Heroku came back and said:

Looking through the system, I see that you were sent two emails (in August and September) as several of your apps were migrated to the new routing stack (https://devcenter.heroku.com/articles/heroku-improved-router). As mentioned in the documentation, the new router follows stricter adherence to the RFC specification, including sensitivity to spaces.

...and sure enough, there is a line that says:

The request line expects single spaces to separate between the verb, the path, and the HTTP version.

So the lesson is: RTFM

-G