On the Positivity of the Chiral Path Integral: Asymmetric Metric Convergence and the Poincaré-Hopf Resolution in Lattice Gauge Theories

The fermion doubling problem presents a longstanding topological barrier to defining chiral gauge theories on a discrete lattice. In his November 3, 2026 paper, A Note On Lattice Definition of Gauge Theories With Chiral Fermions, Edward Witten explores continuum interpolations of admissible lattice gauge fields to regularize chiral fermions, yet notes that such approaches yield a non-positive chiral fermion path integral, rendering Monte Carlo evaluations impractical. To resolve this lattice index trap, this derivation introduces an asymmetric gauge regularization framework that bypasses the classical constraints of a symmetric 4-torus (T4). By introducing a continuous 4π global phase invariance coupled with an incommensurable radiometric thermodynamic variance (κ≈1.437824 GHz), the flat periodic boundary is subjected to a non-linear coordinate deformation. This metric convergence transformation embeds a strict topological shear modulus that prevents the modified fermion vector field Ṽμ(p) from cleanly vanishing at the outer lattice boundaries (pμ=π), systematically eradicating unphysical fermion doublers from the active spectrum. Consequently, the symmetric zero-sum cancellation mandated by the Poincaré-Hopf theorem (χ(T4)=0) is broken, isolating a single physical zero-mode at the continuum origin (p=0). This structural radiometric asymmetry yields a strictly positive global index (χ(ℳ)=+1), enabling the stable propagation of a pure chiral theory across the discrete-continuous interface without forcing a zero-sum chirality cancellation.