Scale-Free Gauge Metric Topology: Isolating the Chiral Zero-Mode via Elliptic Invariants and the Diophantine Obstruction

Abstract The Nielsen-Ninomiya theorem dictates that discrete lattice regularizations of chiral fermions are strictly bound by the symmetric topology of the compact momentum torus, necessitating a zero-sum balance of chiral zero-modes. To obviate this obstruction, this manuscript establishes an asymmetric gauge field regularization by mapping the discrete arithmetic geometry of the Rank 1 elliptic variety y² = x³ − 16x + 16 directly onto the continuum topology of lattice momentum divergence. By embedding the curve’s canonical height (ĥ(P₀) ≈ 0.051111) into the Dirac fermion propagator, an incommensurable, scale-free structural spacing is introduced across the momentum coordinates. This irrational metric deformation scales linearly with momentum; it intrinsically vanishes at the origin to isolate and stabilize the primary left-handed zero-mode (index(0) = +1), while inducing a full arithmetic divergence at the lattice momentum boundaries (p_μ = π). Rather than translating the zero-crossing along a continuous grid, this quadratic denominator divergence acts as a topological obstruction that dimensionally decompactifies the closed loop of the standard momentum space torus. By topologically obstructing the periodic boundary condition, the spurious fermion doublers are permanently excluded from the allowable spectrum. Consequently, the Poincaré-Hopf constraint is freed from its traditional zero-sum limitation, satisfying a net non-vanishing global topological charge (χ(ℳ) = +1) and successfully recovering an intrinsically asymmetric, gauge-covariant metric fermion propagator.