I came back from a break to find it in a hard lockup. Well OK, this is inconvenient, but at least with Linux a hard crash usually doesn't have bad side effects like mangled system files, unlike a certain other inexplicably popular but frail operating system.
Ctrl+Alt+Delete didn't work, so I hit the power button. When it came back up it looked like this:

It is true that my vision isn't getting any better with age, but it's not that bad. So I says to myself oh dear, I have a problem. I couldn't take time out of work to deal with it, so I moved to another computer-- my house is loaded with computers, as it should be-- and used ssh to log into the broken computer:
$ ssh -Y xena
This accomplished two things. Since I was able to log in and cruise the filesystem, it told me that whatever was broken was probably limited to the video subsystem, and the -Y switch let me use the remote PC as a graphical terminal to the sick computer. So I had easy access to all of my documents and email the easy way.
I did take the time to swap out the digital video cable and try the analog cable. That made no difference, and a few minutes later the display was even worse:

But no, I had to take the long way. After work I spent some quality time with it. I connected the monitor to a different PC and it worked fine. OK, so it's not the monitor. I put it back and rebooted one more time hoping it would magically heal itself, when something I should have noticed right away got my attention: the boot screen was just as mangled as when Kubuntu came up. So duh, that means the video card is at fault because no drivers are loaded yet, and I already know the monitor is fine.
So I fetched my trusty rechargeable 14.4-volt Ryobi flashlight and prepared for surgery. That is my second-favorite flashlight in the whole world. It is bright and it stands up, and it is perfect for illuminating computer innards. I poked around, trying to look wise even though I had only the dogs for company, and I noticed the case fan was not spinning. Oops. The Nvidia card faces downwards, so I reached underneath to feel its fan with my finger. Oops again. Both fans use the same power connector, so apparently it was dead. Tried a different power lead, and they fired right up. (Individual power leads can fail, though in my experience it's rare.)
I gave it some time to cool and connected the monitor again. Nope, Nvidia is fried.