So there is a new paper. Presented here: http://arxiv.org/pdf/1502.04135.pdf Published here: http://www.nature.com/nature/journal/v528/n7581/full/nature16059.html And broadly described here: http://phys.org/news/2015-12-quantum-physics-problem-unsolvable-godel.html Here is an excerpt from the abstract: This suggests that there may be cases where you can experimentally determine whether there is a gap, but that, in theory, there is no way to determine this through analysis. Is this true, and are there actual examples? For example, is there a specific superconductor that: - relies on no spectral gap - where there is enough known about its structure where a computation can be attempted - but the computation is demonstrably undecidable. ??