I don't know why you wound up with the settings you indicated... the limit must have been set at 6400, otherwise it would not have been at 4000. Most cameras allow you to set the base/minimum ISO and minimum SS as well. But I cannot think of a combination of settings that would give the unusual settings choices you wound up with.
Typically in Av with auto ISO a camera will first attempt to reach the minimum ISO setting; when that is achieved it will increase the SS. But if the SS reaches the minimum setting first it will use a higher ISO in order to prevent going lower.
I checked the ISO limit, which has not been changed, and you are right. It's 6400.
AI claims this camera is so smart, it may crank up the shutter speed if it thinks a subject is likely to move. I didn't see that coming. It says Sony prioritizes preventing blur from subject motion, so it goes nuts on shutter speed and jacks the ISO to compensate.
The ISO AUTO Min SS is set to Standard because I didn't know it existed, and Standard must be the default.
AI is now telling me that Sony cameras try to stay close to the minimum auto shutter speed instead of just treating it as a floor, so if I had used it, I probably would not have had values of 1/4000 and 1/1250.
Given that the results were pretty good, I now wonder if I should put the minimum at 1/250 to reduce the camera's ISO choices, but then I might have to remember to adjust the floor in certain circumstances. Or maybe not, since it worked surprisingly well at 1/4000.
I suspect I need to make this my default setting and record it as a custom mode I can go to whenever I want. Is that reasonable? I don't see myself needing extremely low shutter speeds except in unusual circumstances, so I should be able to go to manual or something on those occasions.
These fricking machines are too smart. I had to use a machine to understand this machine.