The problem of Ciallos
This calendar has been documented here quite well, but the former writeups in this node fail to address the fact that the intercalary year does not completely correct the calendar such that it aligns perfectly with the solar year; even with an intercalary 30 day month every 2.5 years, there is still drift between the Coligny calendar and the solar year; without the correction, the calendar is 11.25 days short. With the intercalary month, the average year's length is 366, which overcompensates 0.75 days per cycle. In ten years, this is a 7.5 day drift. In a lifetime, the calendar will drift forward two entire months.
One moon cycle lasts 29.5 days -- there are six 29 day months, and six 30 day months. In total these lunar months average to 254 days in a lunar year -- here is the 11.25 year discrepancy between the lunar and solar year. To calculate the offset of the discrepancy, I used the following calculations:
My calculations
(Days in three Coligny years with adjustment) == 354 days per year * 2.5 years + 30 intercalary days = 915 days per 2.5 years
(Days in 2.5 solar years) == 365.25 days per solar year * 2.5 years = 913.125 per 2.5 years
... and processing these numbers ...
(drift in 2.5 years) ==915 Coligny days - 913.125 Solar days = 1.875 days
(drift in 1 year) == (drift in 2.5 years) / 2.5 = 0.75 days
(Solar years per Coligny year) == 915 / 913.125 = 1.0020533881
(drift in 80 years) == (Solar years per Coligny year) * 80 years = 0.164271048 solar years = 60 days!!!!!!
In simpler terms, there is a .75 drift from the solar year even with the adjustment, which adds up over time. As you can see, this drift, seemingly small, can be very problematic. We don't know how the Gauls corrected this drift; presumably a mathematician (in this case a druid) added or removed days here and there to keep the calendar in line. The majority of people in celtic Gaul were illiterate, there was no Gaulish script, and the Coligny calendar was discovered in Romanized Gaul, written in Latin; thus, no surviving record of the method of correction exists. All we can do is interpret and correct this issue ourselves.
A seemingly obvious (but poor) fix
The obvious potential fix is to add an intercalary 33.75 days every 3 years. This will keep the months at 29.5 days in length, but adjust the average year length to align with the solar year. While a fractional day can't be added every three years, the .75 could be could be added as three leap days every 12th year.
(fixed days per three years) == 354 days per lunar year × 3 years + 33.75 days = 1,095.75 days.
(fixed days per year) == (fixed days per three years) / 3 years = 365.25
The problem with this hypothetical fix is that, once a intercalary year has occurred, the months no longer begin on the new moon. For example, lets say that at the calendar's inception, every month begins and ends on the full moon. The first day of Samonios after an intercalary year is a waxing crescent moon, as do the following months. If I were to throw the three leap days, 12 years after inception, let's say I hypothetically throw them at the end of Ogronios, all months following Ogronios (Cutios, Giamonios, etcetera) the first days of those months will begin on a full moon. Additionally, at the beginning of the twelfth year, before the leap days, the months begin on a waxing gibbous.
trying to improve it
The 33 day intercalary months could be divided into an 11 day month every year, with an additional three leap days every twelve years. On paper this is the same, but in practice, with this system, between the twelfth years, a full half cycle is completed every four years; at inception, the calendar begins on the new moon. On the fourth year, it begins on the full moon. This could be a satisfying fix, it would free us from complete irregularity and bring some balance, but the leap days ruin it every twelve years, and suddenly the full and new moons are irregular once more.
The three leap days can be divided as one leap day every four years, it seems a little more natural that way, but the problem still persists.
Conclusion
I have spent my entire day and most of yesterday playing with this calendar, trying to get it to work. I can't come up with a fix that preserves months beginning and ending on full moons, likely because the entirety of my math knowledge stops after college algebra, and that was years ago. I will keep pondering this and see if I can eventually find a way, but if anyone here would like to give it a try, go right ahead. If you can fix it, shoot me a message please and I will add it to this writeup (with credit of course!)