You could, I've done it before. Options? You walk away from the kid that has known you as dad apparently all his life. Talk to an attorney in Cleveland that does custody work for more answers.
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The universal concept at work here is that if you were slipped in a child by someone else, you shoulder the financial responsibility on an ongoing basis anyway. This is the "psychological father" principle that binds you to the child and gives rise to ongoing child support requirements.
From an emotional point of view, that is the proper result. I have my doubts that finances are linked to emotion. The state gets matching funds for support enforcement so you can expect an order of support no matter what the facts are. This means you need to know that your child is actually your child from the very beginning, not ten years out. However, this does not mean you are precluded from paying for a DNA test to be sure. Then the question becomes whether a child support court will terminate your support obligation if you have one. I doubt a typical court will.