Magnetic Winding Number
Magnetic Winding Number: τ-value w = 2, deviation –.
Prediction
τ-Formula
w = dim(T²) = 2
Derivation
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
Source
This prediction is derived in the Physics Ledger (Chapter 64 — black-hole-topology), Books IV–V of Panta Rhei.