Fiddling around with alternative expressions of the basic ideas, I've come up with an entirely distinct way of conceptualizing the basic idea.
Take general relativity as it is. And, specifically, examine a forming singularity, in terms of a 3D Cartesian set of axes set some short distance away from the singularity, with each axis being (for the sake of argument) a meter long. Align Z with the forming singularity, such that it represents distance, and X and Y orthogonally.
As the singularity forms, X and Y curve around it. Z shortens.
When it has formed, X and Y close - that is, they form a sphere. Z, meanwhile, has been reduced in length to 0, meaning it has also closed. We have a 3-sphere of radius 0, wrapping around the singularity.
Add more mass. I submit that the axes turn inside out - that is, they become negative closed dimensions. I submit further this is equivalent to the transformation of each axis into a logarithmic spiral, where arc length is equivalent to distance in that dimension. Treating X and Y as measured along Z, they continue to form a sphere (a negative dimension measured over a negative dimension appears positive), and Z forms a logarithmic spiral in two, possibly three complex dimensions.
One complex dimension is equivalent to curvature - we can derive the equation sin(ln(distance))/distance from simple trigonometry.
An additional complex dimension may correspond to time, and might be given by cos(ln(distance))/distance.
The third complex dimension, if it exists, may correspond to the Kaluza-Klein dimension, which is no longer truly closed.
If this is geometrically correct - I am uncertain - I think the value of b in our Z logarithmic spiral may be pi^3. I don't know why yet, but this gives feasible values for curvature. I suspect Euler's Identity may be at work, if any of this works at all.
This is a promising approach, providing roughly the expected values for a unified field theory, for what may be entirely geometric reasons. I'm still working on it.
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