Thesis The Republic never answers Glaucon’s challenge; it changes the question
Its coordinates: 6.4.1 Distributive Justice
Where it stands
In Book II of the Republic, Glaucon tells the story of the shepherd who finds a ring that makes him invisible, and uses it to seduce the queen, kill the king and take the throne. Then he sharpens the challenge. Imagine the perfectly unjust man with a spotless reputation, and the perfectly just man thought to be the worst of criminals, who in Jowett's translation "will be scourged, racked, bound" and at last crucified. Show us, he says, that the second man is better off.
My thesis: Socrates never meets that challenge. He answers a different question.
What he shows, across Books IV and IX, is that justice is the health of the soul, each part doing its own work, and that the tyrant's soul is in chaos. Suppose we grant all of it. Glaucon's just man is still on the rack. To say he is better off, Socrates must claim that inner order outweighs any amount of torture and disgrace. He asserts this; he does not argue it. And in Book X he asks to have the rewards of reputation back, the very rewards he agreed to set aside, before ending with the myth of Er.
I taught this for years as a triumph. I now think it is a magnificent change of subject. Persuade me otherwise.