The five postulates are:
Over time it became increasingly apparent that we live in a non-Euclidean universe & that makes it easy to apply Euclid's work to d&d where magic can not only tell physics to bend over but to twist it into a pretzel while doing so. & that gets into how light seems to do the same on top of always moving at the speed of light. This series 1, 3, 4, 5,6 does a great job of delving into some of those mysteries & really starts to get complicated, but some of the theories are rather simple in what the complex math later tries to prove.... This one for example says "light as a disturbance caused by rippling in the higher dimension just beyond human perception, similar to how fish in a pond can only see shadows of ripples across the surface of the water caused by raindrops. While not detectable, it would indirectly imply a connection between seemingly unrelated forces." and we built the Large Hadron Collider trying to gather evidence for it (among other things it is trying to gather evidence for).1. A straight line segment can be drawn joining any two points.
2. Any straight line segment can be extended indefinitely in a straight line.
3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
4. All right angles are congruent.
5. If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate.
Euclid's fifth postulate cannot be proven as a theorem, although this was attempted by many people. Euclid himself used only the first four postulates ("absolute geometry") for the first 28 propositions of the Elements, but was forced to invoke the parallel postulate on the 29th. In 1823, Janos Bolyai and Nicolai Lobachevsky independently realized that entirely self-consistent "non-Euclidean geometries" could be created in which the parallel postulate did not hold. (Gauss had also discovered but suppressed the existence of non-Euclidean geometries.)
That simple explanation of light being the ripple from a higher dimension opens up an easy door for the d&d universe/"Shared multiverse" to step through if you simply say "The effects of magic are the result of disturbance caused by rippling in higher dimensions or planes just beyond human perception. Much like how a fish can make ripples by swimming at the surface or even leaping across the surface of a pond, humans are able to do influence these higher dimensions through traditional arcane & divine means of spellcasting." Conveniently, I believe that applies to every setting and explains things like faerun's spellplague or Athas' broken state simply by saying that a slightly different arrangement of those dimensions causes those difficulties. In Spelljammer, when visiting a sphere where the caster's deity has no influence, something still answers up to a point allowing the casting of low level spells (up to 3 iirc?) & you could easily explain the something by saying that it is not until you start reaching higher levels of divine spells that the arrangement of those dimensions being different starts adding up beyond what will still work.
That divine magic spelljammer effect seems like it would justify manifest zones too and by extension Ravenloft. If you say that ravenloft has a very unusual arrangement that affects the prime within its sphere then it explains much of it
In settings like forgotten realms/greyhawk/etc you might have the eleminster types who privately have a very high level understanding of the way all that works, but there are really only a couple settings where that knowledge might be known by more than one or two people. most of them are protected by plot armor or the mists of time. The halfling empire of athas almost certainly had a deep understanding of it before they fell & Athas became the wasteland it is known for being causing their knowledge to be lost. "Arcane", the maker of spelljammer ships likely has a firm enough grasp on it in order for them to make spelljammer ships. "The mind flayer empire" that seems to have no remains also undoubtedly had an extremely deep understanding of it. And then you have eberron where a number of things set it apart from all of those others. Eberron is separate from the other spheres in having its own planes. Given Athas' lack of undestroyed planes, it is entirely possible that it too once had its own planes like eberron. Aside from Eberron, the other advanced groups all share the same planes on the great wheel putting them on a spot where a "good enough" understanding is all that might be needed.
With Eberron, the very fabric of how magic works might be different once you cross into/out of those other spheres & as a result its inhabitants have a way to experiment & test their theories about the nature of reality in a version of reality that is both entirely different, capable of supporting life, and most importantly someone else has already worked out a lot of the basic elementary building blocks of how magic works there. Nobody can find evidence of the mind flayer empire, nobody knows where Arcane is from & they don't teach outsiders, Athas is a hellhole nobody wants to visit where archaeological research into the fallen halfling culture is deadly & difficult under the best of conditions. Eberron along with all of those civilizations seem to have roots in their own non/not quite-great wheel starting point. Faerun might once have had a different arrangement, but that arrangement was lost when the new one shifted into being to deprive them of joining the cool kids club of civilizations able to step into a reality with different rules of the universe for expanding their understanding.
Back to Eberron though & there are several groups who would have had knowledge into that deep understanding of the universe. The Dragons, the Giant empire of Xendriik the Dhakaani,the Gatekeepers each of the Dragonmarked Houses, Korrenberg, and the Arcanix seem to be the big players. I'm hoping for some sanity checking on this next bit but had to get through rolling out the foundations above first.
- The Dragons are unquestionably have one of the most (if not the most) advanced understanding of the whole thing, the prophecy might even be the result of that understanding. they probably don't need much more than that since we are crunchy & taste good with ketchup. Their laser focus on The Prophecy might limit their understanding of how the universe works to good enough even though that good enough is still far beyond everyone else aside from Khyber Eberron & Siberyis.
- The Giant empire of Xendriik: We know that The Dragons taught them tthe secrets of what they themselves knew. The giants later used it to move DalQuor in desperation leaving The Dragons to destroy them in horror. Not wanting to risk someone digging up an old spellshard with teachings from Ourelonastrix that the could not be trusted in the hands of lesser races, they went a step further & put cursed the entire continent of Xendriik so anyone rediscovering too much would destroy themselves somehow.
