Unusual Leak

I have a very unusual water leak. I'm not a licensed plumber, but I am a retired hands-on engineer, with quite a lot of experience with electronics, hydraulics, and all kinds of mechanical systems. Plus I've done quite a few plumbing repairs through the years.

MY SITUATION
I am on a deep well system, unmetered. I have an addition with a second bath (vanity, shower, toilet and bidet). The piping to the 2nd bath runs underneath a concrete slab, exiting underneath the vanity, where it branches and goes three ways: (1) to the shutoff valves for the vanity, (2) into the wall, up the wall, over the ceiling, and to the shower; and (3) back into the concrete, under the bathroom floor, and up to the bidet. There is another branch, somewhere in the wall, for the toilet. Both the bidet and the toilet have their own shutoffs. None for the shower. The only shutoffs for the entire room are at the source, in the basement of the original raised ranch house.
MY PROBLEM
In the bathroom (20 plus feet from the original house), and also in the basement, next to the oil furnace, there is a very audible sound of water flowing through a pipe. Not a drip, but a flowage. I've isolated it to the cold water line.

I've also discovered that if I first shut off the cold water source in the basement, open the lavatory faucet, drain the line, close the faucet, and turn on the source, initially there is no sound in the bathroom. But, there is a sound in the basement. Then, if I open the faucet, relieve the air pressure and let water flow, and then shut it off, the sound reappears.

There is no evidence of water leaking anywhere in the structure or the fixtures. The toilet is not leaking or siphoning. Faucets are tight. Shutoffs are tight. No water on the floor, wall or ceiling.

At first I thought maybe the pressure balancing valve in the shower was somehow allowing cold water to feed into the hot water line, but I don't know if that is possible.

MY APPROACH
1. After draining the cold water line, and closing the faucet, and turning the supply back on, in that sequence, the air in the line prevents the water from getting to the lavatory faucet, ergo no sound of water flowage in the bathroom.
2. But, since the sound is audible in the basement Before the faucet is opened, the leak must be between the basement and the bathroom. It is clearly audible in the bathroom after the faucet is opened and closed again with the water supply turned on.

THE CONCLUSION
I've concluded that it is leaking under the slab. But if there is a leak, how is it possible to build air pressure?
And, if there is a leak, where is it? How do I isolate the leak in a 1/2" copper line, covered with protective foam, and buried 9" underneath the top surface of the slab? The pipe runs about twenty four feet before it comes out.

Help!!

A stethoscope has not been helpful. Any advice?

Thanks.

toilet running

This is usually a toilet or bidet running very slow. Have you shut off the stop to the toilet & bidet to see if that stops the sound? I evaluate your other information differently. When you shut off, bleed, drain and turn the lavatory faucet back on I think you don't hear it for a little bit because it is building it's pressure back up and not because of air stopping circulation to the faucet.
I also do not think you can identify the location by where you hear the sound. Sound travels so well in water and piping that identifying more common sources is often more accurate than starting at the sound.
If it is a slab leak then you can air pressure by capping the line off and installing an air valve at your shower head, where you cap it off or some other location. Isolate the different lines and test. Piping is usually run the shortest distance.

Not the toilet

Thanks for the reply.
Both the toilet and bidet have been ruled out.
The tip about the air valve on the shower head may be useful, and I will check it out. The lack of shutoffs is a bit frustrating, but I have no choice but to break into the lines and install shutoffs, or, temporarily, caps.
Sure hope it's not under the slab. It's covered with carpet, which is not so bad, but there's a grand piano which will have to be moved, and that's not a do it yourself job.
Wish me luck, and thanks again.