In Conway's game of life, take the initial position to be two infinite diagonal lines of live cells, with a single cell in common. Does this thing converge to a stable configuration? I.e., is the state of each cell (or finite region) eventually periodic?
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What I get from an X of size $11121\times11121$ at just around the point where information travels to the tips. Even from Xs ten times as long, there is Methuselahlike ebbing and flowing of debris near the center amid a pool of still lifes and blinkers still thousands of generations on. Just going from experience working on the Busy Beaver of 5, I would imagine this question might be enormously difficult to settle, owing to the globally fractal and locally random nature of the picture. 

