Alright, this is my first post, and I don't know how to work with spoiler tags or proper quoting, but I have a pretty good theory to propose! There will be spoilers for WoK (mild) and Mistborn: Secret History (strong spoiler), so prepare, or leave, as you see fit.
To the Theory! First relevant quote:
Now, Brandon said this wasn't necessarily important, but I think it is. Sanderson rarely includes this much detail for something with no importance, and the fact that it was RAFO'ed means there's something more there.
[spoilers!]
Reading Mistborn: Secret History, we see another sphere that reminded me of the sphere in the pouch the Radiant had: the sphere the Ire has, that can capture the power of a weakened Shard and contain it.
This made me think of another quote, about the black sphere Gavilar has in the beginning of WoK:
(Emboldening my own.)
"Something like that."
I don't think think Odium invested in the sphere, but rather part of his power was taken from him and placed in the sphere. I thought of this largely when I saw this question:
Odium is referred to as the "Broken One", in the epigraphs. How is he broken? I propose he was broken by the Radiants stealing a portion of his power using the sphere--possibly given by the Ire?--shown in Dalinar's vision. With the power of Odium in it, it turned black, and became the sphere Gavilar had.
This would explain why Gavilar needs to keep the Sphere out of Parashendi hands. If they give the sphere to Odium, breaking it as Kaladin did in Secret History, and giving the power, Odium might become un-"Broken". This might even show why there hasn't been a desolation in so long, as Odium is Broken and doesn't have enough power.
It seems difficult to prove or disprove this with the amount of information we have now, but it feels right, and it seems like a Sanderson thing to have!
Also, thanks for enduring my first post.