Proposition: Same visibilities, different discrepancy

For the family (eq:phasefamily),

The copy classes, branch probabilities, and all pairwise visibilities on $S$ are independent of $\theta$. At $\theta =\pi$, every nonempty Möbius element vanishes on $S$, so that subsystem is fully classical for the hierarchy.

Proof. Use the normalized branch vectors

Pairs with different $b$-values are screened by inaccessible $K_2$. The two surviving split pairs are $(0b,1b)$, and

Each has trace norm one. Therefore

Its eigenvalues are $\pm |1+e^{-i\theta }|/4$.

On $S^{\prime }$, the two contributions occupy orthogonal $A$-blocks:

Each block has eigenvalues $\pm 1/4$, giving total discrepancy $1/2$.

At $\theta =\pi$, the singleton $Z_{\{1\}}$ vanishes on $S$. Every element involving record $2$ consists of pairs differing on inaccessible $K_2$, hence is screened. All nonempty reduced elements therefore vanish.