(1) You have to be careful in specifying exactly what ‘cycle’ means here. Perhaps you mean something like: “The configuration can be partitioned into non-interacting subsets each of which is either a ‘still life’, an ‘oscillator’, or a ‘spaceship’.”
(2) If we’re talking about an infinite game of life board, randomly populated, then it won’t generally be true that ‘sooner or later we have a cycle’ even in the above sense.
(3) Even if there are only finitely many cells switched on, if the board is infinite then the configuration may never begin ‘cycling’. Proof sketch: One can build a Universal Turing Machine in the game of life. Therefore, one can build a dovetailer which executes all possible computations in parallel. Therefore, if this was destined to ‘cycle’ after a finite time then we could solve the halting problem. ETA: Simpler proof: A glider gun!
(4) So your result about cycling really only applies to a finite game of life board.
(1) You have to be careful in specifying exactly what ‘cycle’ means here. Perhaps you mean something like: “The configuration can be partitioned into non-interacting subsets each of which is either a ‘still life’, an ‘oscillator’, or a ‘spaceship’.”
(2) If we’re talking about an infinite game of life board, randomly populated, then it won’t generally be true that ‘sooner or later we have a cycle’ even in the above sense.
(3) Even if there are only finitely many cells switched on, if the board is infinite then the configuration may never begin ‘cycling’. Proof sketch: One can build a Universal Turing Machine in the game of life. Therefore, one can build a dovetailer which executes all possible computations in parallel. Therefore, if this was destined to ‘cycle’ after a finite time then we could solve the halting problem. ETA: Simpler proof: A glider gun!
(4) So your result about cycling really only applies to a finite game of life board.