What causes 4013 error

My phone randomly rebooted last week. Now it just cycles through showing the apple logo, then a flash of a green stripe, then the screen goes dark. I have connected it to iTunes and attempted a restore countless times. iTunes reports a 4013 error.

I took the phone to the "Apple Genius Bar". They said there is a hardware problem. Then they tried to sell me a new phone. They seemed perfectly at ease that it was normal for Apple hardware to fail after 14 months. Does anyone actually believe this sounds like a hardware problem? This sounds like defective software. The screen, battery, processor seem fine. But apple's staff don't seem willing to debug their own defectively written OS software.

Thoughts?