Part 8A: Aperiodicity and the Absence of Topological Necessity for Fermion Doubling

The Nielsen–Ninomiya theorem establishes that under standard assumptions—including locality, hermiticity, chiral symmetry, and discrete translational invariance—fermion doubling is topologically enforced on periodic lattices. The proof relies fundamentally on the compact toroidal structure of momentum space generated by lattice periodicity.

In the present work, we investigate fermionic operators defined on embedded aperiodic graphs in four dimensions. In the absence of global translational symmetry, a globally defined compact Brillouin torus is not available, and therefore the specific topological mapping argument employed in the original doubling proof cannot be directly formulated.

To complement this structural observation, we examine spectral diagnostics of the positive operator

H = D† D

including low-energy spectral density, near-zero mode multiplicity, and inverse participation ratio scaling.

Direct comparison with periodic hypercubic lattices reveals a clear structural distinction. While periodic lattices exhibit dense near-zero clustering associated with the familiar doubling structure, the aperiodic graphs examined here do not display growth of near-zero multiplicity with system size.

The present work does not claim a universal absence of additional fermionic modes. Rather, it establishes that within the class of aperiodic embedded graphs examined here, the specific toroidal momentum-space topological enforcement mechanism underlying fermion doubling in periodic lattices is not structurally available.