Ok, Time Machine is spinning and spinning its wheels, unable to make any progress on a backup.
If I kill it, then my Macbook continues to run in high-power mode, fans awhirring at full speed and then if I try to restart, then the system hangs.
Applecare is, for the first time in my dealings with them, utterly unable to solve my problem short of wiping the drive and restoring from the last good Time Machine backup.
Anyone have any thoughts that can save me from this fate?