%
\label{cor:holographic}
The Central Theorem is an \textbf{exact holographic correspondence}:
the $1$-dimensional boundary data
(characters on the algebraic lemniscate $\mathbb{L}$)
completely encodes the $3$-dimensional interior data
(holomorphic functions on the fibered product $\tau^3$):
\[
    \dim(\mathbb{L}) = 1
    \qquad\Longleftrightarrow\qquad
    \dim(\tau^3) = 3.
\]
The information on the boundary \emph{is}
the information in the interior.
Nothing is lost in the passage from interior to boundary.
Nothing is added in the passage from boundary to interior.
The two descriptions are the \emph{same thing},
viewed from different vantage points.

Explicitly:
\begin{enumerate}
    \item[\textup{(i)}]
          Every holomorphic function on~$\tau^3$
          is uniquely determined by its boundary restriction
          to~$\mathbb{L}$.
    \item[\textup{(ii)}]
          Every idempotent-supported character on~$\mathbb{L}$
          extends uniquely to a holomorphic function on~$\tau^3$.
    \item[\textup{(iii)}]
          The passage from boundary to interior
          and from interior to boundary
          are inverse operations.
    \item[\textup{(iv)}]
          The passage preserves all algebraic structure
          (ring operations, bipolar decomposition,
          tower grading, $\iota_\tau$-calibration).
\end{enumerate}

This is a direct restatement
of Theorem~\ref{thm:central-theorem}
in the language of information content.
Statements~(i) and~(ii) are the two directions
of the isomorphism;
statement~(iii) is the mutual inverse property;
statement~(iv) is the structure preservation.