Q. What is the earliest date in the secular calendar that Rosh HaShanah can occur?
A. The dates range between 5 September and 5 October and are governed by the previous Pesach. Just as the possible dates of Rosh HaShanah are up to one month apart, so are the possible dates of Pesach, which can be in late March or even in late April.
Two problems affect the situation – the discrepancy between the lunar and solar calendars and the discrepancy between the Jewish and Gregorian cycle.
To ensure that Pesach, governed by the lunar calendar, falls in the spring, which is determined by the solar calendar, there is a corrective mechanism which inserts an extra month into the year seven times in every nineteen-year cycle. The extra month creates a shanah m’ubberet, a “leap” (literally, “pregnant”) year.
The second discrepancy entails a difference of a couple of hours between the Jewish and Gregorian cycles. The result is that 1 Tishri (Rosh HaShanah) becomes very slightly later each year, but no-one notices the difference from one year to the next and it will take 1900 years for the slight differences to add up to a week.
Calendrical experts will work out what to do, but one has to presume that long before it becomes a practical problem the Messiah will have come and the whole issue will be superseded.