Is the coming Caliphate the Caliphate of the Mahdi, and what is the proof of that, and is there a Hadith stating that there will be 14 caliphs before him?

Imam Ahmad narrated from Hudhayfah, may Allah be pleased with him, that the Messenger of Allah, peace and blessings be upon him, said: "Prophethood will remain among you as long as Allah wills it to remain. Then Allah will remove it when He wills to remove it. Then there will be a Caliphate on the Prophetic methodology, and it will remain as long as Allah wills it to remain. Then Allah will remove it when He wills to remove it. Then there will be a biting kingship, and it will remain as long as Allah wills it to remain. Then Allah will remove it when He wills to remove it. Then there will be an oppressive kingship, and it will remain as long as Allah wills it to remain. Then Allah will remove it when He wills to remove it. Then there will be a Caliphate on the Prophetic methodology." Then he remained silent. Both types of kingship, the biting and the oppressive, have occurred in the Muslim nation, as well as a Caliphate on the Prophetic methodology (the Caliphate of Umar ibn Abd al-Aziz). There is no authentic narration specifying the time of the Mahdi's appearance, nor is there any prophetic hadith indicating that there will be fourteen caliphs before his emergence. It is authentically reported from Jabir ibn Samurah, may Allah be pleased with him, that the Prophet, peace and blessings be upon him, said: "The affairs of the people will continue to be set straight as long as they are governed by twelve men, all of them from Quraysh." Ibn Kathir said that these twelve caliphs will be from Quraysh and will rule justly, and it is not a condition that they be successive. Four of them have already appeared in succession (Abu Bakr, Umar, Uthman, and Ali).