%
\label{def:e0-layer}
\begin{equation}\label{eq:ch05-e0}
    \Elayer{0} \;=\;
    \bigl(\,
        \mathrm{Obj}(\tau),\;
        \text{NF-addressability},\;
        \Phi,\;
        \mathcal{O}(\tau^3)
    \,\bigr).
\end{equation}
\begin{enumerate}
    \item \textbf{Carrier} $= \mathrm{Obj}(\tau)$
          with canonical NF addressing.
          Every object has a unique normal-form address
          in $\mathbb{Z}/M_d\mathbb{Z}$ at each finite depth~$d$.

    \item \textbf{Predicate} $=$ NF-addressability.
          An object is $\Elayer{0}$-admissible
          iff it lies in the image of the hyperfactorization map.
          At $\Elayer{0}$, every object of~$\tau$
          satisfies this predicate (I.T04).

    \item \textbf{Decoder} $= \Phi$:
          the peel map $\Phi(x) = (A, B, C, D)$.
          The canonical ABCD decomposition,
          earned in Book~I, Part~IV.
          At each depth~$d$, it factors through CRT:
          \[
              \Phi_d(x) \;=\;
              \bigl(\,\pi(x),\;\gamma(x),\;\eta(x),\;\alpha(x)\,\bigr)
              \bmod M_d.
          \]

    \item \textbf{Invariant} $= \mathcal{O}(\tau^3)$:
          the holomorphic structure of the total space,
          characterised by the Central Theorem~(II.T40):
          $\mathcal{O}(\tau^3) \cong A_{\mathrm{spec}}(\mathbb{L})$.
\end{enumerate}