LS swap it.
I have seen some water pumps that leak ever so slightly out of the "weep hole" that the coolant evaporates before collecting on nearby parts and lets air sneak into the pump--the only evidence it leaves might be a salty looking stain on the pump housing and nearby parts..
However,you say a new pump was installed--its not common for a new one to leak or be defective..(but anything is possible--rebuilt ones can be suspect)..
I hate to say it but the head gasket is likely the culprit on an engine with over 100K on it--seen some fail much sooner,depending on the type of use and maintainence the engine saw..

So I finally got around to doing the chemical test and it came up negative.
I'm going to try another radiator cap but that's been tried once already.
Can a waterpump push air into the system without leaking? Like I said my mechanic pressurized the system and no leaks showed up.
Kinda lost here
). Does this car have a pressurized overflow bottle or an unpressurized bottle? A leaky unpressurized bottle will lose coolant every time the car runs hot, and also not show up in a pressure test. 
Such a leak would show up in a leak test, no?
I know this does sometimes happen, but I would consider any car that loses a head gasket before 100k miles to be a certified lemon.![]()
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No. You can't add air to a system that is full of water without displacing the water. If you are actually getting more air in the system (and not simply burping out air from the water pump job), and you're continuing to be able to add coolant, the water is going somewhere. If you pass the pressure test the problem may be with the cap (though it's unlikely that you'd get two bad caps in a row.). Does this car have a pressurized overflow bottle or an unpressurized bottle? A leaky unpressurized bottle will lose coolant every time the car runs hot, and also not show up in a pressure test.
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Its an unpressurized overflow. Before burping I make sure the overflow is at the proper level. So far, after I burp it, the heat at idle lasts for a week or two. However the overflow begins to... overflow. It seems air is pushing the fluid out and up to the cap and its evident its coming out while I'm driving. Eventually I lose heat at idle again. At 3k rpms you can see the bubbling in the overflow.

