Translating Dirac Zero-Modes to Bernoulli Corrections via Boundary Deformation

The Origin of Spectral Asymmetry: Lattice Artifacts vs. The Nilpotent Coupling Tensor Within the orthodox framework of chiral gauge theory, as recently regularized by Witten, the transition of chiral fermions onto a discrete computational lattice inherently breaks continuous chiral symmetry. This methodology attempts to circumvent the Nielsen-Ninomiya fermion doubling problem by interpolating discrete, admissible lattice gauge fields over a continuum. However, this discrete-continuous interface invariably generates unpaired Dirac operator zero-modes localized at the topological boundary. In this abstract gauge space, the resulting spectral asymmetry is fundamentally treated as a numerical artifact, a topological defect necessitating external gauge-metric resolution via the Atiyah-Patodi-Singer (APS) η-invariant and anomaly inflow.