See part 1 here.

Quite rigid doesn't tell you at all what you need to do to mess with them and reprogram the brain to do something different.

It's theoretically possible that you can change some mental patterns if you exert strong enough stress. People are certainly possible to switch up their circadian rhythms after having jet lag produced through a intercontinental flight.

I was unclear. There are a number of ways to influence your circadian rhythm. These do allow you to change time zones, etc. It is not clear that these can be taken advantage of for short polyphasic sleep. However, speculation is not necessary, as Dr. Stampi addresses this point in his book (p. 174-175):

Circadian rhythms do not appear to be affected (with the exception of minor phase-shifts found in some studies) during polyphasic sleep schedules, but more detailed studies are required. Preferred times for sleepiness bouts and the "forbidden zone" to sleep tend to occur at circadian times similar to normal monophasic conditions. Anchor-sleep periods at constant times definitely appear to improve stability of circadian rhythms. Data presented in this review suggest also that in designing polyphasic schedules the timing of sleep periods should respect the underlying dynamics of biological rhythms.

The data available verifies my view that a successful polyphasic schedule must respect the circadian drive. Schedules like Uberman seem much less plausible, and other schedules with a "core" period more plausible, but unlikely to be better than monophasic or biphasic (long night sleep with afternoon nap), as those schedules respect the circadian rhythm by their nature.

There one theory not yet covered in our discussion. It possible to imaging sleep as a garbage collection process. After N hours of being awake the body needs N/2 hours of sleep to get sort through all the information stored while being awake. It's also possible that it needs (N^2)/32 hours of sleep to sort through all the information.

Both formula suggest a monophasic sleep schedule of 8 hours for a 24 hour day but the second one also allows Uberman sleep to work. I'm not aware that the academic sleep literature proves that the relevant formula is linear and not quadratic.

This is an intriguing suggestion. Stampi discusses this possibility as well (p. 18-19):

In other words, the recuperative value of sleep on performance may not be linearly correlated with sleep duration; this is suggested by many studies presented in this volume. Indeed, even under sleep deprivation, short naps normally produce remarkable recuperative effects, disproportionate to their duration.

First, I am not aware of any work in the academic sleep literature that directly addresses this question aside from the few studies into short polyphasic sleep. Based on my quick reading of a few sections of Stampi's book, these studies seem to have a number of methodological problems, mainly that they had small sample sizes, but I highlight another major one later in this post (sleep inertia).

I'm happy to see others mention a counterexample (that staying awake for a long period of time requires proportionally less sleep to recover) I was detailing as well. I'll agree that this is not so convincing, as it necessarily involves sleep deprivation. However, it does show that sleep can be more recuperative under certain circumstances, which may be good or bad for proponents of short polyphasic sleep.

Stampi discusses a few other factors in his book that must be factored in. These include the minimum amount of time required for sleep to have any recuperative value, sleep inertia (post-nap grogginess), and the time to fall asleep. It is possible that whatever process that occurs during sleep could be made more efficient in some sense by breaking sleep into multiple instances. That is speculation, however, and what is not speculation is that naps have significant overhead costs associated with them. Stampi seems to recognize that falling asleep fast is difficult if you are well rested and don't have the help of your circadian drive, as well as the very real effect of sleep inertia. He proposes that people might be able to train themselves to fall asleep or wake up fully on demand. Either would be remarkably useful to people with insomnia or hypersomnia, and I am not aware of any technique that's fast enough to work here (the closest I can think of is the Bootzin technique for conditioned insomnia).

In Stampi's book, it's suggested that none of the testing done on subjects on short polyphasic sleep schedules were done within 20 to 30 minutes of them waking from a nap. This strikes me as a significant source of bias in those studies.

There also is the issue that it appears sleep serves multiple purposes, only one of which may be garbage collection. I think it's likely that some purposes are better served by polyphasic sleep in line with the argument you've just made, but others are not.

(I should also note that this analysis totally ignores the circadian rhythm. The recuperative value of sleep and feelings of alertness are both functions of location in the circadian cycle.)

That makes a difference for the spread of memes, if you are interested in why the meme spreads.

With respect to polyphasic sleep as a meme, what bothers me so much is that many rationalists did not do their homework before diving in. I don't know about others, but I at least do some fact checking before doing anything odd. Polyphasic sleep should have set off a few red flags right off the bat, yet best I can tell, even something as easily shown as the problems with the REM claims never were mentioned on LessWrong.