The French constitution declared three conflicting articles on the beginning of the year, which can be read here or here
Article premier: L'ère des Français compte de la fondation de la république, qui a eu lieu le 22 septembre 1792 de l'ère vulgaire, jour où le soleil est arrivé à l'équinoxe vrai d'automne, en entrant dans le signe de la balance à 9 heures 18 minutes 30 secondes du matin, pour l'observatoire de Paris.
In short: the year starts with the day of the autumnal equinox as seen from the observatory in Paris.
Article III: Le commencement de chaque année est fixé à minuit, commençant le jour où tombe l'équinoxe vrai d'automne pour l'observatoire de Paris.
In short: same as article 1, the beginning of the year is on the day of the autumnal equinox as seen in Paris.
The last article - 16 - though "fixes" the leap-year to every 4th year:
Article XVI: Tous les quatre ans, ou toutes les Franciades, au jour de la Révolution, il sera célébré des jeux républicains en mémoire de la révolution française.
The equinoxes do not follow this strict rule.
So, there are 4 variants to consider:
- equinox: only obeying article 1 and 3, follow the astronomical event.
Most years will have 5, some will have 6 complimentary days.
- Romme: 6th complimentary day - leap day - is at the end of every 4th year, in years 4 * n
- continuous: leap day before every 4th year or century starts, thus years 4 * n or 400 * n, but not n * 100 or n * 4000
-> 3, 7, 11, 15, .., 96, 104, ..., 296, 304, ..., 396, 400, 404, ..., 796, 800, 804, ..., 3996, 4004, ...
- 128-year: leap day in years divisible by 4 but not divisible by 128
According to Wikipedia article French Republican Calendar, variant 2 (Romme) would have had the entire 3rd century (from Gregorian year 1992 - 2091) start each year on September 22nd in the Gregorian Calendar.
Also, variant 4 has the benefit that for the Gregorian years 1920 - 2047, the French Republican years all start on September 23rd.