– 23 – finite-lens effects are not important. (See below.) Neutron Stars and Black Holes: We included more massive stars in our simulation, and kept track of those binaries in which a neutron star or black hole orbits a main sequence star within roughly 0 . 5 AU. The evolutionary pathways that produce such orbits are far less certain than the evolutionary channels producing close white-dwarf/main-sequence binaries. Nevertheless, we know that systems with main sequence stars with mass in the range 0 . 8 − 2 M ⊙ orbit neutron stars and black holes. When the orbits are close enough (with semimajor axis of a few solar radii), such binaries are detected as low-mass x-ray binaries. In our simulation we considered all binaries with primary mass larger than 8 . 5 M ⊙ (the lowest mass star producing a neutron star remnant) and secondary mass between 0 . 8 M ⊙ and 2 M ⊙ . We then assumed that ∼ 1 / 6 − 1 / 5 of the initial orbital separations could be compatible with evolutionary channels that produce close neutron-star/main-sequence pairs (see, e.g., Kalogera & Webbink 1998; Kiel & Hurley 2006). Although we did not follow the detailed evolution of individual systems, our approach yielded a binary fraction of neutron stars in close orbits with main-sequence stars or their remnants of ( < 0 . 5 AU ) of 5 × 10 − $^{5}$. We found that Kepler could detect 1 − 2 lensing “events” by neutron stars or black holes corresponding to approaches close enough that the compact object transits the companion star. In most cases, the neutron star or black hole would be too dim for eclipses of it to be detected. We started by considering evolutionary models for KOI-74 and KOI-81. Our results agree with those of vK2010, which suggest that each system may have evolved to its present state through a process of stable mass transfer. Our approach differs from theirs by using a different prescription for the radius of the donor star prior to envelope exhaustion. Whereas vK2010 used a formula for the radius which depends on only the core mass of the donor, our formula also incorporates the dependence on its equilibrium stellar mass M ∗ , prior to the last phase of mass transfer. This has two effects. First our approach tends to predict somewhat smaller values of the white dwarf mass. This will be tested through radial velocity measurements, and Equation (1) can be adjusted if needed, based on the observations. Second, it allows the Kepler’s period measurement to constrain both the white dwarf mass and the mass of the donor, thereby providing useful input for detailed evolutionary models. We also take seriously the possibility that the masses of KOI-74b and KOI-81b are in the brown dwarf range, so that the formalism we and vK2010 have applied is not valid. We 5. Conclusion 5.1. Results