Is this divorce certificate issued by an Islamic judge valid?

Is this divorce certificate issued by an Islamic judge valid? A husband in a heated argument with his wife very angrily on her demands writes a divorce note for the wife. After they calm down, they regretted what had happened and later they both visit an Islamic educator who reconciles them on the basis of the Hadith in Ibn Majah. The exalted prophet is reported to have said, “la talaqa fil Ighlaaq”. There in absolutely no divorce under any kind of duress, coercion, suddenness, anger etc. Both the husband and wife reconciled and willingly agreed to go back to their marriage and lived together as married couples.

After a few moths they had another quarrel. Can the wife now use that same old divorce note that was written months back and take it to an Islamic judge and seek a divorce certificate? Is that divorce note still valid even after they have been reconciled and consummated the marriage for months?

And if the judge issues this woman a divorce certificate based on that note without consulting or communicating in any way with the husband and stating in the divorce certificate that the husband has divorced his wife. Is this divorce certificate issued by the judge valid?

Thanking you again in advance for taking your time to read and respond.