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I decided to write a why Cornell essay as I was unable to fit the reasons why I wanted to attend the school in the 500 character box they provide on the application. My question is whether I should simply put "wrote why Cornell essay" in the box they provide or if I should instead briefly state the characteristics I discuss in my actual essay in the box.

Yes, that's fine. Someone posted on reddit that they asked this very same question during a Cornell admissions webinar. Simply putting "please see attachment labeled 'Why X'" is perfectly acceptable.