I'm starting to see possible parallels with Penrose Tilings. They may be self-similar (patterns at one scale re-occur at larger scales), but when dealing with distance-sensitive stuff such as underlies the quantum world that could make (or break!) the existence of energy gaps in the substance. As could the 'quantum solution' of different sub-sections of the same quasi-crystal whole.
Thus you could have a 'homogeneous' mass of material that does/does not exhibit the behaviour you want. Cleave it in two (without 'damaging' it, except for the cleave-line) and the two apparently similar sub-masses are entirely unalike in whether they do/do not, themselves, exhibit the behaviour.
And all the above masses are finite, so can be solved for whether they have the spectral gap, or not. It just remains insolvable as to whether the theoretically-infinite super-mass (i.e. not just limited to the original mass) has such a quality or not.
And of course there are far simpler constructs that we can say do or do not exhibit it, at any practical size or scale. The problem is generalising to all possible (i.e. counting somewhere in the high Alephs of theoretical combinations) substances, that would remove the need for experimental assessment of a wide-range of generalised compounds and leave us with the possibility of plugging "I want this, given these constraints" into the formula and getting a range of possible answers out of it that we can then experimentally assess to confirm our complicated mathematics...
Or I might have the wrong end of the wrong stick. I probably need to go read the original article properly, first.