Wifi data usage skyrocketing with no attached devices

We have a Centurylink Xyzel C2100Z modem/router combo.

There are no devices connected by ethernet.

Lately we have seen the internet and wifi lights on the device cranking away even when my wife and I have our phones disconnected, airplane mode on, and then batteries pulled from the phones. So we changed the password multiple times but it is still happening.

So through the modem's admin utility I look for other connected devices and it shows nothing. It has a monitor of how much data is uploaded and downloaded, I reset it to zero. Something is downloading 30+ gig of data per day and uploading about 10 gig of data per day.

So yesterday verified no other devices were connected, I reset data counter again to zero and I had my wife take my phone with her to visit her mom, which is miles away. We do not own any other internet capable devices, no laptops, tablets, thermostats, nothing.

She returned this morning and I turned on my phone and logged into the modem. 113 gig downloaded in the past 36 hours and 39 gig uploaded during the same time. I called Centurylink support and they were no help, verifying we weren't using the default password and then suggesting we change our wifi password and the admin password, both have been done before.

They did however confirm that we produced about 113/39 gig of traffic somehow without a device attached. They could not "see" what was attached and using the data.

SO.....how do I find this invisible device? Our neighbors are all at least 1000 feet away. If I could see a device attached I would think it was a device left by family or friends who we gave the old password to, but no one has the new password. Plus any device not plugged in would have lost power within a day or two. This has been going on for a month.

Without sounding too creepy conspiracy theoryish I am wondering if my neighbors have somehow invisibly tapped into my wifi. Our son noticed they had a "Netgear Nighthawk Extender" hooked up in their shed window nearest to our property, but how could it be connected and not show up in the Xyzel/Centurylink utility?

I should also add that when my son came home to visit he had Avast on his computer which scanned the network and found nothing attached to the network other than his laptop, but during the 30 minutes he was on more than 2 gig of data processed through the modem. For what it is worth Avast also said the network was safe and secure, but I have no idea if that has any value.

Thanks for your help in finding my ghosts!