I'm not aware of any evidence that the ancient Romans thought the ideal society would be tripartite; are you?
Maybe there's a language bareer there : I was more thinking about
idéel than
idéal. In short, not ideal in the sense of "good, advisable", but more in the sense of conceptual, abstract
I won't dwell too much on the rest of the first part of your post, but you're misunderstanding the trifunctional theory there : it's not about arguing it did survived protohistoric times and directed the social organisation of historical societies (as you said, and especially with the development of Roman state and society, it became more interwebed), but that ideal and litterary concepts did survived as mode of toughts there and there, and influenced the vision that historical societies could have from their own origins and/or justifications.
With all due respect, if you're going to complain about people "twisting your point", you should probably try writing more clearly so it's actually possible to understand what your point is.
I don't want to be antagonizing there but I'm not responsible of any selective quoting and re-writing one can do.
In this paragraph, for example, it's not clear whether you're saying that the Romans got the idea of a dual monarchy (I assume you're referring to the consuls here, since there's little evidence that the actual kingship was held by two people) from Sparta (which, even if it's true, isn't really relevant to the question of Indo-European tripartite ideology in Rome)
There, for exemple, you replaced "dual magistrature", referring to republican rome and the dual repartition of consuls and praetors or duumviri, with "dual kingship" in Rome. Regardless of your intent, it's an obvious rewriting which doesn't owes anything to clarity or lack thereof : I only mentioned
exploring the possibility of a dual kingship in Latium.
Scrolling the thread back up to the first post, I said "One could see a parallel" between dual magistrature in Rome and dual kingship in Sparta, but a parallel that might be "
accidental". I brang the point only to say that a maintained royalty in Rome could go trough a similar evolution from what happened in Rome and Sparta IOTL : there's no much other exemple of dual rulership on a roughly equal footing in the ancient world (well, you have Carthage with the passage from "kings" or tyrannic suffets), and while possibly accidental (as in, not only not sharing any kind of historical influence; but not coming from similar causes either) there's room for fructuous comparison there for an allohistorical development.
While the Titus Livius' affirmation that Titus and Romulus shared a dual kingship should be taken cautiously (probably an late addition) it could be used as an exemple on how an early dual kingship could have been justified in the Vth/IVth century.
TL, DR: we have an exemple of dual kingship in a relative close place from Rome, that could be interesting to use when it comes to maintaining kingship in Rome that used a dual magistrature. Nothing more, nothing less.
or whether you think that there's some link between tripartite ideology and dual monarchy (which isn't as obvious to me as it seems to be to you).
It's not really obvious to me either on how you can spot this in my post : you're making a mountain, so to speak, about what was an aside about how a dual kingship coming from original tribal situation may not be entierly usable for Rome, given the heavily mythologizied history. In short, not justifying the appearance of a dual kinship because of a tripartite distinction of early Roman society, but because this was probably not historical to begin with, but conceptual.
Or :
1) We have exemples of dual institutions in Rome and Sparta
2) Maybe it could be useful for this ATL
3) But the parallel could be accidental (as in neither coming from similar causes or influence)
3a) Furthermore, Roman history tends to be heavily mythologized at this point, and the institutional development
as described should be taken with caution.
I would just add that the duplication of Tites, Luceres and Ramnes as anterior and posterior tribes might be another discussion point there.
Nor am I sure where you get the idea that tradition "can associate Tites, Luceres and Ramnes respectively with cattle, war and direction/religion," given that the actual tradition as it's come down to us contains no hint of such a division (on the contrary, the Romulean army was supposed to have been drawn from all tribes equally)
Ramnes's association with direction is probably the easiest, trough Romulus and the link with power and divine. Luceres are said to have prepared for war the quicker IIRC (Lycmon being possibly a military function than a name and is labelled as a general, which is not entierly the case for others), and Titus power is said to come from wealth (cattle for Propertius for exemple, but as well the wealth carried by his soldiers).
Note, again, that it doesn't meant it was the case, and doesn't collide with a more or less equal share (that was more stressed between Titenties and Ramnes, tough : I'm under the impression that Luceres aren't that taken in consideration), as we're talking of an idealized distinction (rather than division) of early Roman society.
and that even Dumezil himself ended up abandoning this three tribes = three functions idea.
Not exactly : whqt he acknowledged was to have confused imagination and mode of tought from one hand, with social reality and historical organisation from another.
In this case, he agreed that the primitive tribes of Rome weren't organized along functional lines, but possibly tought as such as an idealization, again not unlike the three orders of society in the Xth and XIth western Europe : society wasn't really organized into three determinate groups; but it was tought having been the idealized case.
I'll stress again, that said, the part about "non-functional" triad : not that it's not a debatable point (it's a minor position even for comparativists), but I think it highlights the notion of artifact and mode of tought, against any temptation to placate tripartite division as fully functional and historical.