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While I don't think polyphasic sleep is as obviously wrong as homeopathy, it is clear that many of the claims short polyphasic sleep proponents make conflict with things known about sleep.
A polyphasic schedule that ignores the circadian rhythm requires sleep deprivation (a bad thing) for some of the naps to happen. Circadian rhythms seem quite rigid for many people. For example, orally taken melatonin is often prescribed in an attempt to shift the circadian rhythm of a patient with delayed sleep phase disorder. The longest study I could find on the subject suggested that melatonin does not work long-term for delayed sleep phase disorder (added 2015-06-30: jacob_cannell pointed out that I misread the study, so I retract this sentence). Indeed, a recommendation I've seen for people with delayed sleep phase disorder has been to not fight their circadian rhythm if they don't have to, and sleep offset from most other people. Based on what I've read from polyphasic sleepers, it seems to me that they usually can't fight their circadian rhythm either, as schedules with a "core" night sleep tend to be more successful (though most people still can not adapt).
Polyphasic sleep proponents also make many assumptions and false claims regarding sleep architecture. To summarize points already made, common claims from short polyphasic sleep proponents include that REM is all you need or the more sophisticated argument that light sleep can be skipped. More than REM is important, in fact, there are a few studies (e.g.) that correlate objective measurements of sleep architecture with subjective measurements of sleep quality, and they consistently find that SWS/deep sleep is the most important. It's also not clear that the lighter stages of sleep are necessarily unimportant (e.g., the K-complex that occurs in stage 2 sleep may play a role in memory formation). Certainly, if someone gets too much light sleep they could benefit from reducing that, but it's not clear to me that polyphasic sleep necessarily does that, and no one knows for certain how much is adequate (I believe greater than zero light sleep is optimal).
I have a copy of Stampi's book now. Skimming through the book, I'm somewhat struck to learn there's a fair bit more academic research into polyphasic schedules than I imagined. In chapter 10 (table 10.1), Stampi cites 11 studies that looked at what he calls "polyphasic schedules with sleep reduction". There are other studies that did not reduce sleep. I'll have to examine this more closely when I find the time.
Schedules with the core sleep don't seem to equate that being awake for 6 hours in a row completely screws you up for a day. That's why they are a lot more practical than Uberman if Uberman works as advertised.
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