Sorry Fred. I can't stop thinking about this puzzle now.
I figure the softening system is on the supply side of the water heater.
Water mixed with air is coming from the well. It goes through the softener then branches into hot and cold circuits. Hot circuit delivers water with air into heating vessel. Air separates and expands creating back pressure. Pushes heated water back through softener. This is why you sometimes get hot salty water from cold tap.