The fifth book of the Brandon Sanderson's Stormlight Archive, Wind and Truth, comes out later this year, so I figured I'd share some of my theories before the book comes out and proves them wrong (or right).
This will be a pretty deep cut into the series, and I doubt it will make much sense if you haven't read the first three books, and I will be spoiling all four of the current books. But let's begin.
Ba-Ado-Mishram is one of the Unmade, powerful spren (the spirits representing ideas) corrupted by the god Odium. Or perhaps Splinters of Odium, representing passions other than the Hatred that he's so tightly tied to. When the Fused and the Voidspren were banished from the world by Taln's sacrifice, she gave forms of power to the singers, giving them the ability to wage war even with the Oathpact in place. Then, shortly after the Radiants imprisoned her in a gem and reduced the singers to nearly mindless parshmen, there came the Recreance. The Radiants broke their oaths and abandoned their spren, destroying the Radiant Knights.
Why? Why did the Radiants break their oath? Oathbringer suggested that it happened when the Radiants discovered what binding Ba-Ado-Mishram to the gem did to the singers, at the same time they discovered that the singers were the original inhabitants of the world, and the humans were invaders, who had fled their homeworld when they had destroyed it with Surgebinding. But Rhythm of War suggests something more, that both Randiants and spren decided to break their oaths in order to prevent something worse. What? It's suggested that in binding Ba-Ado-Mishram they somehow changed the world, but how?
My theory is that Ba-Ado-Mishram had somehow insinuated herself into the Nahel bond, Connecting to the spren through it in a similar way that she had Connected to the singers. Maybe it was something she did herself, or maybe it was a side effect of the way the Radiants had imprisoned her in the gem. The Radiants feared that she might be able to use that Connection to corrupt the Radiant spren, and maybe the Radiants themselves. If that was the case, the obvious solution was to break the bond itself, before Ba-Ado-Mishram, even imprisoned, could make use of that Connection, perhaps using it to corrupt the spren or to escape, or both.
However, something they didn't anticipate is that this would turn the spren into deadeyes. Why? Likely the same reason imprisoning her turned the singers into parshmen. It took away their Connection and their Identities (OB, Chapter 17 and RoW, Chapter 73). No longer Connected to their Radiants, the Nahel bond now only connected to Ba-Ado-Mishram. It was like getting a constant busy signal in the part of their souls that allowed them to Connect with anyone--human or spren, or even themselves.
But what would happen if Ba-Ado-Mishram were freed? The Radiant spren would regain their minds, but they would be Connected to Ba-Ado-Mishram. It sems likely they would become like Sja'anat's Corrupted Radiant spren, only they would be Connected to someone far more violent and hostile than Sja'anat. And Ba-Ado-Mishram would be connected to the singers again, giving them another source to draw from in order to gain forms of power. I'm not sure they need her now that there's the Everstorm, but it could possibly make the singers more dangerous. Or it could lead to divisions among them.
I like this theory for its explanatory power, as it explains the actions of a lot of the actors.
- The Radiants taking part in the Recreance. If the Nahel bond itself had been corrupted by Ba-Ado-Mishram, breaking the bond is not just logical, it is necessary. Windrunners, for example, must protect people from themselves, and that oath can only be fulfilled through breaking it.
- It explains why Nale believed that the formation of the other Radiant orders would cause another Desolation. If the Nahel bond is corrupted, the existence of more Radiants increases the chance of corruption and the likelihood that Ba-Ado-Mishram would be freed. It doesn't necessarily follow that more orders would increase the odds, as opposed to just more Radiants, but perhaps there's reason to believe a wider variety of Nahel bonded spren would be more likely to allow her to escape. Or perhaps that's just an expression of Nale's madness.
- It explains why Kelek started an organization whose goal was to start a Desolation in order to bring back the Radiants. That's not really what Kelek himself wanted. His true goal was to free Ba-Ado-Mishram, as he believed that was the only way to restore the deadeyes--which could very well lead to the refounding of the Radiant Orders, though perhaps not if the spren were Corrupted. However, doing so would also restore forms of powers to the singers, and start another Desolation, so the divergence between the Sons of Honor and Kelek has more to do with the means than the end result.
- It explains how Sja'anat is corrupting Radiant spren. These are almost certainly former deadeyes, as Rlain's future spren thanks her for giving him his eyes (RoW, Interlude I-2). Sja'anat figured out how to take her fellow Unmade's place in the deadeyes' Connection.
- It explains how Adolin was able to Connect to Maya, whereas a Radiant could not. When a Radiant touches a Shardblade, they hear screaming. Is that because the Blade is screaming all the time, but the wielders can't hear it? Or is the Radiant touching the Nahel bond of the spren, the bond that is Connected to a dead end, that causes the spren to be in that state? Are they screaming at the humans, or at Ba-Ado-Mishram, trying desperately to form a Connection with the only one who could be at the other end? However, humans can still bond with Shardblades. Consider it a separate channel, perhaps a backdoor into the system. It's backwards to the Nahel bond, breathing life into the spren through the human heartbeat. And Adolin and Maya found a way to deepen the bond, again going from human to spren, to make it a real Connection, and thus provide the ability to make Connections again.