You need to read a bit more carefully. The Omega guide very clearly says (bold face added):
Tip-branched and leg-branched thermocouples are four-wire thermocouple circuits that allow redundant measurement of temperature, noise, voltage and resistance for checking wire integrity.
It is can be extremely
important to know if your thermocouple leads are intact, otherwise you could melt down some very expensive equipment.
I guess I should say then, for my purposes, this is a closed circuit used only for logging data.
There isn't any operational behavior based on these measurements.
But I do appreciate the slap in the face, what type of measurement from the reference would I be looking for? An increase in resistance and decrease in voltage?