  We revisit the existence and stability of the critical front in the extended
Fisher-KPP equation, refining earlier results of Rottsch\"afer and Wayne [28]
which establish stability of fronts without identifying a precise decay rate.
We verify that the front is marginally spectrally stable: while the essential
spectrum touches the imaginary axis at the origin, there are no unstable
eigenvalues and no eigenvalue (or resonance) embedded in the essential spectrum
at the origin. Together with the recent work of Avery and Scheel [3], this
implies nonlinear stability of the critical front with sharp $t^{-3/2}$ decay
rate, as previously obtained in the classical Fisher-KPP equation. The main
challenges are to regularize the singular perturbation in the extended
Fisher-KPP equation and to track eigenvalues near the essential spectrum, and
we overcome these difficulties with functional analytic methods.
