Inverse polyphonic chord recognition Polyphonic music transcription does not currently appear to be a solved problem.

How about the inverse of a small portion of the problem.  Are there any kind of spectral characteristics (from an STFT) that can be used to eliminate some musical chords from the probability space?  (e.g. this snippet of sound most likely does not contain any C# chord, or any kind of diminished minor chord, or this is a single note not a chord, etc.)

Assume the audio snippet is more-or-less stationary (transient attack removed, etc.), and that overtones for most or all individual notes are very likely present.  (And this question is not about inverted chords.)
The problem of polyphonic music transcription has received a huge amount of attention in the research community in recent years and I would go as far as to say that for single instrument polyphonies (piano, guitar etc...) the results are very good. Here are a few papers/authors who have looked at this problem deeply. Derry Fitzgerald has done a lot in the area, a lot of his NMF work on source separation produces accurate transcriptions. Anssi Klapuri has looked even more specifically at the problem of chord detection within multi instrument polyphonies. And finally, although not published, Mikel Gainza has developed very accurate chord transcription algorithms for commercial music soon to be released in a guitar based software product Riffstation.
The publications in the linkes here should give you a good idea of how the Polyphonic music transcription landscape currently lies.