Hey Brian,
One thing these Exmor sensors are known for is thermal noise, presumably due to the amount of active circuitry on the sensor. In many ways, this is the only fly in the ointment. It would be very useful, if you get the chance, to check out the following:
1) Black frame at ISO 6400, 1/60th. (No NR in capture)
2) Black frame from (1) with +2EV exposure compensation
3) (1) & (2) after 1 minute of live view
If none of these tests reveal any thermal noise, this is a very good finding! It'd be interesting to see where the noise becomes intrusive. Trying at slower shutter speeds should reveal something.
Obviously, at some shutter speed there is a mandatory and automatic black frame subtraction, which pretty much fixes the problem. The black frame subtraction should probably be an option at high gain settings even at slower handheld shutter speeds.