Ok so!
I just got off the phone, and here's what we talked about.
Theory 1: The definition of Shattered is "(of something abstract) damaged or destroyed." Adonalsium is most definitely an abstract being. Something damaged/destroyed can be fixed, to a certain degree. The main point of this theory is that if someone could combine the 16 Shards, and the 4 Dawnshards, that massive coalescing of power would combine into a sentient being. A New Adonalsium.
Theory 2 (Disproven): Only exact opposite shards can Splinter each other. Disproven because Odium Splintered Honor, Devotion, and Dominion. (Sorry if I missed any)
Theory 3: As shown in the Dawnshard novella, Adonalsium is portrayed in the mosaic as a sun split into 4 chunks (Dawnshards) and those chunks were also split 4 ways (Shards). As such, all Shards are just manifestations of either one Dawnshard, or a combo of 2.
Theory 4: Wit (Hoid, Cephandrius) is gathering parts of different Shard's magic systems to combine it all into New Adonalsium, as discussed in theory 3.
Theory 5: Opposite shards (Preservation, Ruin) are needed to actually create life. The Singers were already there when Honor and Cultivation arrived, as were the humans on Ashyn when Odium got there. But Preservation and Ruin worked together to make Scadrial and its life forms.
Theory 6: As discussed in the Dawnshard novella, Rysn feels from the mosaic different feelings. I forget what they were, and I don't know where to go to find the quote. If someone could do that, and post is as a reply, that'd be great. Back to the point, these feelings are interpreted as what Adonalsium felt during the shattering. And, as Adonalsium died, we theorized that parts of his being also became ingrained on the Commands used to kill him. The Dawnshards themselves.
Thoughts on it all?
Seriously, give me some thoughts. Please. I had way too much fun talking about this, I need more interaction based on these theories.