What is the authenticity of the hadiths mentioned regarding the Rightly Guided Caliphate and Imam Mahdi, and is the appearance of the Rightly Guided Caliphate upon the Prophetic methodology exclusively linked to Imam Mahdi?

The hadiths concerning the Mahdi have reached the level of tawātur maʿnawī (concurrence of meaning), and belief in his emergence is obligatory. His name is Muhammad ibn Abdullah, from the household of the Prophet, peace and blessings be upon him, and from the lineage of Fatimah. Allah will prepare him in a single night, and he will reign for seven years, filling the earth with justice and equity. He will emerge at the end of this Ummah. Allah will send down rain for him, and the earth will bring forth its vegetation. He will distribute wealth generously, and livestock will flourish, and the Ummah will prosper.

It is narrated from Thawban, may Allah be pleased with him, that the Messenger of Allah, peace and blessings be upon him, said: "Three will fight over your treasure, all of them sons of caliphs, but none of them will gain it. Then the black banners will emerge from the East, and they will fight you with a slaughter unprecedented by any people." Then he mentioned something that I do not recall, then he said: "When you see them, pledge allegiance to them, even if you have to crawl over snow, for he is the Caliph of Allah, the Mahdi."

And from Abu Hurairah, he said: The Messenger of Allah, peace and blessings be upon him, said: "How will you be when the son of Maryam (Jesus) descends among you, and your Imam is from among you?" The hadiths have indicated that the Mahdi will be a caliph at the end of time, and Allah will rectify the affairs of the Ummah through him.