sterile, Heavy Neutral Lepton (HNL), that is capable of mixing with the
Standard Model $\tau$ neutrino with a mixing strength of $|U_{\tau 4}|^{2}$,
corresponding to the absolute square of the extended
Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix element. Data from the \babar~
experiment, with a total integrated luminosity of $424~ \text{fb}^{-1}$, are
analyzed using a kinematic approach that makes no assumptions on the model
behind the origins of the HNL, its lifetime or decay modes. No significant
signal is found. Upper limits on $|U_{\tau 4}|^{2}$ at the 95$\%$ confidence
level, depend on the HNL mass hypothesis and vary from $2.31 \times 10^{-2}$ to
$5.04 \times 10^{-6}$ (with all uncertainties considered), across the mass
range $100 < m_{4} < 1300 ~\text{MeV}/c^{2}$; the more stringent limits being
placed at higher masses.
