I know this can be upsetting, but from what you posted it is not anything close to clear that the judge based her ruling at all on the things the mother lied about. And, because the best interests of the child is the standard that is applied, the judge could make the decision to make a custody ruling even if the mother never said a single word at the hearing.
Because it will not sit well with you until you hear more, I am sure, I observe that you might have a local appellate attorney consult with you to see if you get the same kind of response once that attorney is hired by you and he or she reviews the record that would make up the record on appeal. And....good luck to you.
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