You call your guild “The Arcane Brotherhood.” Are you only considering male members?
Yeah, we're really sexist that way... No, we're more than happy to accept female members, it's just that calling it the Sisterhood would have been weird, and that leaves us with the Siblinghood which is just pathetic.

Do not expect that since when you were born an old crone prophesied that you would be a great wielder of the arcane arts you will be an Archwizard within weeks.
This sentence is too long for my taste.
We are wizards... Things tend to get complicated. I was actually careful to not simplify any of the IC information too much for that reason.
Government:
What exactly is your plan here? Is this supposed to be a cumulative government system or a more separated checks and balances system? It's not too clear now.
It's a little more checks and balances-oriented, but has both.
Hierarchy:
Reminds me a lot of the Imperial Scholars. The Imperial Scholars are awesome of course, and you led them once, so that isn't necessarily bad.
Let me assure you that nothing here is even remotely taken from the Scholars. When I was the Archsage of the Empire, the Scholars had yet to form into anything significant, and I haven't really seen them since then. Anyway, even if there are similarities, I expect a lot of things such as ranks to grow and end up having more complexity in time. Like I said, this is just a foundation.
Claim to be the only Wizard Guild:
The Imperial Scholars are wizards too, but they have departments or "Paths" of knowledge. Your guild would indeed be the only current wizards-only guild. However, what is then meant with this statement:
We are united by our desire to pursue knowledge in all its forms, and most notably in the area of the arcane.
So, are you going for wizards-only, or are you adopting the model so graciously provided by the Empire?
Again, nothing here is taken from the Empire or any of it's guilds. I've been around a good deal longer than the current system of the Empire, so even if some of our ideas are similar, you can be sure I thought of them first.

The statement that we are the only wizard's guild is, in my opinion, true. The Scholars are part of the Empire. Dispute it, argue it all you want, but the fact remains that the Scholars exist because the Empire exists. No Empire, certainly no more Scholars. The Scholars can't be separated from the Empire, so in a sense they are more of a sub-guild. I know Sangwa may prefer to word things differently, but unless a great deal has changed in the foundational structure of the Empire since I was a member, that's how things are.
For the rest, the Scholars are just that, scholars. They allow for all forms of study. Sure there are people who study magic there, but the guild(or sub-guild) is not really focused on magic, but in general learning. You might just as well say the Knowledge Seekers are a wizard's guild. It takes far, far more for a guild to be a "wizard's guild" than for it to just have a couple wizards in it.
The Brotherhood focuses strongly on magic, but also encourages other study for a healthy and balanced accumulation of knowledge, and to prevent a skewed outlook on life. Phinehas, for instance, studies magic like it's going out of style, but he has also done a great deal of work in chemistry/alchemy, and dabbled in many other areas of science. Thanks to this he has a fairly balanced perspective on things, though you wouldn't know it to talk to him. Yes, the Brotherhood would accept a member who wishes to study, but does not wish to wield magic, because the pursuit of knowledge in all its forms is the natural passion of a True Wizard.
I know some of that may seem like semantics to some people, but there's not much I can do about that. The reasons that we are the only wizards guild are, to my mind, irrefutable. If it's any consolation, I've even talked to some of the Scholars members, and they did not challenge the claim.
For the rest, it is a very nice guild and we may surely expect great things from it and from it's leader.
Appreciated.