What “proper authorities” are those, if someone finds a $5 note on the street, as has been happening for decades? In 99% of the cases, the finder just keeps the money without making any attempt to find the owner(which will be near impossible to do anyways in a place like Time Square). How do you even determine that that particular $5 note belonged to some particular person? All $5 notes are virtually the same.
Back on point, exactly when did it become “stealing”, when you find a phone that someone has left behind, try repeatedly to return the phone to the alleged owner of the phone(as determined by data on the phone), only for the owner to refuse to accept it, and deny they ever lost any phone?
Referring to them as "notes" clearly marks you as a foreigner. Whatever they do "back home", that's not how it's done here ~