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The judgement creditor does not "report" payment of the judgment to the CRAs.
Judgments are public record information, and find their way into consumer credit reports by way of searches of public record information done or contracted by the CRAs.
That includes updates, such as recording that the judgment is satisfied.
Judgment creditors are requried under the rules of civil procedure and/or state statutes to notify the judgment court within a certain number of days after the judgment has been satisfied. The court does not report that information to the CRAs, they update the posted public record, and it is up to the CRAs to review public record information and become aware of the satisfaction. Thus, you can send notice/dispute to the CRA after the debt is paid that the public record has been updated, and require that they recheck the public record.
As with other types of derogs, payment of the debt does not itself negate the occurence of the derog, and thus does not remove its affect on scoring.
You will either have to wait for the judgment to become excluded at 7 years after date of entry, or motion the court to have the judgment vacated from the public record.
Some states/courts permit vacature of judgments based on satisfaction of the judgment, but most require some error in the judgment in order to obtain vacature.
Check your state statute/rules of civil procedure for any provision for vacature based on satisfaction of the debt.
A judgement means the court has determined that it is valid, and its amount.
From that point, no one cares whether or not you continue to assert that you dont owe it.
Thus, paying the judgment is not a concession that you owe it. The court has already made that legal determination.
The judgment affects score the same while still shown in your credit report, whether paid or unpaid. It must be totally vacated to be removed from scoring, or else it must reach its credit report exclusion date under FCRA 605(a)(2).
Technically, under FCRA 605(a)(2), when a judgment remains unpaid, there is no credit report exclusion date, as the exclusion of a judgment is the LATER of 7 years from date of entry, or until the expiration of the period of enforceability of the judgment. Judgements are usually enforceable for an initial period of ten years, but can readily be extended by motion to the court, so theoretically, a judgment can remain until paid.
If you pay it, then the exclusion period reverts to 7 years from date of entry of the judgement, so yes, paying is beneficial to scoring once it is excluded.