Can't find the water leak

Is the house built on a slab? Do you know if the water pipes were plumbed in the slab? If the pipes are in the slab, the acidic nature of the concrete can deteriorate the copper over time.

Where is the water meter ? Any chance that a previous owner ran a water line to a fountain or fish pond or garden faucet - that has been abandoned & forgotten ? John T.

One additional thought, is there any landscape irrigation on the property? If so, have you thoroughly examined the pipes and vales?

That is a lot of water not to be seen. You may want to check the sewer line to see if it is going down the drain or going to the ground. Any out buildings? Auto fill on a leaking pool?

Could you use air to locate the leak? Shut the main off and connect an air compressor as far upstream as possible. Set the pressure regulator below the pressure rating of the pipes. The air should eventually push the water out and maybe you'll be able to hear the leak. Stethoscope?

Get someone with good ears, they should be able to hear the water running in one of the pipes. Then follow that pipe to the leak.

Acidic concrete? Not in my world :)

Beats me but it will stop if you replace the faulty valve. You gave to close rhe valve too.

Where is the meter? Where is the shutoff in relation to the meter? What kind of soil is under the house? Do ANY of the water pipes after the meter and/or shutoff run either within the slab or underground? Can you get your hands on an ultrasonic flow detector? Should be a plumber in your area that has one, you would think.

As a last resort, shut off the water , drain the pipes, cut the pipes and install 1/4 turn gate valves in strategic positions to be able to isolate certain "zones" and see which sector the leak is in.

Plus he might save some money by draining the system and shutting the heat off if it's in a place where the pipes could freeze.

I saw that the DC transit authority bought 40 $800 noise detectors to hear escaping air from iirc some pneumatic system on subways or busses.

In my previous house, the water entered into the basement where there was a meter and water heater. Hot and cold went upward to 3 bathrooms. But, hot and cold also went under the concrete to the laundry area (basement) and also above that, to the kitchen. In the kitchen you could run the hot waiting for it to arrive, which didn't take too long. However, if you then turned off the water and returned only a few minutes later, the water was no longer hot. I discovered where the heat was going when walking through the basement barefoot. I immediately ran a new run (insulated) along the basement ceiling. Problem solved. So, in you case, some pipes could even be under a basement floor and possibly be leaking.

I used that method to find a leak in an underground pool line. I was actually just going to use it to pressure test the line to see if it was leaking, but when I hooked the compressor up, air came bubbling up out of the ground where the leak was.

Like Ed said, 5 gals an hour is a lot of water to be leaking in a structure. You would think it would have to be underground, or someplace with a direct, hidden path to ground.

I find it interesting the OP has not, as of yet, responded in this thread.

Maybe he's busy trying all that was suggested....

Another idea. Has he talked to the water company? IDK what services, help they offer, but water companies would have leak detection eqpt and they might offer help if he's lucky.

I would be surprised if the water company would help. I would think they would come out and check your meter, but after the meter they would tell you to call a plumber. I guess it all depends on where you live. In California, the water districts are particularly unhelpful.

Had a leak one and could not find the source. The town was here in less than an hour. Glad I'm not in CA.

My water in my first house was being used by the next door neighbor. It was before we had meters. It took me a while to figure it out and the neighbor disagreed until i shut my water off while he was taking a shower.