Thanks, all, for the comments. The carpet is only a couple of years old,
and it has never been cleaned with anything other than my home vacuum
cleaner. There's been spot cleaning with Resolve, and sometimes with a
foam spray. Hardly enough to wet the pad.

I walk along the spotted area in going from my kitchen to my computer,
often carrying a cup of tea. Perhaps there were some drips from a
slightly over-filled cup of tea that left a stick residue as Derby
suggested. The pattern of spots matches that theory. I'll try using some
other solvent cleaner - like rubbing alcohol or an ultra-concentrated
dish washing soap - after first testing on a piece of scrap.