The statute of limitations in the state of NY for the collection of back taxes is 20 years from the date the taxes became due and payable. This is covered in NY statute 1092(3)
The date that the taxes became due and payable would be the due date of the return, or the date that the state made any additional assessments to your filed return, whichever is later.
As far as resolving the balance that is due, if they are not accepting the paperwork you have submitted to show that their estimates are wrong, your only choices would be to either pay off what they say you owe or continue trying to work with them to prove that the amount they assessed is incorrect. You might try making a trip in person to the local NY Department of Revenue office in your area to see if you would have any success in getting this resolved. Sometimes meeting face to face with them can have a more favorable outcome, or at least they should be able to show you why they still think their assessment is correct.
You can also try submitting an offer to the state to try and settle this debt for less than the actual amount you owe. They may take this option in to consideration if they feel it is their only option at ever collecting any of the taxes that are due.
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