I have a solution.
My counter, HP53132A was failing on self-test "FAILED: FRONT END". It was done independently, and I did not run ALL. There are differences between the two. When you do ALL, there is no relay chattering, and it actually works without input connected to the reference out. So my setup was, from 10MHz in rear to a tee, which plugged into input 1, then to input 2. This always failed on this unit. But always succeeded another unit.
I never noticed this, but THIS unit has a rear input BNC. The way it's configured, the input amp connects to front input, then branches off to the rear. So if I'm using front connectors, this wire to the rear acts like a stub. It fails. When we connected the same except using REAR input, it works every time.
According to HP's service manual, it says when rear terminal is installed, performance of front terminal is not guaranteed.
And as to fluctuation on reading, according to the lead tech, it is normal for HP53132A. He believes slight phase shift on internal oscillator is the cause of it, and it is perfectly fine as a unit.
Wow....