(I apologize in advance for the long post, I can be quite wordy and over-explain things. The topic title is the relevant part, the rest is just backstory.
I bought my house in April. For the past few months (essentially since the weather's gotten colder) I've been hearing an intermittent running water noise that I've had trouble identifying the source of. It only runs for a maximum of 4-5 seconds at a time, so by the time I hear it and try to run from wherever I am to wherever I think the noise might be coming from, it's stopped. It'll run anywhere from 1-5 seconds and it does so very intermittently. It must have gone off 15-20 times today starting around noon, but I hadn't heard it all week previously.
It's loudest from the powder room (which is right off the living room). I've come to the conclusion that it's most likely the toilet, even though it doesn't sound like the toilet does when it's filling after a flush, it's at least twice as loud - I can barely hear the toilet filling when I'm out of the room normally, but this is loud enough to hear throughout the entire first floor and the basement, and startle me when I'm in the living room and it's quiet. The only other source of water in that area (other than the faucet) is the water heater which is in the basement underneath the powder room.
I turned the toilet off when I was gone for a few days to see if it would lose water in the tank, but when I came back it hadn't. I turned it off for nearly a week again, this time when I was home, to see if I heard the noise when it was off, and I didn't. It did run for a few seconds this time when I turned it back on, but it was not the same sound as the mystery noise and I think it was due to evaporation - it gets really hot in that room with the door closed as it usually is and the water level in the bowl was a tad bit lower than normal. If it was a leaky flapper, I would think that it would run on a consistent schedule and always run after the water to it had been turned off for a few days. This is a new toilet I installed before I moved in and I have the same exact toilet upstairs, which does not run.
The curious thing is that the old toilet that used to be installed in this location also ran, quite frequently it seems. It was running when I first toured the house; it was running during my inspection (the inspector fiddled with the tank innards and got it to stop but unfortunately I didn't ask what he did); it was running on the morning of my closing and the seller paid a plumber $200 to do an emergency fix. Don't know what he did, though. And now the new toilet in the old spot runs (or seems to...still haven't been able to catch it in the act!)
So I got to thinking the other day, maybe it is a water pressure issue? I don't know the exact psi of the house but it's got to be fairly high because I can run the dishwasher, washing machine, and take a shower at the same time without the pressure noticeably diminishing. The powder room toilet is the first fixture on the water lines, I think, and there's only that, the powder room sink, the kitchen sink, and the dishwasher on the main floor. There's also a pretty noticeable water hammer sound when the mystery water noise cuts off. So maybe the fill valve is the weakest point in the system and is being forced open every once and a while by the high water pressure? Is that possible? How would I know if that's the case and what would I do about it?
The only other idea I have is that it's something related to the heat pump or the water heater, both of which are right under the powder room. But I don't see any water connections on/near the furnace, or a humidifier or anything, and it's not as loud in the basement as it is upstairs. Or maybe my house is haunted by a ghost that uses an invisible faucet to wash its hands...