Is there any demonstrable example of the Real Domain being infinitesimal in size to the Imaginary Domain?
There's a variety from different sources of the lore that indicate that.
The database's lore on Hilbert Effect that I mentioned to CNBA above explains that imaginary domain existences like the Gnosis must be forcefully captured into a
finitely bound realm before they can be interacted with.
When infected by U-DO,
Albedo was going to produce a Matter Shift and Anomaly between the Real and Imaginary Number Domains, which he describes as the lower dimensional creatures obtaining what they lack. Most blatant, he was going to shove the entire 3rd dimension to the imaginary one through a matter shift. An act that he described as
the evolution of mankind and of the world itself. (Episode 2, Final Boss). And Albedo had been causing a Space-Time anomaly at the time, with
the Database entry of Episode III describing it as a
phenomenon where the boundary between real and Imaginary Space warps. The same database also says that such a thing can be caused for example by the Zohar going out of control,
something that generates infinite amounts of energy.
There is a structure in the plot called an Hypersphere that becomes invulnerable to any intervention from the Real Number Domain by simply
being connected to imaginary space. The only power outputs that can pierce through come
from even higher in the cosmology, at a level of energy that can only come from interaction with the Upper Domain and U-DO.
But Imaginary Number Domain beings can cross the surface as they please without extra effort.
It's also stated that
dimensions of imaginary space and time are their own side of reality without beginning nor end, and the researcher Juli Mizrahi has a plot-relevant statement which
describes the process of something shrinking to Planck Scale size before disappearing altogether from the dimension when a shift is occurring, something that happens because the only accessible area between dimensions is planck sized at best. The mass of the Abel's Ark, an U-DO terminal existence created in the imaginary number domain is simply
Immeasurable even when materializing into normal space.
Xenosaga thus approaches the differences between dimensions as infinite or immeasurable in energy outputs, mass and size, the Real Number Domain consistently considered the lower of the two. Only by binding the superior beings to finite limits of the lower one and expanding the area of the Imaginary that is accessible from the physical world is interaction possible. The imaginary domain is also described multiple times as having no material limitation.
Finally, everytime that Hyperspace jumps are executed they must pseudo-construct the Real Number Domain inside the Imaginary Number Domain, to make it possible without the lower domain people imploding upon their own structure the moment they try access.
This is why, when an object that has a complicated structure in the imaginary number domain is to be transmitted, there must be protection from interference from the imaginary domain by using an isolated environment such as hyperspace and a spaceship.
"Hyperspace" is a protective tube so that interference from the imaginary domain is immediately prevented. The image of its boundary is like the inner wall of a shining tube, and it is recognized during transference as the boundary with the imaginary domain. Perhaps it can be thought to be like something that pseudo-constructs the real domain on a tube inside the imaginary domain, so to speak. (— Xenosaga's Perfect Guide, << Substance Transmission through the UMN >>)
In the
Superiority of Imaginary Space Time subsection of the blog, I also explain this all in more detail. If anyone else is interested.