Eruvin 39: When Rosh Hashanah Falls on Rosh Chodesh

The establishment of Rosh Chodesh, based on witness testimony, and the implications of that for Rosh Hashanah, which, by definition, falls out on Rosh Chodesh, as it were. The Gemara says, set up two eruvin as possible for each day that might be the day of the new moon, vs. the day that retroactively will have been determined to be a regular day. But the sages reject this approach of dividing the holiness of Rosh Hashanah into two. With implications to separating terumot and maaserot as a conditional statement. Likewise, the question of an egg that was laid on the first day. Plus: when the shaliach tzibur goes up to daven on behalf of the community, he would also make the davening a conditional statement. But the sages did not accept the conditional approach to Rosh Hashanah. Also, more "real life with the Tannaim and Amoraim": when a deer was captured by a non-Jew on the first day of Rosh Hashanah, and slaughtered on the second day, is that venison permitted? Machloket. And the sages acted in accord with their respective positions. With many elements to unpack. Plus: when the non-Jews wised up to this two-day conditional day, and how they would purchase fresh produce, etc. Note that Rav Sheshet's more stringent approach isn't a matter of strife.