Thanks, Simon. You suggested the only solution left untested. But. At House of Guitars, he re-wired the whole thing. He put in a new single coil right out of the box. He wired in a new 3-way switch. And these steps one at a time, of course. The single coil gives kind of a dull, muted sound, sounded to him like it was out of phase. Problem is, there is nothing for it to be out of phase with. It's just there by itself with only a hot lead and ground. So I brought my one-pickup guitar home last week. The neck pickup works just fine, so I guess that is what we will go with. Thanks to all for your help and encouragement on this project. Photos will follow.