I'm really glad we're having this discussion, as it's causing me to hash out my thoughts in more detail and research more than I would have if I were merely writing a note to myself. I broke this up into two comments.

While that's certainly claimed by some polyphasic sleep advocates there are others who read a bit and who therefore don't make that false claim and still advocate polyphasic sleep.

Even more, those people who do make the claim don't know that they claim something that in conflict with the academic literature on sleep. That's quite different from the case of homeopathy where the conflict is obvious.

While perhaps excusable, I don't think the ignorance of some short polyphasic sleep proponents is a point in their favor.

There are many proponents of short polyphasic sleep who are aware that more than REM is necessary. Take the Polyphasic Society as an example. They make the more argument that polyphasic sleep is more efficient because it reduces time spent in light sleep, but maintains time spent in deep and REM sleep (they even mention the K-complex in stage 2 sleep!). At first, this struck me as plausible, but I see that Stampi has something to say on the subject of sleep architecture on a short polyphasic schedule (p. 172-173):

Summarizing what was presented in this review, it appears that the organization of sleep within a nap under polyphasic schedules is quite different from that occurring in monophasic nocturnal sleep. Naps are indeed "not miniatures of the normal 8-h sleep pattern" (Weitzman et al., 1974), and only rarely are they replicas of the first part of anormal nightly uninterrupted sleep. For example, REM sleep onset episodes are quite frequent during polyphasic schedules, and it is interesting to note that REM sleep and SWS appear to be mutually exclusive under such conditions: they rarely occur together during short naps. Despite sleep architecture being remarkably different, long-duration studies indicated that all sleep stages (and not just SWS) appear to play an important functional role under these sleep reduction patterns. Indeed, after the initial adaptation period in which daily amounts of all stages but SWS tend to be reduced in amount, sleep percentages become remarkably similar to baseline conditions.

The claim that short polyphasic schedules reduce light sleep but don't reduce deep sleep (SWS) and REM is false. And this book isn't unknown among polyphasic sleep proponents. I'm having a hard time believing that they didn't read it, but it seems they have not. At this point, I don't know of any mechanism by which short polyphasic sleep could work, but I'd accept that it works if I saw empirical evidence suggesting so.

I am somewhat floored by this, to be honest. I want to note that I haven't read Stampi's book in much detail due to time constraints, but I'm not finding anything other than this conclusion in there. For those with the book, please point out if I've highlighted an opportunistic passage, as I am not trying to cherry pick; I just have not read the book as fully as I'd like.

I'll detail a few other major problems with one other claim the Polyphasic Society makes. I have not verified all of their other claims, but I became aware of this when investigating long sleep. They cite a study that suggests people who sleep less live longer. There are a number of such studies, and they appear to be confounded by depression and low socioeconomic status. I have not read the study I just cited beyond the abstract, but I should now. This study was not difficult to find, and I'm disappointed that the Polyphasic Society website did not put the effort in to think or find alternative explanations. There also is the issue, as you've suggested at Stack Exchange, that correlation does not make causation; simply changing your own sleep duration may not actually change your longevity. Perhaps people who are naturally predisposed to sleep longer would reduce their longevity if they slept less. That certainly would make sense as short sleep is associated with many health problems. For some reason the Polyphasic Society forgot to mention those studies.

If you know any short polyphasic sleep proponents who make better justified claims on average, I'd be interested in seeing them, as this is the best I've seen.

Continue to part 2 of this post.