Vector Field Neighbors

I have been thinking a lot about the latest work on twist fields.  It has a lot of good things about it, it appears to successfully add quantization and special relativity to a vector field.  It opens up a possible geometry for the particle zoo.

But if this is really going to be workable or provable, I’m going to have to create a simulation, and that has to start with a mathematical basis.  And that wont come until I understand how the vector field operates on neighbors.  Yes, the unitary twist field has the right configuration to make things work, but the actual quantitative behavior is completely dependent on how the field propagates in space and time.  Up to now, the model looks like a sea of rotating balls, each with a black point spot that normally points in an imaginary direction, but can temporarily point in a real space formed by three real basis vectors orthogonal to the imaginary direction.  (Note that this discrete representation simplifies visualization, but there is no reason that the correct solution can’t be continuous, in fact I suspect it is).  If there is a connection between adjacent ball directions, the necessary quantization, stable particle formation, and special relativity behaviors will result.  However, a quantitative specification of these behaviors is entirely and completely specified by the nature of this neighborhood connection.

How does one ball affect its immediate neighbors?  Can a ball affect nearby balls that are not immediate neighbors?  Can a ball move in 3D or is everything that happens solely a function of ball rotation in place?  I see only two possible connections, one I call gear drive (a twist motion induces an adjacent ball in the twist plane to twist in the same (or opposite) direction) and the other I call vortex drive (a ball twist causes an adjacent ball on the twist axis to turn in the same or opposite direction).  Both of these forces could also induce normal twists, for four possible neighbor connections.  Which, or what set, of these neighbor interactions are valid descriptions of how balls move?  And what mathematically is the exact amount of dispersion of twist to neighbors?  Is the field continuous or can discontinuities occur?

Certainly the requirement for continuity is a powerful constraint, allowing discontinuities from the imaginary to any of the real axes, but prohibiting discontinuities between the real axes or in the imaginary direction.

These are the questions I have been pondering a lot.  I have come up with a nice framework but now I have to work out just how the vector field neighbor connection must happen before I make any further progress.



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