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

Description

Overview 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 depends crucially on the compact toroidal topology of momentum space generated by lattice periodicity. In a periodic d-dimensional lattice, momentum is defined modulo reciprocal lattice vectors, and the Brillouin zone has the topology of a compact torus Tᵈ. The doubling constraint then follows from a topological degree or winding argument formulated on this compact momentum manifold.

Aperiodic Geometry and Momentum Topology The present work investigates what remains of this obstruction when global translational symmetry is removed at the geometric level. We study Weyl-type graph operators defined on embedded non-periodic and aperiodic graphs in four dimensions. In such geometries, there is no global translation group, no globally defined reciprocal lattice, and no Brillouin torus of the standard periodic type. Consequently, the specific toroidal momentum-space topological mapping used in the original Nielsen–Ninomiya proof cannot be formulated in the same way.

Spectral Diagnostics and Operator Formulation This structural observation is supplemented by spectral diagnostics of the positive operator H = D†D, where D is a local Weyl-type graph operator constructed from embedded edge directions. We examine low-energy spectral density, near-zero mode multiplicity, and inverse participation ratio behavior. Representative comparisons with periodic hypercubic lattices show a clear structural distinction: periodic lattices exhibit dense near-zero clustering associated with familiar doubling behavior, while the aperiodic graph families examined here do not show systematic growth of near-zero multiplicity within the sampled sizes.

Conclusion and Scope The conclusion is intentionally limited. This Part does not prove the universal absence of additional fermionic modes. It does not construct the Standard Model chiral spectrum. It does not establish exact microscopic lattice chiral symmetry on arbitrary aperiodic graphs. Instead, it establishes a narrower structural result: within the non-periodic embedded graph settings examined here, the specific compact-Brillouin-torus topological enforcement mechanism underlying the original fermion-doubling theorem is not structurally available. Aperiodic geometry therefore provides a legitimate setting in which the standard periodic-lattice obstruction must be reformulated rather than assumed.