  Our new study of the two central compact object pulsars, PSR J1210-5226 (P =
424 ms) and PSR J1852+0040 (P = 105 ms), leads us to conclude that a weak natal
magnetic field shaped their unique observational properties. In the dipole
spin-down formalism, the 2-sigma upper limits on their period derivatives, <
2E-16 for both pulsars, implies surface magnetic field strengths of B_s < 3E11
G and spin periods at birth equal to their present periods to three significant
digits. Their X-ray luminosities exceed their respective spin-down
luminosities, implying that their thermal spectra are derived from residual
cooling and perhaps partly from accretion of supernova debris. For sufficiently
weak magnetic fields an accretion disk can penetrate the light cylinder and
interact with the magnetosphere while resulting torques on the neutron star
remain within the observed limits. We propose the following as the origin of
radio-quiet CCOs: the magnetic field, derived from a turbulent dynamo, is
weaker if the NS is formed spinning slowly, which enables it to accrete SN
debris. Accretion excludes neutron stars born with both B_s < 1E11 G and P >
0.1 s from radio pulsar surveys, where such weak fields are not encountered
except among very old (> 40 Myr) or recycled pulsars. We predict that these
birth properties are common, and may be attributes of the youngest detected
neutron star, the CCO in Cassiopeia A, as well as an undetected infant neutron
star in the SN 1987A remnant. In view of the far-infrared light echo discovered
around Cas A and attributed to an SGR-like outburst, it is especially important
to determine via timing whether Cas A hosts a magnetar or not. If not a
magnetar, the Cas A NS may instead have undergone a one-time phase transition
(corequake) that powered the light echo.
