This morning, I brought it to the dealer, wrapping it up. And would you believe? Just like the tooth ache that goes away the minute you walk in the dentist's office, the watch was running again. I was so thrilled to see it ticking away! I guess the motion from carrying it around was enough to get the power it needed (although it stopped at around mid-point on the PR). I left the watch at the dealer, explaining that the day/date setting could be out of sync.
The dealer called me this evening. After I left, he confirmed with Patek service and was told to set the month, date and then day, in that order. Also, Patek said never set the date on 30 or 31. If you do, the month won't change. (I think he messed up by first setting the date this past Tuesday, which happened to be the 30th.) They then suggested he manually enter 2 dates and watch what happens. He put in April 30 Thu and then Jun 30 Tu. Both times he said the watch did exactly as it was supposed to do. He did not turn the hands manually, but kept an eye on the watch to see if it would do it on its own.
He said that it has continued to run smoothly since this morning and that power reserve is now just a little to the left of middle. He will keep an eye on it tomorrow too, and if it seems to be OK and keeping good time, I can go pick it up tomorrow night.
Without looking at the watch, Patek has no idea why the day moved from Tuesday to Thursday, or why the watch stopped. But they assumed it could be because the whole thing was way out of sync from the beginning and couldn't handle the changes without constant power coming from me wearing it, although the PR was around halfway when the watch stopped. Not sure if I buy this, but I have my fingers crossed, and hopefully I don't have a dud.
Waiting to see what happens.
Ron