  Less than half of the SNRs within 5 kpc have identified central sources, and
only three are identified as the remnants of Type Ia SNe. Here, we report a
systematic effort to search for compact central sources in the remaining 23
SNRs of this distance limited sample.
  In this first paper we report Chandra observations of four SNRs. We have
undertaken a systematic optical/IR identification program of the X-ray sources
detected in the field of each SNR. Foreground and background sources have
identifiable IR/optical counterparts, while the counterparts of neutron stars
(or black holes) are expected to be very faint. We are able to account for all
the well-detected X-ray sources and thus able to state with some confidence
that there are no associated central sources down to a level of one tenth of
that of the Cas A central source, L_X <~ 10e31 ergs/s. We compare our limits
with cooling curves for neutron stars and find that any putative neutron stars
in these SNRs must be cooling faster than expected for traditional 1.35 Mo
neutron stars and that any putative pulsar must have low spin-down
luminosities, Edot <~ 1e34 ergs/s. However, our limits are unable to constrain
the presence or absence of more unusual options, such as relatively more
massive neutron stars, neutron stars with exotic interiors, or quiescent black
holes. In subsequent papers, we will report on the X-ray and optical/IR
observations of the remaining members of the 5-kpc sample.
