  Low-mass white dwarfs (LMWDs) are believed to be exclusive products of binary
evolution, as the Universe is not yet old enough to produce them from single
stars. Because of the strong tidal forces operating during the binary
interaction phase, the remnant host systems observed today are expected to have
negligible eccentricities. Here, we report on the first unambiguous
identification of a LMWD in an eccentric (e=0.13) orbit with a millisecond
pulsar, which directly contradicts this picture. We use our spectra and
radio-timing solution (derived elsewhere) to infer the WD temperature T_eff =
8600 +/- 190 K) and 3D systemic velocity (179.5 km\s). We also place
model-independent constraints on the WD radius (R_WD = 0.024+/- 0.004/0.002
R_sun) and surface gravity (log g = 7.11 +/- 0.08/0.16 dex). The WD and
kinematic properties are consistent with the expectations for low-mass X-ray
binary evolution and disfavour a three-body formation channel. In the case of
the high eccentricity being the result of a spontaneous phase transition, we
infer a mass of 1.6 M_sun for the progenitor of the pulsar, which is too low
for the quark-nova mechanism proposed by Jiang et al. (2015). Similarly, the
scenario of Freire & Tauris (2014), in which a WD collapses onto a neutron star
via an rotationally-delayed accretion-induced collapse, requires both a
high-mass differentially rotating progenitor and a significant momentum kick at
birth under our constraints. Contrarily, we find that eccentricity pumping via
interaction with a transient circumbinary disk is consistent with all inferred
properties. Finally, we report tentative evidence for pulsations which, if
confirmed, would transform the star into an unprecedented laboratory for WD
physics and stellar convection.
