iMac display turns back on unpredictaby without any keyboard or mouse activity.

Perfect! Now we're getting somewhere. Thanks for that.

I called the company, and there are two names I'm dealing with: Life Wireless and Pure Talk.
Life Wireless seems to be the name of the government program I'm using which provides senior citizens free (but limited) service for talk and data, and unlimited text. That's the primary name in the status bar.
Pure Talk is what shows as the carrier name.

I show only two entries under Pure Talk: Carrier Services and Set Up Personal Hotspot.
Carrier Services shows Check Bill Balance and Directory Assistance; also My Puretalk Account, which gets into the sales end of phones and wireless plans for the younger set.
Apparently, the Hotspot thing allows you to share the iPhone internet connection with our computer and iOS devices.

I agree with you about the automatic Wi-Fi thing. On my phone, it's controlled under Settings > Wi-Fi > Ask to Join Networks.

But that's all I've got. Nothing affecting the subject matter of this thread.


FOLLOW-UP:
I just talked to Tech Support at Pure Talk, and after a lot of bouncing around they told me that AT&T (whose towers they use) has not made this feature available. This doesn't compute: You're using Verizon, which I have to assume is not using the AT&T towers, and you get the feature okay; but my daughter is using Verizon, and she doesn't have the feature either -- although, to be fair, she hasn't requested it either. I think I'm just going to go back to bed now.

I didn't request it from Verizon, it's just there. Has your daughter updated to the latest version of iOS? And if she doesn't have any other Apple devices (iPad, Mac) then there are no other devices on which to take calls, so that feature won't be available to her.

All that said, if AT&T isn't supporting it, that should not be the factor waking up your Mac anyway.

I just talked to Tech Support at Pure Talk, and after a lot of bouncing around they told me that AT&T (whose towers they use) has not made this feature available. This doesn't compute: You're using Verizon, which I have to assume is not using the AT&T towers, and you get the feature okay; but my daughter is using Verizon, and she doesn't have the feature either -- although, to be fair, she hasn't requested it either. I think I'm just going to go back to bed now.

Click to expand...

I'm not sure what you have been told by your carrier but that feature "calls on other devices" has been around for a long time. As far as I'm aware, all the big carriers here in the US support that along with WiFi calling. It's also a feature that's available in the UK and many other places. I'm on T-Mobile and have had that available even on my very first iPhone (5c). I do not know what all the other off-brand cell phone providers do but most of them use towers provided by either AT&T, T-Mobile, Sprint, or Verizon.

Daughter is up-to-date, but she has no other Apple devices. She's still using Windows.

I'm going to let this rest as is. Even if it does turn on the iMac's display, it'll go off in ten minutes.

This hasn't been a waste of time, though; I've learned a few things along the way. Thanks, guys.