Entropy, partitions, groups and AI

Last night some of the strands of gamgee inside my skull connected up and I remembered something I learned nearly 14 years ago, which I think is relevant to the discussion of AI in mathematics, and in particular to the Birch test. So I ask you to imagine some counterfactual history. Suppose that the real history I relate below had not occurred, but an AI robot came up with the following. Let A,B,C,D be subgroups of a finite group G. Then |A∩B|·|A∩C|·(|B∩C|·|B∩D|·|C∩D|)3 ≤ (|A|·|B|)2·|D|·|A∩D|·|A∩B∩C|·|B∩C∩D|4. (I hope I typed that right.) What would your reaction be? I expect mine would have been “Who cares?” But suppose that I told you that this was a consequence of two major results in information theory, namely the first non-Shannon entropy inequality for random variables on a finite probability space, by Zhang and Yeung in 1998; and the proof by Chan and Yeung in 2002 that the entropy function of any finite family of random variables on a finite probability space and their…