Angle Maps Are Direction Fields. The half-integer rulebook for plasma polarimetry (MSE and Faraday): wrap-free map processing, an exact equivariance property, and a disclination parity test that flags what a smooth magnetic field cannot produce

Abstract

Plasma polarimetry measures directions, not vectors: the polarization angle of motional-Stark-effect (MSE) emission and the Faraday rotation angle are defined modulo 180deg. The measurement hardware knows this — photoelastic-modulator systems encode the angle in second-harmonic pairs proportional to cos 2gamma and sin 2gamma, and the Stokes pair (Q, U) is the double-angle embedding. Yet at the map level, downstream processing often unwraps to a raw angle and then averages, smooths, or interpolates it — operations that silently fail at the +/-90deg wrap. This article states the missing map-level rulebook as three results, in the language of the framework's sector calculus, where direction fields form the half-integer class [1]. (T1, representation) pi-periodic observables carry only integer 2gamma-harmonics, director-valued quantities only the half-integer class; spatial linear operations are well defined and exactly rotation-equivariant in the double-angle embedding — and only there. (T2, the parity test — the new theorem) If an angle map is the direction of a smooth planar field, the 2gamma-winding around every zero-avoiding loop is even; a half-integer disclination (odd winding) therefore cannot come from a smooth magnetic field: it is a machine-checkable flag for optical, calibration, or genuinely non-field structure. (T3, quantitative) On a synthetic MSE pitch-angle map with an X/O island pair (240x240, 24 frames, 2deg noise), the naive frame-average errs at 8.02deg RMS while the axial estimator reaches 0.41deg — the statistical floor 2deg/sqrt24 — a factor of 19.6; axial smoothing commutes with global rotation to 4e-16. All claims are verified deterministically (14/14). No diagnostic hardware is proposed; the contribution is the rulebook, the parity test, and their checkability.