%
\label{prop:functions-as-objects}
The Pre-Yoneda embedding $y$
preserves the following structures:
\begin{enumerate}
    \item[\textup{(i)}]
          \textbf{Bipolar decomposition.}
          For every $f \in \mathrm{Hol}_\tau(\tau^3, H_\tau)$:
          \[
              y(f)
              \;=\;
              e_+ \cdot y(f_+)
              \;+\;
              e_- \cdot y(f_-),
          \]
          where $f = e_+ \cdot f_+ + e_- \cdot f_-$
          is the Idempotent Decomposition
          \textup{(II.L07, Chapter~\ref{ch:idempotent-decomposition})}.

    \item[\textup{(ii)}]
          \textbf{Regularity.}
          A holomorphic function $f$ is $\tau$-regular
          at a point $p \in \tau^3$
          \textup{(Definition~\ref{def:tau-regularity}, II.D49,
          Chapter~\ref{ch:regularity-positive})}
          if and only if $y(f)$ stabilizes at~$p$:
          the $\omega$-germ $y(f)$
          has a well-defined limiting value
          at the point~$p$
          in the profinite topology.

    \item[\textup{(iii)}]
          \textbf{ABCD coordinates.}
          The image $y(f)$ inherits all four
          ABCD coordinates from $f$:
          the address data (A),
          the B-channel exponent data,
          the C-channel tetration data,
          and the depth data (D).
          A holomorphic function \textbf{has a position}
          in~$\tau^3$.
\end{enumerate}

\emph{Clause~(i): Bipolar decomposition.}
Let $f \in \mathrm{Hol}_\tau(\tau^3, H_\tau)$.
By the Idempotent Decomposition Lemma (II.L07),
$f = e_+ \cdot f_+ + e_- \cdot f_-$,
where $f_\pm = e_\pm \cdot f$
are themselves $\tau$-holomorphic.
The $\omega$-germ transformer respects this decomposition:
at each stage~$k$,
\[
    (G_f)_k
    \;=\;
    e_+ \cdot (G_{f_+})_k
    \;+\;
    e_- \cdot (G_{f_-})_k,
\]
because the idempotent projection
commutes with the stage-$k$ evaluation
(the idempotents $e_\pm$ are elements of $H_\tau$,
which is the codomain at every stage).
Passing to $\omega$-germ equivalence classes:
\[
    [G_f]
    \;=\;
    e_+ \cdot [G_{f_+}]
    \;+\;
    e_- \cdot [G_{f_-}],
\]
which is $y(f) = e_+ \cdot y(f_+) + e_- \cdot y(f_-)$.

\smallskip
\emph{Clause~(ii): Regularity.}
By the positive regularity definition
(II.D49, Chapter~\ref{ch:regularity-positive}),
$f$ is $\tau$-regular at~$p$
if and only if the stage-$k$ values $f_k(p_k)$
stabilize: there exists $N$ such that
$f_k(p_k) = f_\ell(p_\ell)$
under the natural identification
for all $k, \ell \geq N$.
But the stage-$k$ values of $f$ at~$p$
are exactly the stage-$k$ values
of the $\omega$-germ transformer $G_f$ evaluated at~$p$.
Therefore $f$ is regular at~$p$
if and only if the $\omega$-germ $[G_f] = y(f)$
stabilizes at~$p$.

The regularity criterion (II.T34,
Chapter~\ref{ch:regularity-positive})
gives the equivalence explicitly:
$f$ is regular $\Leftrightarrow$
$f$ has a canonical extension to the interior
$\Leftrightarrow$ $y(f)$
has a well-defined profinite limit at~$p$.

\smallskip
\emph{Clause~(iii): ABCD coordinates.}
The ABCD structure is inherited stagewise.
The A-coordinate (address) of $y(f)$
is the address of the stabilization point
of the $\omega$-germ:
the profinite limit of the tower-coherent sequence
$(G_f)_k$.
The B-coordinate (exponent channel)
is $e_+ \cdot y(f)$;
the C-coordinate (tetration channel)
is $e_- \cdot y(f)$;
and the D-coordinate (depth)
is the stabilization stage---the
smallest $N$ such that
the $\omega$-germ has stopped acquiring
new spectral components.
Each coordinate is well-defined
because $f$ is $\tau$-holomorphic
and therefore satisfies tower coherence
and finite spectral support
(II.T32, Chapter~\ref{ch:sheaf-coherence}).