Simoun/WMG

Amuria(f) is Aaeru's grandfather.
Two of the big mysteries that were Left Hanging at the end of the series were the identity of Aaeru's grandfather and the fate of Sibylla Amuria, since they Never Found the Body. They seem to have somewhat similar facial structures, and share an obsession with the Emerald Ri Maajon and the "other world."
 * This would also explain why Neviril was immediately attracted to Aaeru. Must be the blood.
 * It's also odd that Aaeru, who's of a lower social class then almost all of the rest of the cast, would have a grandfather who not only was a sibylla but was a regina. Why would they let a nobody in? Perhaps because she had already preformed the Emerald Ri Maajon, seems like that'd be enough for them to boost her to the top of a chor created solely for achieving the Emerald Ri Maajon.

Dominura is Onashia.
This seems less likely, but both have the same sparkly problem, and both were in Chor Dextra. An alternative but related theory makes Limone Onashia.
 * There isn't really anything against this, just that it would be ridiculously depressing and go against the tone of the show.
 * I assumed that that was the canonical identity of Onashia. Is it meant to be a mystery? I mean, she's a Dextra member who looks exactly like Dominura in that last scene of hers...
 * Not to mention Yun's "Dominura" at the end...
 * It's Fanon. Onashia is actually somewhat physically smaller than Dominura and has different hair.
 * This theory does not account for Rimone/Limone, who was stuck onto Dominura like glue. Also, the "sparkly" problem was a general side-effect of never going through The Spring proper--chances are, it would have happened to Yun, Aer, and Neviril as well.

Aaeru and Neviril are Tempus and Spatium.
They seem to exist in every time and place now, they're a pair just like Tempus Spatium, and Limone appeared to have started the whole phenomenon of identifying "aaeru" with the divine.

Anyone entering the Spring without clearly intending to be female will be assigned male.
Inertia and fear would naturally cause people to tend to prefer to stay female, to the point that society enforces pre-feminist gender roles in order to get enough people to be men. And since someone who's indecisive on that point hasn't demonstrated a clear preference anyway, the interest of society is for them to help redress the potential man shortage.

The Spring is created from the same magical water as the Jyusenkyo springs from Ranma 1/2.
Presumably, no tragic drownings were needed to activate these ones. Or the legends have long been forgotten.

The civilization of Simoun was the victim of a Gendercide.
This results in there being no Y chromosomes left naturally in the population, so the spring tricks the body into thinking it is male so it develops the appropriate physical attributes and allows the reproduction of (female) children. However, this takes years, as shown by Waporif still having and eventually losing his breasts. "The Woman-Nation, First Movement: Ancient Mankind (Track 3)
 * Confirmed! All There In The OST Track Titles:

The Woman-Nation, Second Movement: Collapse and Eradication of Male-Dominated Civilisation (Track 4)

The Woman-Nation, Third Movement: Engineering and Preservation of a New Species (Track 5)

The Woman-Nation, Fourth Movement: Founding of the Woman-Nation (Track 6)"

- Simoun OST 1

Helical motors are designed to harness Spiral Energy
Furthermore, the civilization that created the Simoun fought the Anti-Spirals and loss. The odd system of sex determination was created by a Lord Genome-like person in order to prevent the triggering of the Human Annihilation Program.
 * The Emerald Ri Maajon is actually the final initiation to become a Spiral Warrior. Whoever completes it perfectly, breaks out of the usual dimensions, with an option to go fight the Anti-Spirals in their space.
 * The absence of real males on Daikurikku explains the absence of big humanoid mechas. Which was probably planned by local Genome-wannabe.

