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If you don't mind my asking: why Yale? I also wanted to do cs at Yale for a good while, but in hindsight, I realize that I would have been miserable haha it would be far from fulfilling. Same with Caltech. The size sounds homey, but I'm realizing more and more that bigger tends to be better.
Don't underestimate the prospect of a particularly endowed cs department (Stanford, mit of course, but also schools like CMU, uiuc, and Berkeley that aren't floating near the #1 position overall, but their cs departments are. Big schools especially have lots and lots of resources to offer, and being in the "best" major usually leads to you and your peers being coddled like a king, particularly because YOU are where the funding for the school is coming from!)
Also, be careful about applying to math with the intention of transferring :o it can sometimes be harder than just getting accepted, especially for the top programs. I can't speak for the others, but my school in particular is somewhere in the single digits in terms of cs acceptance (like the other schools), but transferring to CS from a different major has a significantly lower chance (low single digits) of working out because of how quickly the department is still growing. There is also a CS+X(CS + math, astronomylinguistics, etc.) cs program which is in the college of las instead of engineering that (meaning that las manages admissions for it. The cs department still handles the transfers, though) is easier to get in to, and many students will get deferred from eng cs into the las version. They are pretty much the exact same, though, with less physics/chem and more math and x classes. Once that fills up (pretty new), acceptance rates will be another story :oo
In any case, best of luck! Haha Sorry for the rambling