There are "garden of eden" patterns which can only exist as an initial configuration: there is no pattern which evolves into a garden of eden, hence there can be no constructor capable of building them https://en.wikipedia.org/wiki/Garden_of_Eden_(cellular_autom...
I wonder if you could bypass this with a level of indirection.
If the GoL wizards can construct an expanding and initially configurable 'GoL in GoL' grid [1], they could configure the 'garden of eden' as the initial configuration for their 'GoL in GoL' simulation.
It's not the same thing, of course, but it would allow you to run Garden of Eden patterns still starting with just your initial 15 gliders.
To construct a garden of eden pattern, you must first construct the universe :)
Sure, we can build them with "meta cells"; but that's just emulation (the same way a SNES emulator doesn't make my laptop a SNES).
We already have GoL turing machines (with "tape factories" that travel faster than the read/write head); we could likewise use those to emulate GoL with arbitrary input, just by setting up an appropriate "tape". The result is far less pretty though ;)
If we limit ourselves to glider interactions, the article links to the following patterns which cannot be constructed (including garden of eden patterns): https://conwaylife.com/wiki/Category:Patterns_that_can_not_b...