I ran AppleJack using "applejack auto reboot". AppleJack was doing its thing; it got through the "repair disks" phase (no problems found) and then it got to step 2, "repair permissions." The CPU halted and the system shut down during this process. That was freaky—the repair permissions process hadn't finished, and moreover I didn't tell AppleJack to shut down the computer. I started the computer again, tried applejack auto reboot", and the computer shut down again. I noticed an error message before the CPU was halted:

shutdown: halt by root: Shutting down due to power loss!

Power loss? The only power loss was the computer shutting itself down. I have my Mac plugged into an uninterruptible power supply (UPS), with a USB cable from the UPS to the Mac. The cable allows the UPS to communicate to the Mac that it's running off of battery power and how much battery time remains.

Although having the UPS plugged into the Mac had never caused this problem to happen before, naturally I decided to unplug the USB cable. Success! AppleJack was able to get through the whole cache cleaning process and clean an individual user's cache. After the computer restarted, I plugged the USB cable back into my Mac and it booted normally into the Finder.

I'm not certain whether something changed in the OS or whether the UPS is communicating differently with the Mac now. I'm currently running Mac OS X v10.4.10 on the PowerMac, but I'm pretty sure I've run AppleJack since upgrading to 10.4.10.

I have seen this issue on Power Mac G4s (Mirror Drive Doors Dual 1.25GHz) and an iMac G5 that are using APC UPS Battery Backups with the USB data cable connecting the Computers with their own APC UPS.

I will try upgrading to 10.4.11 and see if the problem still exists.

I hope this helps someone.