The simple answer is not to be concerned with proof of adultery. As New Jersey is a no-fault divorce state, a divorce filed on the grounds of "irreconcilable differences" or separation, if you have been separated for the required period, will have the same effect in a court of law as a claim based on adultery. The division of assets, or how a court will treat your wife, will not differ if the cause of action is based on adultery; moreover, since the boyfriend appears to have entered the picture after your separation, any possible effect of the "adultery" would be further lessened.
Filing for divorce based on the other grounds, or even based on extreme cruelty, will be much easier to prove. Instead of attempting to prove a cause of action for adultery, it is better to spend your time and resources on achieving proper separation of marital assets and arrangements surrounding parenting time and custody of your children.
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