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The Earth and the dark star that is the second foci of the 12th Planet's orbit do not rotate around each other any more than the planets in your Solar System rotate around each other. The reason for the latter is that the Sun dominates the planets, and their influence on each other becomes the lesser voice.
In like manner your Sun and this dark star, of a comparable size, are caught in a larger net and are essentially motionless within your Galaxy. This net exists for all the stars in your Galaxy, as elsewhere, and is the reason the stars in the sky do not lose their position and float toward each other. It is not that they are so far apart that they do not influence each other. Influence, however slight, is always there. It is rather that influences have been balanced to where an equilibrium is reached. To you, who see that distance is maintained, it looks like the lack of influence. It is balanced influence. Were you to have seen your galaxy born, clumping into masses with these masses first attracted and then to some degree repulsed by each other, motion initiated as a result of these opposing forces, you would intuitively understand that large bodies that cease motion do so not because there is no influence upon them and not because they were not at one time in motion, but because they came to a situation where they essentially are in a dither. The influences upon them are balanced.
This second foci of the 12th Planet has not been located by your astronomers because it is dark, not lit, and does not happen to block any view your astronomers are particularly interested in. They think it empty space. Unlike the Sun, this dark twin never lit. Although comparable in size and mass, its composition was subtly different, and it has no potential for becoming a lit sun under the present conditions in your part of the Universe. It has no planets of any size to mention, though is orbited by a lot of trash. Should one wish to search for it, it stands at an angle of 11 degrees off the Earth's orbital plane around the Sun, in the same direction we have given for the approach of the 12th Planet. Not being a luminous body, and not giving off any radiation detectable by human devices, you will be unable to locate it, but this does not mean that it is not there. Do you, like a child with his hands over his eyes, think that if you cannot see something that it does not exist?