I've read that Conway's Game of Life (CGOL) can have unbounded growth from a finite initial number of alive cells (e.g. a glider gun). However, if CGOL is played on a torus, space (the number of cells) becomes finite, and glider guns are guaranteed to eventually destructively interfere with themselves.
Because of this, I wondered about the maximum proportion of alive cells on a torus. To be specific,
What is the maximum stable density of alive cells in CGOL on a torus, where density means proportion of alive cells out of all cells, and stable means this density occurs infinitely many times (as opposed to the recurrence of position: a glider might have nonperiodic position, but stable density)
Please let me know if my question is ambiguous, poorly phrased, or self-contradictory - I struggle to be precise with my questions.