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If your wife was legitimate overpaid and received money that she did not earn, she has a legal obligation to pay it back and her former employer can file a lawsuit against her for unjust enrichment and conversion to compel its return. In court, they would bear the burden of proving their case, which means proving by a "preponderance of the evidence" that your wife received money that does not belong to her. Since the employer would have to supply this evidence in court to get their money via civil judgment, they should have no problem providing that same information to your wife now, informally. In fact, if there is any doubt as to whether there is an overpayment, it would be perfectly reasonable for you to demand as you have that proof be supplied.
You could simply refuse to pay until you get adequate proof. Your wife would risk the possibility of being sued, but if you make it clear to her employer that you will pay once adequate proof is supplied, I have a hard time believing their choose litigation over simply providing you with the same evidence they's have to produce in court anyway. You may wish to try negotiating some sort of a repayment plan. Employers are generally open to that sort of thing when it comes to overpayments.
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