Yosef was born and died on the first day of Tamuz. It was exactly nine months after he was conceived, which was on Rosh Hashana, as per the Gemara in Rosh Hashana, which explicitly states that Sarah, Rochel and Chana all conceived on Rosh Hashana. Rosh Hashana was the day that the first man was created, and it was the day he sinned, creating the need for a Moshiach ben Yosef to come to rectify that sin. Rosh Hashana is a day of potential for Moshiach ben Yosef, and it was that day that not only was Yosef conceived, but he also completed his difficult trials and was miraculously raised to the post of second in command to Paroh on this day. But that day was only the beginning for him. It was a point in time of completion of his potential, and a point in time of the beginning of the seed starting to strike root. It would take time for that potential to become actualized. At first, only the Egyptian people knew of his greatness and humility. He ruled over them with great wisdom and concern for their welfare. This care became apparent as he continued to rule in his humble manner. It would be a significant amount of time until he would rule over his family, the children of Israel, and reveal himself to them. Only in the end would he finally reveal himself to his father. Similarly, it can be surmised that Rosh Hashana is a significant day for Moshiach ben Yosef. It is a day of potential being actualized, and it is a day that is the beginning of a greater revelation.
It is significant to note that Yosef's conception both for his birth and for beginning his role as leader occurred outside of the land of Israel. The trials that would lead him to Egypt began in Israel with his brothers selling him, but the main bulk of his trials and rise to greatness were outside of Israel. This would seem to coincide with the idea that Moshiach sits at the gates of Rome - this means that since Moshiach ben Yosef's purpose is to rectify the fall of Adam, so he enters into the physical realm, the darkest places, the depths of physicality and distance from Hashem, in order to bring everything back to its former state of perfection and equality. Interestingly, Yosef's first interaction with Esav was when Yakov and his family were returning from Lavan's house, on their way to Israel. It would be interesting to explore where that interaction took place, whether it was in Israel itself, or not.
It is interesting to note that Yosef Hatzadik was not born in a leap year. This is clear from the fact that he was born on the first of Tamuz, which is nine months after Rosh Hashana - without the leap month of a second Adar. This would seem to indicate the concept that when Yosef is in potential mode, he comes into the crass world we exist within - but in a very inconspicuous way. He is not born in a leap year where his presence is clear, where the idea of the 'ibbur' - the bringing down of the higher dimension - is apparent. Just as the Maharsha says that Yosef does his work in Av - in the month that Esav/Satan believes he has control, so too, Yosef enters the world in this secretive manner, with no show of glitter or glamor. His entire essence is pure humility. His brothers are even unaware that Yosef is leading them to their tikkun. He accomplishes his goal with the cover of darkness. His goal is never greatness or grandeur. His leadership is only a means to a pure end - to connect the people of Israel, and all of mankind, to their Creator. When that is the goal, there is no room for ego. Yosef excelled in this area, and was a paragon of egolessness, of pure recognition that everything is from Hashem, and his talent was given to him to serve mankind.
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