** it seems like a theromstat to me **
While a thermostat would control the on and off of a refrigerator, if only it was defective, the appliance should run too cold.
The appliance will run until both sides' temperatures are satisfied. If the freezer isn't *extremely* cold after running continually, it is quite possibly not cooling correctly.
You can read about common causes of cooling problems in refrigerators at the following link. I might guess it could be a 'sealed refrigeration system' problem but that would require at least some disassembly of the appliance to see the tell-tail signs.
LINK >
My frost free refrigerator is not cooling properly. What might be the problem?
JMO
Dan O.
www.Appliance411.com
The Appliance Information Site
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