The infinite lattice doesn't represent a "real" physical processes, it's just mathematical technique for closing a (fundamentally) quantized combinatorial sum o

The infinite lattice doesn't represent a "real" physical processes, it's just mathematical technique for closing a (fundamentally) quantized combinatorial sum over millions of interacting elements. The gap problem exists in the limit. For real systems the spectrum can be measured (in principle) by probing the ground state. The computational paradigm is incredibly general but only within what's apparently a pretty atypical thermodynamic regime (the ordered universe).