John Brabender, a GOP consultant and Rick Santorum’s longtime adviser, who knows the state, figures the race has narrowed from about 8 percentage points over the summer to about four, and could perhaps settle at one or two in the coming weeks. The key for Mr. Trump, he says, is understanding the seams of Pennsylvania’s political geography.

Democrats hold a registered-voter advantage of about a million people, 49% to 38% (12% are “other”). In 2012 the split was 50% to 36%. These Democrats are concentrated in Allegheny County, Pittsburgh’s seat; smaller cities like Harrisburg and Scranton; and above all the machine city of Philadelphia.

The Republicans who dominate the most sparsely populated rural regions in Pennsylvania’s central and western tiers are very conservative, politically and otherwise. Mr. Obama’s 2008 remarks about the people who “get bitter” and “cling to guns or religion or antipathy toward people who aren’t like them” were inspired by his experiences in “these small towns in Pennsylvania.”

Democrats tend to capture the urban areas and Republicans the rest of the state—Mr. Obama won 13 of 67 counties for a 5.4-point margin in 2012. But elections are decided by the swing voters in the four large “collar” counties around Philadelphia proper—Bucks, Chester, Delaware and Montgomery. These voters—more women than men—are predominantly white, well-educated, suburban and higher income. Their politics are moderate or liberal culturally, moderate or conservative fiscally. Of the Quaker State’s 5.6 million votes in 2012, 1.2 million were cast there.