If so, would this hurt my chances at Cornell?
Yes and no it would not.
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Sure. EA to MIT or Caltech is completely unrestricted (i.e. no restriction on what other schools you can apply to, and no commitment to enroll if admitted). However, if you’re admitted to Cornell ED, and one (or both) of the other schools EA, you may regret your ED decision since it’s binding (unless, of course, Cornell is your first choice).
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