Simoun takes place about thirty thousand years after A Canticle for Leibowitz
The Quo Peregrinatur Grex led to the development of FTL and time travel. Eventually the human colonies on Daikuuriku were cut off from the rest in some disastrous war (a third Flame Deluge?), at the same time falling victim to Gendercide. They harnessed the power of the Spring (see above WMG) to trick some of their bodies into being male to ensure the survival of the species, a Divine boon that, combined with their dim memories of Roman Catholic ritual and space-faring civilisation, led to the creation of the Tempus Spatium religion (through the time loop engendered in the show). This explains, among other things, the Gratuitous Latin, the themes of recurrence, the existence and nature of Simulacran civilisation, and possibly even what happened in the past to destroy the Precursors.
 * Chapter 24 of Canticle (bold emphasis added): "He spun the globe again until the axial mountings rattled; 'days' flitted by as briefest instants--In a reverse sense, he noticed suddenly. If Mother Gaia pirouetted in the same sense, the sun and other passing scenery would rise in the west and set in the east. Reversing time thereby? Said the namesake of my namesake; Move not, O Sun, toward Gabaon, nor thou, O Moon, toward the valley--a neat trick, forsooth, and useful in those times too. Back up, O Sun, et tu, Luna, recedite in orbitas reversas....He kept spinning the globe in reverse, as if hoping some simulacrum of Earth possessed the Chronos for unwinding time. A third of a million turns might unwind enough days to carry it back to the Diluvium Ignis. Better to use a motor and spin it back to the beginning of Man. He stopped it again with his thumb; once more the divination was wild."
 * So...can we declare this confirmed at least as a Shout Out?

Simoun Gems are computers.
One of there purposes is to regulate the two Helical Motors the Simoun uses. This is why Simoun Gems aren't used in any of the crafts that run on a single motor.

Simulacrum did not get partitioned in the end.
It just allied with Plumbum, which seems to harbour less fundamental ideological hatred for it than Argentum. This is why Vyuraf has to specify that he was born abroad to explain why Argentum is drafting him. Plus, the Plumbish Military Governor (the 'we were all maidens once' guy) was a defector to Simulacrum waiting to happen already.
 * Simulacrum still has plenty of Simile and Simoun in the skies in the Distant Finale, even though they supposedly all got carted off to Plumbum. After seeing what Aaeru and Neviril did, the Plumbish officials decided to conclude a more equal treaty with Simulacrum, pissing off Argentum and leading to the current crisis. Since Plumbum has most of the Simoun now, Argentum is pretty much setting itself for a huge clusterfuck here, which (being apparently an entire nation of Jerkasses and all) is as it should be.
 * Word of God is that Simoun was supposed to be a fundamentally uplifting show, which lends credence to this WMG (and takes some away from the Dominura-is-Onashia theory).
 * Plus Yun's still wearing the same Empress/Pope getup that Onashia had. If it looks like an Empress/Pope, talks like an Empress/Pope and presides over the Spring like an Empress/Pope...

The Spring is directly connected to Tempus Spatium
Hence why Yun can apparently see the 'legendary pair' Dominura and Limone when she looks into it.

Amuria is a Shout Out to the The Girl Who Leapt Through Time
At least two versions (the 1983 movie, and The Girl Who Runs Though Time manga) of The Girl Who Leapt Through Time associate time leaping with lavender. Amuria has lavender hair and escaped from almost certain death by traveling though time.

Mamiina became an archangel upon her death
The mutiny of the Plumbish priestesses and outright Heel Face Turn (Heel Faith Turn?) of the Plumbish government are textbook eucatastrophe and required exploiting several flaws in the Arcus Prima superstructure that didn't seem to be there before (an open Maaju pool, breakable windows). The intervention of Mamiina's soul as opposed to someone else (or Tempus Spatium Herself) seems more likely given the Eyecatch that has her as an angel and the fact that, honestly, it just seems more Mamiina's style.

The helical motors contain Rynax entities
In a long-forgotten past, Rynax entities were locked inside spiral structures which were placed into round containers to facilitate making use of their special capabilities. Since Rynax are binary beings, always existing in pairs, placing two of them together will make sure they function at their best: one manipulating the temporal structure of space-time, and the other the time structure. The Rynax could have been forced into those spirals, or they could have done so out of their free will, so they can enjoy the "vast world" by interacting with the beings they find so fascinating. Perhaps Rynax even manipulate people unconsciously to place the right containers together. They work the best, or even at all, when the people manipulating their power care about each other as well, so the Rynax can project their pair-relationship onto them.

When Dominura looks into one of the opened containers, she stares straight into the memories of a Rynax that may have been around for thousands of years already, which overloads her brain.

The best way to unlock the Rynax powers is by performing the Emerald Ri Maajon. This not only causes a shift in time-space, but permanently imprints the Rynax pair-relationship onto the people performing it. This explains why Limone and Dominura form an inseparable bond, which resembles the one between Kurau and Christmas in Kurau Phantom Memory.