Where should the engine ground wire connect?

Hi, Joe, thank you so much for your response. The wire is only about 5-6 in long. It cannot reach anywhere like the firewall. Please see my first picture above. It is bolted on the engine and is located under the EGR. I am attaching another picture here. The only two parts it can reach are circled. But there is no place to bolt it on these two parts. I am just learning to work on a car myself. I followed a YouTube video to replace the gasket. When I put everything back, I noticed this wire but don't know what to do with it so I just leave it there. I tested driving the car. It has stopped leaking, but I keep getting a P1520 code. Cleared it and it came back again and again. I looked up the code. Autocodes. Com says something like "Faulty Intake Manifold Runner Control (IMRC) actuator". But I looked at one YouTube video "DTC Mazda P1520 Short Explanation" (youtube. Com/watch?V=kOQ7n5LnNIg) It says "the exhaust solenoid control circuit is shorted to voltage, ground, or open when solenoid activation is commanded ON for 2.5 seconds. The exhaust solenoid supply voltage circuit is shorted to ground or open when activation is commanded ON for 5 seconds." So, looks like the code is related to electric circuit. That's the reason I came back to look at the wire. I also did continuity test using a multimeter to the two circled parts and the front IMRC actuator. They are all good. They are all connected well. I really don't know what to do. Right now, the wire is the only thing I can think of. Maybe I am in the wrong direction.

Thank you so much, Joe. I did what you said. I used a multimeter red pin to touch the #3 and black pin to touch battery negative. The meter shows 1 so not good. Then I used another wire to connect the unknown wire to battery negative and tested #3 again. Still showed 1.
As for the 2nd picture, I am not quite sure if I understand it I looked at the marked place on the picture, there is no junction connector there. I took a picture in that place (driver side of the engine).
Thank you again.

Thank you so much, Joe.
I am terribly sorry. I tried another multimeter. There is continuity on #3. I used the red pin to touch #3 and black pin to touch the battery negative. The number kept changing and finally became 0.00 (not 1 anymore). I went to get a third multimeter to try, also keep changing and finally ended with 0.03.
Then I joined the unknown wire to a piece of long wire and then connected it to the battery negative and repeated the above continuity tests again. No difference and the multimeters showed the same numbers.
Sorry for making things complicated.

Yes. That is a bolt. I thought about putting the unknown wire on it before because I feel it should go there. But that does not make sense because if so, both ends of the wire will connect to the same piece of valve cover.
I looked around many times in the last few days for the bolt hole but cannot find any.
The unknown wire itself cannot reach transmission because it's only 5-6 inches long. (Unless I join it with another piece of wire like I did above.)

If #3 is good, does that mean the elec connection to the IMRC has no problem?
Thank you again.

Wait a minute, Joe. I just used the multimeter that I used this afternoon to test #3 again. There is continuity now. So, there is a possibility that the multimeter plug was not plugged in well this afternoon. But there is another possibility. After dinner tonight I was trying to trace the IMRC harness to try to find out where it comes from. It passes through the middle section of the engine and then to the passenger side and then to the black plastic long board in the back. When I used my hand to move along the cable, I touched many wires and connections. It is possible that one loose connection was put back by my hand accidently. I had the hard time to remove the back valve cover in order to replace the gasket because of the harnesses around. Maybe when I pulled out the valve cover, a connection became loose and now it is back to be connected. That can explain there is no continuity this afternoon and now there is. Let me put back everything and do a test drive tomorrow and will let you know.
Have a good night. Thank you.

Hi Joe, I test drove the car today. P1520 still came out as before. I tested #3 immediately. There is continuity. So, it looks like it is not the reason.
I went to Mazda dealer today. There is an old gentleman in the part department. He is always super nice and very knowledgeable. I explained everything to him. He said the code has nothing to do with the unknown wire. Most possibly the other end of the wire should go to the stud on the valve cover --- the same place you mentioned last night. I said it does not make sense because the other end is connected to the engine. But he said it makes sense that one end to the bottom engine and the other end to the top cover. Second, I asked him if there is any way to connect the unknown wire to EGR and solenoid because they are the only parts the 5 in wire can reach. He said no because both of them have "self-closed circuit" (something like that, I cannot remember the exact words he used).
I described the whole process to him. He thinks the code may not be related to IMRC, solenoid, or EGR. His guess is most possibly I loosened a connector on the back of the engine when I pulled out the back valve cover. It may not drop and just become loose. So, what I need to do now is to check all connectors there. It makes sense to me because it was a very hard time for me to pull out the back valve cover. Had to shake and pull. I will take off more parts in the coming weekend so that I can have more space to reach out to the back to examine all the connectors. I will let you know if I can find anything. Thank you and talk to you later. Have a good night.

Hi, Joe, I checked the back. There is a plastic long rectangle black box Wires come out from inside. I did not open the box and just checked all connectors around it to make sure they were tight. Everything looks fine.
Still get the same P1520 code. There is a pattern. After I cleared the code. When the first time I started the car and drove it, I can drive a long time without getting the code. After I stopped the car and when the 2nd time I started the car, the code will come out within five seconds. It has nothing to do with the engine temperature. Even if I started the car and stopped it within one minute when the engine is still cold, the code will come out the 2nd time I restarted the car.
I tested continuity with all the IMRC pins, all including #3 have continuity except #1. Is this a problem? If #3 has continuity, do you still think there is a short along the wire to the IMRC?
Have no clue now. What else can I check? Maybe I should replace the IMRC.
Thank you so much!

Hi Joe, thank you for the detailed instructions. I believe I can do it. Actually, I have to do it. The situation is I will have a safety inspection in June, and I will not be able to pass it with the engine light on. The car's value is only about $1,000 It's not worth it to go to dealers / shops. It happened to me many years ago with an old Honda that was worth about $800 At that point of time, I had no idea about fixing cars and I had to post at craigslist and explain everything. A handyman came to take it for $300.00 I started to learn working on car a few months ago by watching YouTube. I replaced several cars' brake pads and drums and replaced one Honda Civic's timing belt and crankshaft / camshaft seals. So now I have some experiences. It's the first time I am working on electricity. I want to gain some experience in this area.
I followed the instructions. I also ordered some tools like the breakout box that have not arrived yet.
At step 5, there is a voltage. The solution is to replace the harness between terminal 2 and the main relay terminal D. I am going to work on it. I think instead of replacing the harness, I will cut the plastic tube and tape around the harness and visually check every inch of the wires inside to see if I can find the broken section and then fix it if possible.
At step 6, there is no continuity right now.
Thank you very much. Will let you know.

Hi Joe, I did it! I did two things. Most possibly the 2nd fixed the issue. I checked the wires and couldn't find any problem this afternoon. I had no idea and just thought about anything I could do. First, I opened the four big connectors and spray electronic cleaner on both sides (picture 1). Second, I looked at a part (Picture 2). I do not know what it is. It has a 2x0.5in metal plate at the bottom. There is a screw hole at the end. What I did was I used a bolt to mount it to the top of the plastic coil pack. Suddenly I realized maybe the metal plate is this part's ground. I should put it to the bottom of the coil pack so that it can touch the valve cover (Remember, the unknown ground wire before. Now this wire connects the valve cover to the engine). After I did those two things and put everything back, the code is gone. Before it always came out the 2nd time I started the engine. Tonight, I have started the engine about eight times, and it has never come out again.
I will test more in the next few days. Hopefully the issue has been resolved. Thank you very much. Without your help and encourage I cannot do this work.