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Here is another style of design with a 48 month leap year cycle https://www.youtube.com/watch?v=YERho3Q4Abo

I am trying to imagine an even simpler mechanism. Stack 3 cams on the same shaft. The first cam rotates once every 24 hours and allows a lever to drop up to 4 positions every night at midnight. The second and third cams limit the depth that the lever can drop. The second cam rotates once every 31 days. The depth is 1 position deep on most days, 4 positions deep on the 28th and 2 positions deep on the 30th. The third cam rotates once per year but only advances in steps on the 27th and 29th days.

The combinations of all three cam depths determines how many days to advance the calendar. Days 1 through 27 will only advance by 1 day because of the second cam. The second cam will allow an advance of 4 days on the 28th only if the third cam allows it in February. All other months will be limited to a single day advance on the 28th by the depth of the third cam. Another decision is made on the 30th day. The third cam allows for 2 days of advance during some months and only 1 day advance on the rest.