I have used (1.)  SONAR's mute (i.e. -inf gain), (2.) -inf clip and track envelopes, and (3.) "remove silence" --  on clips that have essentially no level to begin with (recorded with no input selected) and then exported them and they give the results shown above:
 
Exported 16-bit wave file has a -65dB noise floor
Exported 24-bit wave file has - 112 noise floor
Exported 32-bit wave file has undetectible noise floor - below -138 dB
 
But when the 24-bit and 32-bit exported files are converted to 16 bits in Soundforge, the noise becomes undetectible - below -138 dB, just like a SONAR-exported 32-bit wave file.
 
I agree, it is not the interface as I tried two different one with the same result.  There is no hardware noise, no hum, no pick-up noise, no ground-loops, no sound, no nothing - until I convert the file for 16-bit export in SONAR.  I've been recording audio since 1971 and digital audio with Cakewalk since it came out in 1995 or so.  I know what those noise sources/symptoms are and what causes them.  This is noise created solely by and during SONAR's conversion from a 24-bit recorded file to a 16-bit exported file.
 
End of story for me.  I just won't do that.  Thanks to everyone who chimed in to try to help.

What's the noise level if you disable dither? 

I just did a test here. 16 bits with square dither gives peak noise of ~-112dB. 16 bits with Powr-3 peaks at -65 dB.
 
Look at the frequency distribution of the noise. What differentiates the square vs. Pow-r algorithms is the former distributes the noise evenly across the spectrum while the latter skews it to the upper end where it's less audible. If you look at just the maximum peak value it appears surprisingly high, but the average is down around -126 dB.
 

I stopped using Dither a while back, All my exports are 16 bit and dead quiet. Whatever dither can do to imporve my songs it's not worth the bother.