%
\label{prop:enrichment-iteration}
The self-enrichment of~$\tau$ iterates:
for each integer $n \geq 0$,
there is a strict $n$-category $\tau_n$
with the following structure.
\begin{enumerate}
    \item[\textup{(I1)}]
          \textbf{$n$-cells.}
          For $0 \leq k \leq n$,
          the $k$-cells of~$\tau_n$
          are elements of iterated hom-objects.
          Specifically:
          \begin{itemize}
              \item 0-cells: objects of~$\tau$.
              \item 1-cells between 0-cells $A,B$:
                    elements of $[A,B]$.
              \item 2-cells between 1-cells $f,g$:
                    elements of $[f,g] \subset [[A,B],[A,B]]$.
              \item $k$-cells: elements of the
                    $k$-fold iterated hom-object
                    $\underbrace{[\cdots[}_{k}A,B]\cdots]$.
          \end{itemize}

    \item[\textup{(I2)}]
          \textbf{Tower coherence at each level.}
          Every $k$-cell has a tower-coherent
          stagewise representation.
          At stage~$m$, the $k$-cell restricts
          to a map on the stage-$m$ version
          of the $(k-1)$-fold iterated hom-object.
          The system is compatible
          under the primorial tower projections.

    \item[\textup{(I3)}]
          \textbf{Bipolar decomposition at each level.}
          Every $k$-cell $\sigma$
          decomposes as
          $\sigma = e_+ \cdot \sigma_+ + e_- \cdot \sigma_-$,
          with the two channels independent.
          The idempotents $e_\pm$
          are central in~$H_\tau$,
          so they commute with all compositions
          at all levels.
          Consequently:
          \[
              \tau_n
              \;\cong\;
              \tau_n^+ \times \tau_n^-
          \]
          for every~$n$,
          where $\tau_n^\pm = e_\pm \cdot \tau_n$
          is the $n$-category restricted to one channel.

    \item[\textup{(I4)}]
          \textbf{Holomorphic structure at each level.}
          Every $k$-cell is $\tau$-holomorphic:
          it satisfies the split-complex
          Cauchy--Riemann conditions
          in the appropriate iterated hom-space.
          This is because each iterated hom-object
          is a $\tau$-object
          and therefore carries holomorphic structure
          by the results of Parts~VI--VII.

    \item[\textup{(I5)}]
          \textbf{Strict $n$-category axioms.}
          At each level~$k$:
          \begin{itemize}
              \item Vertical composition
                    of $k$-cells is associative.
              \item Identity $k$-cells exist.
              \item Horizontal composition
                    with $(k-1)$-cells is functorial.
              \item The interchange law holds.
          \end{itemize}
\end{enumerate}

We proceed by induction on~$n$.

\emph{Base case $n = 0$.}
$\tau_0 = \tau$ is a plain category.
All five properties hold vacuously for $k$-cells
with $k > 0$ (there are none),
and for $k = 0$ they reduce to
the fact that $\tau$ is a category
with NF-addressable objects
and $H_\tau$-valued morphisms.

\emph{Base case $n = 1$.}
$\tau_1 = \tau$ viewed as an enriched category.
The 1-cells are morphisms,
which are $\tau$-objects (II.D54),
tower-coherent (by construction of self-enrichment),
and bipolar (II.P11).
Composition and identities are given.

\emph{Inductive step $n \to n + 1$.}
Assume $\tau_n$ is a strict $n$-category
satisfying (I1)--(I5).
We construct $\tau_{n+1}$
by adding a new layer of cells.

The $(n+1)$-cells between $n$-cells
$\sigma, \sigma'$ are the elements of
$[\sigma, \sigma']$
in the iterated hom-object.
Self-enrichment (II.D53) guarantees
that this is a well-defined $\tau$-object.

\textbf{(I2):}
Tower coherence at level $n+1$
follows from tower coherence at level~$n$
plus the fact that hom-objects are tower-coherent
(II.D54 gives this for the first iteration;
the same argument applies at each level
because the hom construction
is internal to~$\tau$).

\textbf{(I3):}
Bipolar decomposition at level $n+1$
follows from II.L07 (Idempotent Decomposition Lemma).
The iterated hom-object at level $n+1$
is a $\tau$-object valued in~$H_\tau$,
so the decomposition
$\sigma = e_+ \sigma_+ + e_- \sigma_-$
applies.
Channel independence follows from
$e_+ e_- = 0$.

\textbf{(I4):}
Holomorphic structure follows because
$\tau$-objects carry holomorphic structure
(Parts~VI--VII).
The iterated hom-object is a $\tau$-object,
hence holomorphic.

\textbf{(I5):}
The strict $n$-category axioms at level $n+1$:
\begin{itemize}
    \item Vertical composition at level $n+1$
          is composition in the endomorphism space
          of the level-$n$ hom-object.
          Associativity follows from
          associativity of function composition.
    \item Identity $(n+1)$-cells
          are the identity maps
          in the endomorphism space.
    \item Horizontal composition
          is given by the action of the composition map~$\mu$
          on the new level.
          Functoriality follows from
          the enriched category axioms of~$\tau$.
    \item The interchange law
          at level $n+1$
          follows from the functoriality of~$\mu$
          at the previous level,
          by the same argument
          as in Definition~\ref{def:two-category}.
          \qedhere
\end{itemize}