Halakhah sa Levitico 23:2: Mishneh Torah, Sanctification, Shev Shmateta & Shulchan Arukh, Yoreh De'ah

It is a positive commandment from the Torah upon the court to calculate and know if the [new] moon was seen or was not seen; and they should question the witnesses [to ascertain this] until they sanctify the [new] month. And they must send and inform the rest of the people which day is Rosh Chodesh (the first day of month), so that they would know which days are the holidays – as it is stated (Leviticus 23:2), "which you shall proclaim as holy occasions"; and it [also] states (Exodus 13:10), "And you shall keep this statute at its appointed time."

A court which sanctifies the month, whether by accident, whether they were mistaken, whether they were forced, it is sanctified and everyone is obligated to fix the festivals on the day they sanctified. Even though one knows that they made a mistake, he is obligated to rely on them for the matter is only given over to them and the one who commanded to keep the festivals is the one who commanded to rely on them as it is said (Leviticus 23:2) "that you call them…".

Witnesses that saw the month go to the court to testify, even on Shabbat, as it is said (Leviticus 23:4) "that you call them at their appointed time" and anywhere it is said "appointed time" the Shabbat is pushed off. Thus, we only violate the Shabbat for the first of the month of Nissan which contains the festival of Passover and for the first of the month of Tishrei which contains the festivals of the Day of Atonement and Tabernacles in order to fix the festivals which are described with the term "appointed time". And in a time when the Holy Temple exists, we violate the Shabbat for all the months because of the additional sacrifice of every first of the month which pushes off the Shabbat.