People you are taking this to a really complex level.
The math is very simple. Please let me outline it here.
1.
where in the first step we made the change of summation variable
, and in the second step, we made use of the fact that any sum over all
terms is equivalent to a sum from 0 to
.
Now natually you will if you want to minimize your starting location seperation as N approaches infinity in the following series
in other words, since you don't plan on creating an infinite amount (theorectical impossible except with hacking SEIV and changing the default player number by increasing it beyond the current level of 20) you really have the following formula -- which is an approximation that becomes more accurate with the larger the number of systems:
There is a difference in spread between Sprial arm Quadrants (when maximizing the number of threads) and the Torus Quadrants-- in additiona the Sprial Arm Quadrant will limit starting locations based on a number of factors.
For this we look at the Fourier thereom as applied to Duration and Bandwidths as Second Moments (using the normalized second moments of the squared magnitude:
where
As you can all see this explains my point in an elemantary model that is easy to read and follow.