Here's my feelings on some of the submissions thus far.
The Heart of Stall: Now, everyone's said they love this idea so far, but I want to throw in a different view. I still like this idea a but, but it might be best seen as an approach to defining how a defensive/support Pokemon can be made so that Stall can use it better than Offense. Or, what is it that makes a Pokemon fit better in Stall, Offense, or Balanced teams? What does each need its members to bring to the table, and what does each not want? That said, this one might be difficult to make, since support/defensive Pokemon can have a variety of uses. But some CAPs that fail at their concept give us greater insight into their purpose than a success might've.
Climate Control: This could be neat, but I'm not sure what we'd learn from it (or what, rather, we're setting out to learn). The weather wars are so important to win for a weather team that I just feel like this would be seen on almost any weather team. What are we trying to learn about the metagame as a whole from this?
The Big Dipper: Neat, but very, very general. Learning about what defines a core is a fantastic goal, though. It might be best to pick a core (or set of similar cores), or a type of core, like "Offensive" or "FWG". Otherwise I feel this could either lose focus quickly, or because a standard "annoyer" support Pokemon.
Number Cruncher: Math is behind everything in this game, so this one's really admirable. While the most obvious numbers are crunched at the end of CAP design, we have mathematical analyses about what is a "good" type or ability or immunity and such. I think the only issue is that without custom stuff, I'm not sure how eliminating/controlling some number-altering parts is possible. But even if we couldn't figure that out, it'd at least be a noble failure.
Chill Pill: Sort of a more focused version of Heart of Stall that I think would work a little better. I think we could look back at slower metagames (ADV or, God forbid, GSC) and figure out what made them that way, and how one CAP might be able to try to bring that back. That said, I'm not sure if this could single-handedly meet this objective, and it also feels like it's creeping a little close to CAP 1's goal of understanding and controlling momentum.
Compact Movepool: I have almost nothing to adjust here. It's a great idea in which we try to learn about what simply defines movepools as we know it. What causes "four moveslot symdrome?" What are the truly amazing moves that make for a great movepool on a Pokemon? How can typing, ability and stats affect the viability of a movepool. Plus, as stated in the post, it does a nice contract to CAP 4.
More later if I have time.