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\label{def:angular-sectors}
Let $k \geq 1$ be a stage index.
An \textbf{angular sector} of~$\tau^3$ at stage~$k$
is a clopen subset defined by a solenoidal fiber constraint:
\begin{enumerate}
    \item[\textup{(i)}] The \textbf{B-sector}
          at stage~$k$ with value $b \in \mathbb{Z}/p_k$:
          \[
              \boxed{S^+_k(b)
              \;:=\;
              \bigl\{\, (D, A, B, C) \in \tau^3
              \;\big|\;
              B_k \equiv b \pmod{p_k}
              \,\bigr\}.}
          \]
    \item[\textup{(ii)}] The \textbf{C-sector}
          at stage~$k$ with value $c \in \mathbb{Z}/p_k$:
          \[
              \boxed{S^-_k(c)
              \;:=\;
              \bigl\{\, (D, A, B, C) \in \tau^3
              \;\big|\;
              C_k \equiv c \pmod{p_k}
              \,\bigr\}.}
          \]
    \item[\textup{(iii)}] A \textbf{combined angular sector}
          is an intersection $S^+_k(b) \cap S^-_k(c)$.
\end{enumerate}
At the boundary $\omega$,
B-sectors degenerate to the $e_+$-lobe;
C-sectors degenerate to the $e_-$-lobe.