Lately I have been thinking a great deal about the nature of real dimensional spaces. That is spaces whose dimension can be any real number not just an integral value. A method by a Yurosevich Koloskov is to use a kind of stochastic metrization of the space which is something I don't entirely understand but I think is a very interesting thing to try and look into. I think however that very probably in order to be able to build up the geometry of a probability space that would give rise to the kind of appropriate structure to describe Scroedinger wave type behavior we will need to be able to handle geometries of spaces with non-integer numbers of dimensions. This is backed up in some small degree by the practice of dimensional regularization which is a renormalization technique which assumes a non-integer number of dimensions in order to obtain renormalizations that otherwise could not be achieved. If the geometry that could be used to describe the quantum world is ultimately (even in the limit of the multiverse) necessarily stochastic (a kind of quantum gravitational bell's theorem?) then integer dimensional spaces should suffice to describe its dynamics I am not certain that the stochastic coordinate system mentioned above would be sufficient to encode these stochastic spaces as well but then part of the beauty of the idea of having parallel realities is that they make the geometry static instead of stochastic. The stochastic perception then comes from the constraint of path selection in a probability space which very likely is governed by a kind of action minimization multiplicity peak where the multiplicity of paths with close to minimum action occupying a kind of thermodynamical peak in much the way of the configuration of greatest entropy. At this point I think we finally have enough of a framework to begin to try and make headway on the predictions of this sort of world view.
I recently learned that the gyromagnetic ratio of a nucleus is dependent on the nature of the self interactions of the nucleus with the surrounding vacuum fluctuations. Since a singularity would constitute a kind of point charge with spin then it stands to reason that the gyromagnetic ratio of a singularity would be dependent on its charge its mass and its g-factor which perhaps could be calculated. using QED in much the same way that the calculations are done for the nuclei and point charges like the electron. If there were a reliable way to make a measurement of the charge and the rotation of some black hole as well as the magnetic field strength at a known distance from the singularity then we could test if the gyromagnetic ratio matches the predicted value. That could be a good place to try and get quantum gravitational observational measurements. black hole NMR
Also the g-factor of the singularity which we would be measuring might very possibly be dependent on the number of dimensions in which we are carrying out the measurements. which could be a good way of trying to figure out something about the geometry of spacetime on quantum gravitational territory.