Is there something like an anti-Noether theorem associating every bit of information in a system with an asymmetry?
Yes I get the Shannon angle, and I want a relationship to Noether angle. If an obvious one existed I wouldn’t be asking 
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Well any symmetry breaking would take us out of Noether land. I guess that's how this is an "anti-Noether" theorem
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For some reason you have me thinking of Nambu-Goldstone modes https://en.m.wikipedia.org/wiki/Goldstone_boson …
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I haven't thought this through carefully, but I can imagine that there might be something analogous to Goldstone modes that arises from certain types of noise on a channel.
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