You could do a simple sketch proof I suppose by noting that almost all random infinite grids will contain (your favourite methuselah with lifespan x) somewhere, so the whole thing must have lifespan at least x. Since x was arbitrary, the total lifespan is larger than any finite number.

You make a good point about the median vs. mean thing – I’ll definitely create median data if/when I post the results of the toroidal version of this script.

And I completely meant to attach the source code to this post (as I try to do with all my coding posts), but I forgot and it unfortunately is sitting on a harddrive halfway around the world right now. I’ll update the post to include the source code in two weeks time

Also, just as an FYI in case you’re curious – I tried something semi-similar to what you described about trying to create methuselahs based on observing past long-lived patterns and evolving the density used. Instead I tried things like taking the top half of one methuselah (possibly modified by one or two cells) and the bottom half of another methuselah (also possibly mutated a bit) and checked whether that was long-lived. It was basically a standard cross-breed and mutate genetic algorithm, but unfortunately it yielded results that were no better than a uniformly random search, due to the high volatility of methuselahs. Silly Life.