The Gregorian date of Rosh Hashanah moves within roughly a five-week band, and the movement is the visible output of the intercalation described for Passover.
In an ordinary Hebrew year of 354 days, the festival falls about eleven days earlier in the Gregorian calendar than the year before. In a leap year, with a thirteenth month of thirty days added, it jumps about nineteen days later. The result is a sawtooth: three or two years of drifting earlier, then a correction back. Over the nineteen-year cycle the average holds, and Rosh Hashanah stays in September or early October indefinitely.
There is a slow residual drift. The Hebrew calendar's mean year is about 365 days, 5 hours and 55 minutes, roughly six and a half minutes longer than the mean tropical year, so the festivals creep later against the Gregorian calendar by about a day every 216 years. This is a known feature and is discussed in rabbinic and scientific literature; it will eventually require attention, on a timescale of millennia.
The four postponement rules also produce a visible effect: Rosh Hashanah never falls on a Sunday, Wednesday or Friday. Anyone checking a range of years will find those weekdays simply absent. The rule exists so that the Day of Atonement does not fall on a Friday or Sunday, which would produce two consecutive days on which no cooking or burial is permitted, and so that a particular festival does not land on a sabbath.
The calendar is therefore constrained by liturgical practicality as well as by astronomy, and the constraints are visible in the output.