"So, is the soft shot staying true and just flowing different through a given barrel."
I think the difference is, more likely, in flight characteristics.
"I'd wonder if softer shot got deformed by some manner or another,---"
The most likely time/place for shot deformation is at peak pressure/maximum acceleration. Soft-er shot would be expected to suffer more deformation than hard-er shot.
"---it may hold a certain distribution at shorter ranges, but that may not translate to the same distribution only larger pattern at long distance."
I don't think so. An accurate rifle and a much less accurate rifle will each make groups that conform to the Rayleig distribution. Obviously, the inaccurate rifle will make larger groups than will the accurate rifle. So it is with a shotgun; more choke effect makes smaller patterns than less choke effect. Wider or narrower spread does not change the distribution --- I suspect that some are confusing the size of the pattern with the distribution. It is not so. If you compare a cyl pattern at about 15 yards to a full at about 30 yards, you will find the same pattern. The 15 yard full would not look like the 15 yd cyl nor would the 30 yd cyl look like the 30 yd full. Yet, all will have a Rayleigh distribution if the individual shot pieces are grouped according to radius from the center of impact. Please, post back if anyone doesn't see this yet.
Maybe time in flight would allow air resistance to act differently on spheres and randomly distorted spheres."
Very likely. Good thinking. Do you see that the distribution remains constant as the pattern blooms?
DDA
