The Fast of Gedaliah does not seem to be related to the Ten Days of Repentance, and only by coincidence does it fall on the third of Tishrei. As such, the main message often attributed to him is "that here is an opportunity to reduce calories after the holiday". This is of course true, but there is a deeper message:
The murder of Gedaliah ben Achikam finalized the destruction of the First Temple. After the burning of the Temple, a depleted remnant of the Jewish population remained in the Land of Israel; the punishment that followed the murder uprooted the last Jews from the Land of Israel and made the exile final. A closer look reveals that it was precisely this tragic event that caused growth.
Until then, the Jews toyed with the fact that the destruction was not final and, in any case, did not gather the strength for a renewed return. It was precisely the recognition that the First Temple era was over that enabled Zerubabel and his people several generations later to immigrate to the Land of Israel and start to establish the Second Temple era.
The recognition that must be internalized after Rosh Hashanah 5784 is that the era of 5783 (2022-2023) is over. That is sad because there were good things in our lives in the previous year, and therefore we fast; and it is precisely from this that we grow. We are facing the year 5784. The year 5783 will not return and it is a good thing that we will fast during the Fast of Gedaliah ben Achikam, and we know that "Gedal-Ya" – God is raising us and "Achi-Kam" – He gives our brothers and us the strength to rise for the year 5784.
Have a good week!
Have an effective fast and Shana Tova!