In the Gregorian calendar, January 1 falls on a Saturday in any year that is a multiple of 400. (This follows from any correct day-of-the-week algorithm, such as https://firstsundaydoomsday.blogspot.com/2009/12/quick-start....) Since each 400-year cycle begins on the same day, rather than balancing out, any imbalances would actually accumulate.

To check, see this 5-line brute-force calculation in JavaScript: https://news.ycombinator.com/item?id=42024474

Ah, the number of days in 400 years is 400 * 365 + 97 = (57*7+1) * (52*7+1) + (14*7-1) = 0 mod 7, so indeed it doesn't work out.

Exactly.