This result is really beautiful! At the same time, it's like GoL has been conquered, and in that it leaves me a little sad. But just a little bit. Congratulations!
* Is there an oscillator of every possible period? (We have them all except 19 and 41.)
* If you start off the entire plane in a random starting state, does its density tend to a limit as time goes to infinity?
* Is there a 'phoenix' oscillator, in which every live cell dies every generation, of period greater than 2?
* Can every pattern be destroyed by bombarding it with gliders?
* Is there an indestructible pattern?
At the moment people are working on building a spaceship which is only 1 cell tall in its starting state: https://conwaylife.com/forums/viewtopic.php?f=2&t=2040.
(... and encounter other AIs that act as hegemonising swarms, eg. that populate the plane with copies of themselves)
What's been done with GoL is fascinating, but I think the next emergent layer of fascination for me is universal constructors which can handle a certain level of constant distributed noise or interference. A lot of times people just shrug and say, "well this pattern will always be critical/vulnerable in these locations, and cannot be made robust. But to me that opens up the door for entire classes of patterns which measure, embed and repair state of surrounding entities.
Conway's Life design work is kind of like building robots out of masses of subcritical uranium. Everything's fine until two robots unexpectedly bump into each other... which means you have to start out with everything very carefully balanced, such that that never happens.
So I guess one fairly obvious insight is that real-world physics supports more reliable and less explosive low-level structures than Conway's Life does, and those low-level structures can then safely be used as the basis for new levels of organization -- atoms -> molecules -> DNA -> bacteria -> eukaryotic cells -> multicellular organisms -> colonies of organisms -> ecosystems.
It's not clear how those higher levels of organization would work in Conway's Life. If they're possible, then they seem to be far beyond our current ability to simulate them -- though there's some recent research vaguely along these lines, about self-replicators that might be able to exert some control over the space around them: