OK, for the sake of good documentation (and in case anyone is interested)...
I took this to Norton Email Support to see if they could shed some light. Their response was that they had never heard of this problem before, and turned it over to Norton phone support, Priority.
The specialist called and requested a remote session to take control of, and check my computer. I agreed, and immediately ran LiveUpdate and declared that the definitions were indeed, up-to-date. I explained the liveupdate he ran removed the "evidence" of the problem and it would just happen again in 48 hours. He scheduled another call in 3 days to see what I meant. This call never came.
I complained to Norton Support and was promised a HIGH PRIORITY call from a Supervisor.
The Supervisor called and said everything was working fine. I showed her (in another remote session) that automatic LiveUpdate was was scheduled, but not running, and that the definitions were not updating. She thanked me for calling and asked if she could close this case. I said no, it was not solved and she said she could see no problem. She then promised yet another call today. In today's call she said there was no problem and asked to close the case. To be absolutely certain we understood each other, I repeated each point of my complaint and she verified that:
1) Automatic LiveUpdate does not run automatically. Automatic LiveUpdate will only run when "forced", during the "PC Security & PC Tune-up" task. Otherwise, LiveUpdate needs to be run manually.
2) The Definitions will only update for 48 hours from the last LiveUpdate. At that time, the "Definitions Last Updated" notice will revert to the time of the last LiveUpdate (either manual or forced).
3) Even though the pulse updates continue to run, the definitions will no longer update 48 hours after the last Liveupdate, as in (2) above.
4) The ONLY way to keep my definitions up to date is to manually run LiveUpdate every other day.
She verified that this is absolutely normal operation for Norton 360 and is the way the software is designed to work.
Since this comes from the very top of the support food chain, I have accepted it as gospel, and have no choice but to mark this as "Solved -- operator error".
Many thanks to all who have taken the time to read this thread, puzzle over it, and especially to those who tried to help.