What is the maximum "alive density" of cells in Conway's Game of Life when played on a torus? 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)

In the specific case of a pattern that is fixed under the rules of the Game of Life, the maximum density is 1/2, as shown by Noam Elkies in The still-Life density problem and its generalizations.
Elkies also presents (on page 22) a simple example of a period 6 oscillator with maximum density 3/4. Here are all its phases:
\begin{array}{|c|c|c|c|c|c|c|c|} 
\hline 
1&1&0&0&0&0&0&0\\ 
\hline 
1&1&0&0&0&0&0&0\\
\hline 
\end{array}
\begin{array}{|c|c|c|c|c|c|c|c|} 
\hline 
0&0&1&0&0&0&0&1\\ 
\hline 
0&0&1&0&0&0&0&1\\
\hline 
\end{array}
\begin{array}{|c|c|c|c|c|c|c|c|} 
\hline 
1&1&1&1&0&0&1&1\\ 
\hline 
1&1&1&1&0&0&1&1\\
\hline 
\end{array}
\begin{array}{|c|c|c|c|c|c|c|c|} 
\hline 
0&0&0&0&1&1&0&0\\ 
\hline 
0&0&0&0&1&1&0&0\\
\hline 
\end{array}
\begin{array}{|c|c|c|c|c|c|c|c|} 
\hline 
0&0&0&1&0&0&1&0\\ 
\hline 
0&0&0&1&0&0&1&0\\
\hline 
\end{array}
\begin{array}{|c|c|c|c|c|c|c|c|} 
\hline 
0&0&1&1&1&1&1&1\\ 
\hline 
0&0&1&1&1&1&1&1\\
\hline 
\end{array}
