\label{def:tau-native-abstract-tm}
A \textbf{$\tau$-native abstract Turing machine} (TATM) is a tuple
$\mathcal{M} = (Q,\, \Sigma,\, \delta,\, q_0,\, q_{\mathrm{acc}},\, q_{\mathrm{rej}},\, \mathcal{T}\!\mathit{ape})$
whose components are $\Elayer{3}$-diagrammatic objects
(Definition~\ref{def:diagrammatic-sector-e3}):
\begin{enumerate}
    \item \textbf{Tape} ($\mathcal{T}\!\mathit{ape}$):
          a $\tau$-presheaf on the primorial tower---a compatible
          family of finite functions
          $\{f_k \colon \{1, \ldots, \operatorname{Prim}(k)\} \to \Sigma\}_{k \geq 1}$
          with $f_{k+1}|_{\{1,\ldots,\operatorname{Prim}(k)\}} = f_k$.
          The tape is the inverse limit, not a function from a completed~$\omega$.

    \item \textbf{State set} ($Q$):
          a finite $\tau$-object at depth~$k_Q$
          with $|Q| \leq \operatorname{Prim}(k_Q)$.

    \item \textbf{Alphabet} ($\Sigma$):
          a finite $\tau$-object, similarly bounded at some depth~$k_\Sigma$.

    \item \textbf{Transition function} ($\delta$):
          a $\tau$-morphism
          $\delta \colon Q \times \Sigma \to Q \times \Sigma \times \{L, R\}$,
          a finite lookup table with $\tau$-address entries.

    \item \textbf{Computation}:
          a diagram in the $\Elayer{3}$ diagrammatic sector---a
          compatible family $\{\mathcal{C}_k\}_{k \geq k_0}$,
          where $\mathcal{C}_k$ restricts the computation
          to the tape segment $\{1, \ldots, \operatorname{Prim}(k)\}$.
\end{enumerate}
The TATM is genuinely abstract: no $\Elayer{1}$ hosting is required.
Yet it lives within~$\T$'s diagrammatic sector, not within ZFC.