  Pick a formal system. Any formal system. Whatever your favourite formal
system is, as long as it's capable of reasoning about elementary arithmetic.
The First Spectral Gap Incompleteness Theorem of [CPGW15] proved that there
exist Hamiltonians whose spectral gap is independent of that system; your
formal system is incapable of proving that the Hamiltonian is gapped, and
equally incapable of proving that it's gapless.
  In this note, I prove a Second Spectral Gap Incompleteness Theorem: I show
how to explicitly construct, within the formal system, a concrete example of a
Hamiltonian whose spectral gap is independent of that system. Just to be sure,
I prove this result three times. Once with G\"odel's help. Once with Zermelo
and Fraenkel's help. And finally, doing away with these high-powered friends, I
give a simple, direct argument which reveals the inherent self-referential
structure at the heart of these results, by asking the Hamiltonian about its
own spectral gap.
