N27 — Magnetic winding number $w = 2$

N27: Prediction

The RM winding theorem (V.T227): $w = (T^2) = 2$. Faraday rotation measure near BH jets encodes toroidal winding. Experiment: SKA, ngVLA, ngEHT polarimetry. Timeline: 2030–2035.

Derivation Context

The Faraday rotation measure (RM) around the shadow of a $T^2$ black hole exhibits winding number

The toroidal magnetic field on the minor $S^1$ cycle causes two sign changes per azimuthal circuit, compared to one for a radial/dipolar field on $S^2$. The winding number $2$ is a topological invariant of genus$(T^2) = 1$.

The toroidal B-field wraps around the minor $S^1$ (Definition (def:ch42-toroidal-bfield), Chapter (ch:book5-ch42-eht-reread)). As the line of sight traces one azimuthal circuit around the shadow, it crosses two sectors where the toroidal field component $B_$ reverses sign: once at the top'' of the torus and once at thebottom.’’ Each reversal produces a sign change in the RM integral $RM = n_e \,B_ \,dl$. For $S^2$ with a radial or dipolar field, only one sign change occurs per circuit. The winding number is $w_RM = |number of sign changes|/1 = 2$ for $T^2$ and $1$ for $S^2$.

The Stokes $V$ circular polarization winding number around the shadow is