Sounds like your tank is not cycled yet. Ammonia should not be spiking. Are you doing a fishless cycle?
If not, the fish you chose to put in the tank are not ideal to start the cycling process. Tetras and corys can be rather sensitive, and goldfish just produce too much waste.
It quit spiking after the 4th week. The water has tested great the last 2 weeks.
So you did do a fishless cycle?
What do you mean by "tested great"? Did you see any nitrates, and did they increase over time?
I use API Master Test kit. I also had the local tech and the store test the water. everything was with desirable levels. I don't remember them all, but the only thing that test just above was the water hardness GH
I keep them in the bag till the temp is even and then add a 1/4 cup of tank water every 10-15 minutes
Did you add anything to the aquarium to help the cycling process, prior to adding the fish?
Ammonia? Something like Tetra Safestart? Filter media from one of your other tanks?
You have not answered my question about whether you did a fishless cycle.
If your tank has cycled, you should see nitrates. Testing the water without any fish in it won't show anything, unless you are doing a fishless cycle.
As your tank cycles, you should see ammonia rise, then fall, followed by a rise in nitrites, which then fall as nitrates rise. When the tank is cycled, ammonia and nitrite will be zero, and you will have measurable nitrate.
This is if you are adding ammonia daily, or have fish in the tank.
Sure sounds like you're not quite cycled yet and you're getting the right advice here.
What were you doing those first three weeks when you say the tank was "cycling well"?
Once that local tech helped you with the 80% water change what did you do when you "let it run a few days" and what do you mean the "water was excellent" after that? Did it look good or did you use your API master test kit? If the tank was just empty running for a few days with no fish and no source of ammonia, then all you had was a water container essentially.
If you've got that test kit, go use it and report the readings. For a tank to be cycled you should be getting 0 ammonia and 0 nitrite and at least some measurable reading of nitrate. Confusingly enough a tank that is not yet cycled may also show 0 ammonia and 0 nitrite if you've not added a source of ammonia (which is the main part of the fishless cycle) and typically in this scenario you'll also see 0 nitrate. so to truly ensure your tank is cycled you need to dose ammonia to a measurable level and see it return to 0 within 24 hours and also have a measurable nitrate reading.
The fact that you saw your ammonia reading was ever above 0 is a pretty good indication that the tank was not yet fully cycled resulting in those deaths.