- The Dhakaani: We know that they were far more advanced than khorvaire today in some ways (metallurgy/a wildly different grasp on magic/etc). Vvaraak might have given them limited understanding into parts of what was taught to the giants. The same might have happened when she was teaching the Gatekeepers. Needless to say, the Dhakaani used this knowledge to great advancements thanks to their eusocial bond. That Eusocial bond might even be part of why so much of that knowledge was so completely lost... the dragons no doubt helped it along after the Gatekeepers solved the Daelkyr problem.
- The Gatekeepers: we know that the various druidic traditions split off from the gatekeepers, possibly even starting due to disagreements on how far is too far during the daelkyr invasion. We know for a fact that the Gatekeeprs were trained by the dragon Vvaraak & the magic they used to imprison the Daelkyr in khyber does not exactly scream "standard druidic spell list magic". It seems reasonable & almost certain that Vvaraak Taught the gatekeepers a sizable chunk of what Ourelonastrix taught the giants of Xendriik to at least some limited number of individual students who might have later gone on to teach others a small fraction of that knowledge. Either way, at least some of the most powerful Gatekeepers seem likely to have understanding of the very knowledge that caused the Dragons to destroy Xendriik. The Gatekeeper focus on keeping out the blighted energies & influences of beings & energies from other planes might very easily fit a role specifically designed to help protect the prophecy. That protection being given knowingly or not is irrelevant to the fact that it might be enough to keep the Dragons from pulling a Xendriik on the Gatekeepers
- The Dragonmarked Houses: The dragonmarks themselves might simply be a hereditary way of influencing some of those higher dimensions in the ways we all know. That explanation conveniently explains why dragonmark focus items only work for dragonmarked heirs & why they are so much more powerful than non-dragonmark focus items usable by anyone. It's not simply a matter of being tied to a genetic lock with a different power source as a matter of being able to skip a big chunk of the arcane manipulation of those higher planes simply by doing something like channeling energies of the device through a dragonmark. The individual houses all made great advancements during the last war & no doubt have unraveled quite a bit of how the universe works, but have done so within their own individual area of expertise leaving them still far behind the Dragons but catching up to the Dhakaani/giants in some areas or simply standing on their shoulders to advance further in other areas that make the Dragons concerned enough to be watching Khorvaire closely.
- The Twelve: No doubt that the houseshave conferences between their theoretical physicist equivalents to share knowledge & gain a better understanding of the shortcomings & alternate solutions in their own models.
- The Library of Korrenberg: They want knowledge & might even sponsor some of those events that the Twelve attend in order to justify their own researchers being involved. We know that the Gnomes stole some of the elemental binding techniques from Xendriik so they might even be further along than the Twelve simply by looting the knowledge from the ruins of Xendriik in a manner that slips through a loophole in the curse. The gnome (lets call him Allen) passing the knowledge via sending stone to khorvaire hasn't the foggiest hint of understanding & the gnome on the other end(Bob) is outside the curse's effect. Even if the curse stomps Allen into goo before he can leave, Bob is still free to share the knowledge he too does not understand with the researchers from the library paying him. It doesn't matter if the curse wipes out Allen's notes because Bob has at least a sizable chunk of them.
- The Arcanix: I think they would operate similar to the library of Korrenberg in hosting their own scholarly conferences in addition to being a prime source of training researchers up to what is known. With their overlord adjacent location, the Arcanix likely makes occasional breakthroughs in understanding it or putting it to use making it into an important player in that understanding.
- Mordain: He was part of house Phiarlin & dabbled in forbidden Daelkyr magic, it seems plausible that he was a researcher into this kinda fabric of the universe stuff & made one or more discoveries that opened some very dark doors later papered over by his peers by saying "nope! Daelkyr magic & insanity lies that way". Interestingly enough is the fact that some considered solutions to the 5th postulate in that same kind of lovecraftianesque box.
- Lady Erandis Vol: Vol sits in an utterly unique position among all of the players & might even be beyond the level of understanding that the dragons have. She's half dragon & at least began getting an education from both elves & dragon(s) until her dragonmark manifested. Her father did not simply abandon her though & went on to create the Emerald Claw to help protect/serve her so it's entirely possible that he could have taught her all of what the Dragons know. Having once had the ability to use thew mark of death, she also had the ability to directly influence those higher dimensions/planes without needing to resort to normal arcane means. While she can't test her theories involving the dragonmark given her status as a lich, she can use those memories to further the foundation that her father might have taught her. With both the need to hide and a father who might have told her what lines not to cross, she could even have the ability to step far beyond those lines along with the wisdom to know it would be suicide to do so & as a result focuses on trying to restore her mortality in order to take some infinity gauntlet power level step denied to her as a lich. Restoring her dragonmark might just be a secondary benefit that she can overtly pursue on the down low without the Dragons stomping her out of existence. The Blood of Vol seekers who perform research into preserving their divine spark might even be unwitting well meaning folks who are a useful source of inspiration & advancement for her.
present day Khorvaire probably has not yet discovered a way to travel the spheres in a way that would allow them to advance their understanding in the presence of a similar but different dimensional arrangement. If they have, the unlucky folks who crossed over might have found themselves stuck. Imagine an airship crossing into Athas (or worse) & immediately falling out of the sky as the elementals were unbound or lost control. With nobody on athas or the airship itself in a position to undo the damage & get home, a researcher at Cannith/Lyrander makes a note that the experiment failed with the airship vanishing in an untraceable manner. For whatever the reason that Khorvaire has not managed to cross into other spheres, the lack of multiverse spanning colonial trade empire forming or existing is good evidence that something has kept it from happening. Eberron's "magic as a science" foundation would allow them to expand that empire to an alarming degree with extreme speed.