Corollary: Holonomy parity and zeros

Extend the result periodically in $\beta$. Then

On $0\le \beta <1$, its only zeros are $\beta =0$ and $\beta =1/2$. It is negative in the positive covariant angular component for $0<\beta <1/2$, and positive for $1/2<\beta <1$.

Proof. Integer shifts reindex the angular modes. Equation (eq:B3) gives ${\mathsf{B}}_3(1-\beta )=-{\mathsf{B}}_3(\beta )$. The factorization in (eq:main) gives the remaining claims.