For what my opinion is worth, probably not much, I think Len is not quite correct in saying "So it turns out the split part of the sail is used in the same way as on a plane, for slow wind speeds." Quite frankly, I don't think the split itself does anything much at all.
My understanding of the SJR is that it allows the sail maker to build the best possible foil shape into the forward 30-35% of the chord which is the most important part of the sail in regard to lift, and to maintain that foil shape unimpeded on either tack. "Best possible foil shape" also includes an entry which is quite blunt, so it is possibly worth the extra little bit of work in building the jibs using angle shelf foot in order to get that perfect shape right up near the luff, because the jib panels will be unimpeded by the mast. The purpose is to improve the ability to sail to windward (which is not necessarily the the same thing as merely being able to point higher). Whether this objective is actually achieved (better windward ability compared to a cambered contiguous sail) is still open to rigorous testing. All I can say is that my SJR goes well to windward - that is the result also of a good hull shape, and good harmony between the design of the sail and the design of the hull. It is possible the boat would go just as well to windward with a well proportioned cambered contiguous sail - I don't really know for sure, but consider it quite likely.
With all due respect to Paul McK (and I really mean that) - I don't think discussion about slats and slots has much relevance to junk sails. A fixed leading-edge device on an aircraft wing which utilises a gap between the device and the main part of the wing is usually referred to as a "slot". If the device is retractable, it is usually referred to as a "slat". The purpose of either is the same - to allow a very high angle of attack which would otherwise put the wing into a stalled condition. Landing, taking off, flying at a relatively low airspeed or "on the back of the power curve", steep turns - all may involve high angles of attack and one or both wings being closer than normal to a stalled condition. Slats or slots may be useful then, if you want the aircraft to keep flying. I am not sure if sailing to windward is in any way analagous to an aircraft wing flying at an unusually high angle of attack, as in a slow-speed short-runway landing or take-off.
I also don't think there is much relevance to the SJR in the long-held notion about "the slot effect" which is supposed to arise from a genoa sheeted in correctly on a bermudan sloop. I think this is what initially attracts people to the SJR - the bermudan paradigm - it did for me at first anyway. But I doubt it now, and the sheeting angle of a SJR jib panel, relative to its corresponding main panel, is built into the sail and not adjustable anyway.
It is possible that there is a bit of a "slot effect" at times on a SJR - I do not know, but I doubt if the effect is much, if any. And it may be that if the SJR sail is offered to the wind at a very high angle of attack, that the slot might come into effect and the stall may be delayed - but why would you want to do that anyway?
[Answer - you might be doing that on a broad reach and close to running before the wind - I have always felt that my SJR is superb down wind - I never thought of that before. When running and letting the sail go out beyond 90 degrees, the sudden effect of the SJR sail becoming unstalled is remarkable - a sudden lurching heel to windward which once nearly had me overboard ... maybe there is something in that]
The question "split or unsplit?" is actually a bad question, as it focuses the mind on what is the least relevant thing about a SJR - the split itself. I can say that bluntly without (for once!) offending anybody, because I think it was me who framed the question at the beginning of this thread. With hindsight, I think now that it is not a useful question, although it has led to some very interesting comments.
Another thing, and it has been discussed on this thread, the SJR is supposed, in theory, because of the entry shape of the SJR jibs, to give the sail a high "alpha tolerance", that is to say, the angle of attack can vary quite a lot without affecting the windward performance - or to put it another way, the sail should be fairly tolerant of sloppy helmsmanship and still perform well to windward. I doubt if that is actually the case, although it might be just something about my sail, but I have found that careful helming is quite critical with this sail, and if I get a little sloppy on the helm (which often happens, as I am not much of a helmsman) the performance goes from "very good", quite quickly to just "very ordinary" so I am inclined to think that while the windward performance is very good, the alpha tolerance on a split sail is not necessarily all that great. This is consistent with the results of testing that was done some time ago, when it was found that the soft wingsail had very high alpha tolerance but the SJR, while being very good, did not have as high an alpha tolerance as the wing sail. I don't know if that constitutes a proof, but anyway, it leaves that element of performance also at least open to question and further testing.
There are some very interesting points raised in this thread, by others, in regard to attachment of air flow to the surface of the sail - which I found quite interesting and worth a second read. Particularly the comments made by Len, Slieve and Arne in that regard. But I don't know if "split or unsplit" has much to do with those things. In theory it should, at high angles of attack, but in practice? And if so, is that useful when sailing to windward?
Personally, I like the SJR very much (at least, the Amiina Mk 2 version which is the only one I know) for a number of other reasons, mainly due to the high mast balance which the SJR allows. I have an aversion to the antics of a single sail (cat rig) boat when on a broad reach or running, with all the sail out one side - the high balance of the split rig mitigates this to some extent. It also reduces the sheeting forces required - in fact all the forces on the sail which are required to hold it in shape, seem to be less required. A bit of vertical tweaking on the parrel-downhauls is all it seems to need to set well- no hong kong parrels, throat parrels luff hauling parrels etc needed. I like to kid myself that the jibs and the mains both have optimal foil shapes on either tack, never crushed by the mast, but that may be more of an aesthetic advantage and it is quite possible there is no real performance advantage from that. I don't know. The rig inflates and accelerates very quickly between tacks, making the boat very handy when tacking up a confined waterway. Is it better than a contiguous sail in that respect? I don't know. There are other little things too, which might be just imagination driven by personal preference.
So, if you like a high balance sail, then SJR is certainly an advantage. (But probably a waste of time unless the balance really is high, like, around 33%) If other matters dictate having the mast placed a little further aft in the hull, then, again, the high balance of the SJR allows this and SJR then has an advantage. All things considered, I suppose the further back from the bow the better, for a single mast junk - but often there is not much choice in the matter.
These are not huge advantages. There are a few small downsides too. The Amiina rig with its high mast balance and correspondingly low yard angle gives what I believe is probably a better plan form from the point of view of drag - (advocates of the high peaked yard and greater apparent luff height might disagree) - Slieve refers to vortex shedding, but I'm a bit out of my depth at that point so i won't labour it. But the down side is that because of the plan form, a slightly taller mast is required by the SJR in order to carry the same sail area at the same aspect ratio, as compared to a high peaked, lower balance sail like the Johanna sail. On the same mast height, but with a slightly reduced sail area, would the Amiina rig match a Johanna sail, to windward? I'd like to think it would, but who knows? I suspect that the SJR "punches a little above its weight" and that the difference probably would not be much anyway, either way. There is probably a bit more work in making a SJR compared with a making a contiguous sail - but if you are an amateur making your own sail, what of it? Making a sail is an enjoyable task. There's no harm in a bit of extra sewing practice, and it could be argued that you are handling smaller panels. And there is probably little extra time required to loft that brutally simple (but deceptively well-proportioned) Amiina sail, with its replication of panels.
I would like to reach back into an earlier post on this thread, and give the last word on this post to Slieve - especially his last sentence: "Only by building two rigs and fitting to identical hulls and sailing them together would we find the answer. Even then we would still have to evaluate the different physical characteristics.
Until we do that then we will always be asking the question, to split or not to split. Perhaps we should concentrate more on mast position or ease of handling when selecting a rig".