  The $^{25}$F($5/2^+) (-1p) ^{24}$O reaction was studied at the NSCL using the
S800 spectrometer. The experimental spectroscopic factor for the ground-state
to ground-state transition indicates a substantial depletion of the proton
$d_{5/2}$ strength compared to shell-model expectations. Our result supports
the findings reported by Tang \textit{et al.}, from their study of the $(p,2p)$
reaction at RIBF. The overlap between the $^{25}$F and $^{24}$O ground-states
is considerably less than anticipated if $^{24}$O acted as a robust and rigid
doubly-magic core in $^{25}$F. We interpret the results within the framework of
the Particle-Vibration Coupling (PVC) of a $d_{5/2}$ proton coupled to a
quadrupole phonon of an effective core. This approach provides a good
description of the experimental data by requiring an effective $^{24}$O* core
with a phonon energy of $\hbar\omega_2$= 3.2 MeV, and a $B(E2) ~ 2.7$ W.u.,
softer and more collective than a bare $^{24}$O. Both the Nilsson deformed mean
field and the PVC models appear to capture the properties of the effective core
of $^{25}$F, suggesting that the additional proton tends to polarize the free,
doubly magic $^{24}$O in such a way that it becomes either slightly deformed or
a quadrupole vibrator.
