The furnace, which had been working intermittently, finally died yesterday. It was down to 52° in the house this morning. The furnace guy was already scheduled to come out. He got here about 9:30, poked at the dead furnace for 30 seconds to confirm what he thought, from what I'd said on the phone, was wrong, and spent half an hour replacing the control board. And when he plugged it back in, it came right on and worked.
In theory, it ought to be possible to fix the control board, but his insurance doesn't allow him to try, and I don't have the skills. So I'm glad he'd ordered the replacement part when we scheduled the service call (while the furnace was acting flaky but still keeping the house heated).
Now I can go back to my regularly scheduled crisis, the unplanned emergency crisis having been dealt with.