All the stellar-mass black hole candidates mentioned in the text are members of very short-period binary systems. Explain how this makes it possible to detect the presence of the black hole.

First, a fundamental comprehension of a binary system needs to be established. This is a star system comprising two stars rotating around their common center of mass. The most familiar and nearest example of such a system is the Alpha Centauri star system.

Black hole candidates are members of very short-period binary systems. This means that the two elements of the binary system - one being the black hole candidate - orbit their common center of mass within a short period. This hints at a substantial gravitational influence that the black hole candidate casts on its counterpart which could be a star.

The matter drawn from the companion star towards the black hole forms a structure referred to as an accretion disk. This is a flat, disk-like structure of matter (from the companion star or a neighboring gas cloud) that spirals into the black hole. Because of the intense gravity of the black hole, the matter in the accretion disk spirals inward, heating up as it goes and emitting a lot of X-rays. Hence, X-ray emission from these systems can be one of the significant ways to detect a stellar-mass black hole.

Short-period systems are favorable for observational purposes because more orbits can be observed in a shorter amount of time. Thus, the variation in the orbital motion caused by the presence of the black hole can be detected faster - in a short-period binary system - as opposed to a long-period one. The faster orbits provide more observational data, which in turn allow for more accuracy in validating the presence of a black hole. The gravitational influence on the companion star can be observed from the shifts in its spectral lines known as Doppler shifts. From these shifts, the mass of the black hole candidate can be inferred, and if the mass is found to exceed the Tolman-Oppenheimer-Volkoff limit (the maximum mass a star can have before it collapses into a black hole), then one can affirm the presence of a black hole.