How did the messenger judge?

One common argument made for justification for the upholding of Hadith besides the Quran is that it was the messenger’s job to judge, and it is the duty of the believers to obey his judgment. Some of the strongest verses in the Quran used to attempt to justify these points can be found in the following two verses.

4:65 Never indeed, by your Lord; they are not believers unless they come to you to judge in their disputes, then find no hesitation in their hearts whatsoever in accepting your judgment. They must submit a total submission.

فَلَا وَرَبِّكَ لَا يُؤْمِنُونَ حَتَّىٰ يُحَكِّمُوكَ فِيمَا شَجَرَ بَيْنَهُمْ ثُمَّ لَا يَجِدُوا۟ فِىٓ أَنفُسِهِمْ حَرَجًا مِّمَّا قَضَيْتَ وَيُسَلِّمُوا۟ تَسْلِيمًا

33:36 No believing man or believing woman, if GOD and His messenger issue any command, has any choice regarding that command. Anyone who disobeys GOD and His messenger has gone far astray.

وَمَا كَانَ لِمُؤْمِنٍ وَلَا مُؤْمِنَةٍ إِذَا قَضَى ٱللَّهُ وَرَسُولُهُۥٓ أَمْرًا أَن يَكُونَ لَهُمُ ٱلْخِيَرَةُ مِنْ أَمْرِهِمْ وَمَن يَعْصِ ٱللَّهَ وَرَسُولَهُۥ فَقَدْ ضَلَّ ضَلَـٰلًا مُّبِينًا

The error made in thinking these verses support the upholding of Hadith besides the Quran can be understood by asking, “By what mechanism did the messenger judge?” Or put another way, did the messenger judge by the Book of God, or did he judge by some other means that is not contained in the Book of God?

the function of a Judge