Is there any absolute proof that the Sun is not a member of a binary star system?

There are no known binary stars that have separations greater than the average separation between stars in this part of the Milky Way ...about 5 light years. Because we can detect faint nearby stars down to the limit of what we define a star to be, we know that there are no stars nearer than Alpha/Proxima Centauri to the Sun, with masses high enough for them to have had self-sustaining thermonuclear reactions at masses above 0.04 times the mass of the Sun. Sensitive proper motion and parallax surveys have yet to detect any luminous object orbiting the Sun...however...we cannot rule out a distant object a few times larger than Jupiter. However, that would not make the Sun a part of a binary system, only a star having a very distant 'planetary' companion. There have been searches for dead stars such as black holes or neutron stars, but no one has ever conclusively detected any x-ray or radio emission from such bodies.