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\label{def:causal-arrow}
The \textbf{causal arrow} on~$\tau^3$ is the preferred direction
in the primorial tower
selected by the B/C asymmetry:
\begin{enumerate}
    \item[\textup{(A1)}]
          \textbf{Forward direction.}
          Stage $n+1$ is in the \textbf{causal future} of stage~$n$.
          The evolution operator
          $\mathcal{E}_{n \to n+1}$
          propagates data forward.

    \item[\textup{(A2)}]
          \textbf{Asymmetry mechanism.}
          At each stage~$k$,
          the new prime $p_{k+1}$
          enters the CRT decomposition.
          The B-channel absorbs
          the exponent of $p_{k+1}$
          (how often $p_{k+1}$ divides the argument),
          and the C-channel absorbs
          the tetration data.
          Since the C-channel data
          depends on the B-channel data
          (tetration requires exponentiation as input),
          there is a structural arrow
          from B to C within each stage.
          This intra-stage arrow
          is consistent across all stages,
          giving a global forward direction.

    \item[\textup{(A3)}]
          \textbf{Irreversibility.}
          The projection $\pi_k \colon
          \mathbb{Z}/P_{k+1}\mathbb{Z}
          \to \mathbb{Z}/P_k\mathbb{Z}$
          loses the $\mathbb{Z}/p_{k+1}\mathbb{Z}$ component.
          This loss is \textbf{structural}:
          it is not an artifact of notation
          but a consequence of the fact
          that the $p_{k+1}$-component
          carries new spectral data
          not present at stage~$k$.
          Evolution forward creates information;
          projection backward destroys it.
\end{enumerate}
The causal arrow is the unique direction
in which information is created
(refinement increases)
and never destroyed.