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The proof your boyfriend needs is proof that shows he not only supported them but that they lived with him. That is what he claimed when he entered them as dependents on his tax return.
The reason so much is now owed is most likely Earned Income Credit was claimed. He may have only received $700 in actual refund but if you look at the return there is EIC and that either was used to reduce his tax or applied to some other amount he owed.
If the children did not physically live with this person then there is nothing that can be shown that is going to assist him.
The best he can do is request to pay the amount back over time by an Installment Agreement.