I would appreciate someone directing me to a source that states/indicates/suggests whether the visibility of the new moon outside the Land of Israel may be used to begin the Jewish month in that region. Basically, if the Calendar had not been pre-calculated, what would keep an observant Jew living in Egypt or Spain or Cleveland from using the local new moon to determine when the new month begins? (Another way of stating this question is this: Is there anything in the Tanakh or any Rabbinic text that says the new moon that starts our Jewish months must be the new moon observed in Israel?)
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The Mishna in Rosh haShana (3:1) states that even if all the Jews and the Beit Din saw the new moon, but didn't manage to declare the new month before the end of the day, then the (previous) month is 'full' (ie has 30 days and the new month would begin the next day).
In other words, Rosh Chodesh is set by declaration of the people via Beit Din, not by when the new moon was seen. So even if you saw the new moon in Cleveland, you wouldn't know if anyone had testified to Beit Din and that they declared the new moon.
A few other points: the Talmud relates (Rosh haShana 25a) that the power is entirely in the hands of the Beit Din to the extent that even if Beit Din declares Rosh Chodesh on the wrong day on purpose, the declaration is valid and the new month takes effect.
The Talmud in Brachot (63a) relates that certain Rabbis declared the new moon while not in Israel; however they only did so because there were no other Rabbis in Israel as qualified as they were to make the declaration. The Rambam writes in his Sefer haMitzvot (Aseh 153) that the current calendar only works because there exist currently Jews that live in Israel. Were to be (chas veshalom) that no Beit Din was left in Israel, nor any Beit Din that received Semicha in Israel but has since moved to the Diaspora, then the current calendar would somehow stop. The Ramban there argues on this, but all agree that while the formal declaration of a new month could happen in theory outside of Israel, it rarely did. Israel is clearly the focal point of the whole operation.
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