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Good question and I don't think a failing thermostat sets any codes?
You can read the actual engine temperature in real time? I use the custom PID pack and Torque Pro.
We don't have any temperature numbers and that might help here?
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Yes, monitor all the temps through the OBD port with a wireless device. Oil, Coolant, and Boost psi. Just did another block test and it passed
Regarding oil temperature, assuming your engine is fully up to temperature your oil temperature should read at least 30F higher than coolant, and if you are pushing hard the difference could be higher than that. Below is picture of my temperature monitor showing over 40F difference.
I can't see how a bad battery could cause the problem you described. Do have a copy of the Service Manual for your car? Did you follow the procedure for refilling the cooling system?
Engine coolant is lower left, engine idling after run