Here is a diagram of the problem switch.
When I remove the wire nuts at the broken fan fixture, I can measure a current at about 25 volts from black to white. On a whim, I reset the volt meter to AC1000 v, AC250v, AC50v and on each of these setting, the needle moves the same distance.

Across the GFCI, it measures 140 v, across the working light switch it measures 140 v, across the switch to the fan is measures 25V, but it is the needle deflection as described above.