Then how is it that you can play one frequency into one ear and a slightly different frequency in the other ear, and the brain will register the beat frequency?
Only for the lowest frequencies (say, under 100 Hz) are there reliable traveling waves to even measure in the brain. Due to the firing refactory period where a neuron resets its membrane potential, most neurons have a max firing rate in the range of 250-300 Hz, so it's not possible for difference tones to explain any binaural beats above those frequencies.
Even for lower frequencies in the range of delta-gamma waves, it seems deeply unlikely that difference tones are the explanation, because the neurons encoding for frequency are separated spatially, so it's not as if two traveling waves reach the same neuron, subtract out, and result in the difference frequency.
However the brain produces the beat frequency, it's nothing like the way signals subtract in other media.
That's not where this is coming from. If, as you say, the brain is processing information in the frequency domain, then how would beat frequencies arise? I'm not debunking what you are claiming. That fits what I know about hearing. I'm genuinely curious as to how!
My primary area of study was perceptual consciousness in the visual domain, so I had to look a little more in-depth into binaural beats to answer your question. So, it appears that frequency coding is the norm for higher frequencies, but not for lower frequencies, which are the only ones we can hear binaural beats from. Unfortunately, I wasn't able to find much relevant neuroimaging literature, so I don't really know.
Binaural beats are an auditory illusion, but I'm not sure of what process. The lower frequencies do record phase information, and use that for localization information (comparing phase differences at each ear for a frequency tell you the angle it came in from), so It could be related. Sorry again!
For binaural beats with low-frequency tones, I can only guess, but you can actually reconstruct phase information from the frequency domain signal depending on your sampling period. As a trivial example, imagine f << 1/T; now you can treat the DC component of the Fourier transform as a time-domain signal that contains f. I imagine that's how binaural beats work, as they occur at sufficiently low frequencies for this to happen. It could also be that the ear transmits low-frequency time domain information as well as frequency-domain information.
This is my new favorite way to describe the process. Gonna use this with my students :)
No. Tinnitus is not caused by a real sound. It occurs somewhere after the phase-erasing Fourier transform performed by the physical structure of the ear. If it were caused by a real sound, or the brain added sounds in the time space (rather than the frequency space), it would work. But neurons are not fast enough to handle time-domain sound processing, so they offload Fourier transformation to the ear and then work with the (presumably phaseless) frequency-domain representation.