Neutron stars and pulsars
Claim in the list: Neutron stars only 10–20 km in diameter have more than the Sun’s mass and can spin up to 1,000 times per second (pulsars).
Verification: The first part is broadly right; the speed overstates the confirmed record. A typical neutron star has about 1–2 solar masses and a diameter on the order of 20 km, though exact radii depend on the dense-matter equation of state. The fastest known pulsar spins about 716 times per second, not 1,000. A pulsar is a neutron star whose emission beams sweep across our line of sight; not every neutron star is observed as a pulsar.
Conclusion: Neutron stars are extraordinarily compact and often more massive than the Sun; confirmed spin rates exceed 700 rotations per second, but 1,000 per second is not an established record.