I’m ambivalent about a 4 day work week but I do have thoughts on the general breakout of the week itself.
From my own experience, a 3 day weekend seems preferable. You need 1 day to recuperate and relax, 1 day to do home chores, and a third day to be active and have fun. With a two day weekend, you lose one of those or do them incompletely. So if we want a 3 day weekend and weekends are 2/7ths of the year, then we would want something like a 21 day “month” with two three day weekends (or a 42 day months with four weekends).
One problem with the week is that we’ve all decided to match everyone up onto the same definition and schedule. This causes a problem when people want to do their chores and home life business, but the workers who would facilitate all of that are all off from work, and also enjoying their weekend. It would be better if there was more of a mix of schedules, so that businesses could easily provide their services to people who are on their break.
On the other hand, having a shared schedule allows us to coordinate social activities and get together large groups to go out and party all night, etc. There is value in having a shared schedule.
From research done on fitness, weightlifting, and athletics, we know that the ideal work structure (for progressive overload-style physical work) is four to six 7-day weeks of hard work with a one 7-day week break of lower-intensity recovery. We also know that the ideal number of days to try and build new muscle is 3-4 days per 7-day week - with both 3 and 4 performing equally well. This somewhat implies that the ideal is actually that you want to workout every other day and that an even-numbered week length would be ideal for workout regimes.
However, as an athlete moves into an advanced stage, they need to add training days in order to keep gaining. My guess would be that what’s really happening is that, once you start hitting the limits of the human body, you need to continue subjecting the body to the stimulus that you’ve trained it to, in order to convince it to maintain the muscle mass rather than converting it into fat, and so you need a baseline of activity for maintenance of capabilities, past a certain point. Under this theory, it would still be maximal to workout for progressive overload every other day, but you would add in maintenance days on the off-days with roughly 1/7ths of your days being a full rest.
If we hypothesize that the brain works similarly to our body then that would imply that tasks that require learning new skills and knowledge could also be maximized by following a similar schedule. Children in schools, for example, might do better if they alternated the intensity of their workload. E.g. intense days might have new subjects and tests, maintenance days might focus on materials review and physical education.
For both children and adults, we might include a mandatory break every four to six 7-day weeks (28 to 42 days).
Now, if we take that all together, we might envision a 42 day month with 10(ish) day weeks.
In a normal week, you have two off days at the end that are shared with everyone else. In the middle of the week, you have one day that is available for chores. You can choose which day that is.
The last week of the month there is a 12-day week. The first half of the week gives you one chore day. The other half is a free, recovery period.
The year would not align with months.
All of the above has zero chance of happening but, I suspect, it would optimize human capability and happiness.