Cohen-Kong Theorem

It is easy to check that the Theory of Everything for Langton's ant is time-reversible—that is, the current pattern and heading uniquely determine the past as well as the future. Any bounded trajectory must eventually repeat the same pattern, position and heading, and by reversibility such a trajectory must be periodic, repeating the same motions indefinitely. Thus, every cell that is visited must be visited infinitely often. The ant's motion is alternately horizontal and vertical, because its direction changes by 90 degrees at each step. Call a cell an H cell if it is entered horizontally and a V cell if it is entered vertically. The H and V cells tile the grid like the black and white squares of a checkerboard.

Select a square M that is visited by the ant and is as far up and to the right as possible, in the sense that the cells immediately above and to the right of it have never been visited. Suppose this is an H cell. Then M must have been entered from the left and exited downward and hence must have been white. But M now turns black, so that on the next visit the ant exits upward, thereby visiting a square that has never been visited. A similar problem arises if M is a V cell. This contradiction proves that no bounded trajectory exists.