Is the fast valid if a person breaks it relying on the calendar, then hears the Adhan moments after breaking the fast?

If the calendar is based on a reliable and knowledgeable religious authority, then your fast is valid, and the muezzin delayed the adhan. However, if the muezzins do not rely on this calendar but rather depend on their own sighting of the sunset, then you must make up for this day because you broke your fast before sunset. In this case, the reliable source is the muezzin, not the calendar. The basic principle is that the day remains until the entry of night is known with certainty, either through direct sighting or by a reliable source.