The 4-Week Leap Month Calendar has 13 months of exactly 4 weeks, i.e. 28 days, in common years and an additional 14th month of equal length in leap years. At 364 days, the common year is a bit too short to be an appropriate solar calendar and the months are also a bit too short to make it a good lunar calendar, but it’s a reasonable compromise of both with a focus on very regular months.
Unfortunately, the Gregorian leap cycle of 400 years does not contain an integer number of 4-week intervals, but the 293-year cycle does, namely 3822, i.e. there need to be 13 leap months per this cycle or about one each 22 to 23 years on average. Therefore, the leap month could be inserted after (or before) each common month exactly once per cycle as in the Thirteen Month Leap-Month Calendar, but for convenience, it’s always appended to the end of the year as the final month.
Like its sibling, the 5-Week Leap Month Calendar, this calendar follows ISO 8601 conventions as far as possible, i.e. all weeks, months and years start on a Monday. The date notation is YYYY-MM-W-D, where MM ranges from 01 to 14, W from 1 to 4 and D from 1 to 7. Alternatively, ±E⁺-YYY… can be used, where YYY is reset and E incremented with every completion of a leap cycle, i.e. the start value of YYY is 000 and the end value is 292. Another option would be ±EE⁺-YY… if YY was reset to 00 whenever [or after] a year contains the leap month, so its maximum value would likely be 22.