Here are the results from my appointment at the Apple Store.  The Genius quickly decided that the phone was broken and looked up the warranty status.  Since this was a replacement for a previously damaged phone, it was out of warranty even though it was under a year old. 
  
The Genius immediately attempted to upgrade me to another iPhone.  I said no thanks and today bought a Samsung Galaxy S4 Active.  Hopefully it will be more robust than the iPhone 3GS phones I had.