Fourth-order closure obstruction and chiral nonlocality in circular kinetic magnetotransport

Abstract

Closing an angular moment hierarchy at the stress level omits a definite back-action from higher Fermi-surface harmonics. For a circular two-dimensional Fermi surface, streaming changes angular momentum by one, so the shortest omitted sequence, $1\!\to\!2\!\to\!3\!\to\!2\!\to\!1$, adds a fourth-order term to the current eigenvalue, $\Lambda(q)=\gamma_1+\nu q^2-\kappa_4q^4+\cdots$, with $\nu=v_F^2/(4\gamma_2)$ and $\kappa_4=\nu^2/\gamma_3$. We call this missing operator term the fourth-order closure obstruction. Its gradient expansion is controlled when $\nu q^2/\gamma_3\ll1$. Circular symmetry carries the same coefficient into a radial bi-Laplacian within each conserved angular-momentum block, and retaining $m=3$ exactly, without a gradient expansion, amplifies higher radial modes monotonically. At zero field, positive collision rates exclude real-wave-number poles and response zeros; an equal-rate tail gives a square-root completion. A magnetic field makes the coefficient chiral, produces a Hall sign reversal, and enhances it when the $m=3$ harmonic is long lived. In the collisionless high-field limit, the complete hierarchy becomes a Bessel pole--zero ladder, while finite closures form rational approximants to it. The result separates a controlled low-gradient coefficient from its geometry- and field-dependent finite-wave-number completion.