Okay so I went back and checked all this stuff as well.
Lostroom mats had this:
The blanking was conducted on December 31, 2017. The invasion of Fantasy Trees began from space, and seven Fantasy Trees fell on the surface of the earth, creating seven Lostbelts. The land of true history disappeared, and the Lostbelts were protected by storm walls. Then, after some time lag, the surface of the earth was blanked all at once. The blanking is considered to have spread radially from each Lostbelt. The branches of the Fantasy Trees that grew from the surface of the earth became a membrane that covered the Earth, creating a huge curtain at the boundary between the troposphere and the stratosphere, the tropopause.
Which implies that the tree branch network is the "dome" that covers the sky, however, in general we should assume the lostroom mats are wrong in any case of conflicting info because they came out very early and don't have any "reveal info", and are written with "pre-story knowledge".
Bluebook says this:
Truth be told, we had time to prepare.Maybe it was only twelve hours,
but it was still time we could have used.
On that New Year's Eve, we lost every satellite around the world in an instant.
In consequence, we were not able to observe a single cosmic ray even as the whole world was reduced to a blank, dust gray sphere.
I know now that it was a membrane in the sky...a dome of sorts made by the tree's branches that was placed over the entire world...
...but we weren't exactly in a position to sit down and take stock of what happened on that day, not with all the crazy things going on.
Twelve hours later...Armageddon.
A total slaughter.
A massacre on an unimaginable scale.
Uncountable...tree-things descended on the earth,
like so many tentacles from space.
They crawled across the world, finding and exterminating all forms of life with unerring accuracy and unnerving tenacity, killing almost every human on the planet over the course of three months.
After that, the trees just...disappeared, returning to the sky and leaving behind a blank, desolate world.
Which again makes it sound like it's the tree branch network creating the dome, but at the same time the trees (in his world, cuz if you don't buy into his world being an alternate one at this point you're lost, man) descended after the "dome".
However, we know from Ophelia's data that the trees came after the Bleaching:
---Fantasy Trees and the Bleaching.With the exception of us, the Crypters, and the Alter Egos, the only means the Alien God has of interacting with the planet are the Fantasy Trees.
That is why we come to this conclusion.
The bleaching of the earth was due to the descend of the Fantasy Trees.
But can we be sure that is the case...? I once again observed and analyzed the data collected from the surface of the world.
As it turns out, we may have misinterpreted the facts.
---It was after the bleaching of the world that the trees appeared. This order of appearances is very important.
I don't know who the Crypter who will view this file is, but I hope it is useful.
and we know the Bleaching happened after we dove into INS, meanwhile the blocking of cosmic rays was caught way earlier during the prologue:
Announcement:
Warning! Warning!
An ■■ has occurred in observational data.
No past observational data found.
Statistical response, forecast, and prediction abilities compromised.
Unable to detect anomalies in observational data.
Unable to detect any form of electromagnetic waves.
Unable to detect cosmic rays reaching Earth.
Satellite visual feed lost.
Mauna Kea Observatories transmissions lost.
There are currently no observable celestial objects on Earth.
So the blocking of cosmic signals came first, way before anything else, and we know it's caused by the dome, and the trees came after the bleaching which happened after the cosmic blocking, ergo the dome was there before the trees and could not be made by the trees. Whether Bluebook is just wrong (we still don't know everything about his version, to be fair. For instance, he says the trees left, but he notices changes in the sky above the areas where Lostbelts are in our worlds, but how can there be Lostbelts if the trees disappeared?), or it's just different, who knows.
Finally, there's this from Atlantis, which is explicit enough that I could've just posted this, but that's less fun:
Pepe:
So I take it that means all the branches connected to that canopy covering the sky are from the Atlantic's Tree then?
Kirsch:
Good grief. I really can't get anything past you, can I, Pepe?
That's correct. My Fantasy Tree is the only one receiving the blessings of the Foreign God's Earth-covering canopy.
Which, like I said, is pretty explicit about the tree connecting to the canopy/dome, ergo the dome and the tree are not the same thing.
So all in all, you basically have two "branch networks". One is the one formed by the big branches of the trees, which acts as an information network, and is all but explicitly said (iirc) to exist for the sake of feeding leftover energy to the Atlantic tree. The other is one formed pre-trees, which is blocking out all cosmic signals (but the sun is still ever-shining, however that works), and gives a "blessing" to the trees that connect to it (which is just the Atlantic tree, again).