Is this narration authentic: "Umar or Abu Bakr (may Allah be pleased with them both) came to Ali (may Allah be pleased with him) when he was presiding over a judgment, and said: 'I saw so-and-so commit adultery.' Ali then said: 'If you do not have four witnesses, then remain silent and say nothing'"?

Adultery is only proven by the testimony of four men. If the number falls short, silence is obligatory; otherwise, the punishment for slander (hadd al-qadhf) will be imposed on those who spoke, because testimony regarding adultery should not be used as a pretext to defame people's honor.

As for the story of 'Umar ibn al-Khattab, may Allah be pleased with him, seeing a man and a woman committing indecency, and his inquiry about establishing the prescribed punishment based on that, it is a story mentioned by Al-Ghazali in "Ihya' 'Ulum al-Din" for which no chain of narration (sanad) has been found. Furthermore, what has been reported from Abu Bakr and 'Umar, may Allah be pleased with them, contradicts this narration, as they held the view that a ruler does not judge based on what he knows without witnesses.