\label{thm:bridge-ledger}
Let $F\colon \operatorname{Cat}_{\T}(\Elayer{2})
\to \operatorname{Mod}(\mathrm{ZFC})$ be the bridge functor
and let $\{M_{1}, \ldots, M_{5}\}$ be the five forbidden moves
(Definition~\ref{def:five-forbidden-moves}).
The eight Millennium-scale problems partition as follows.
\begin{enumerate}
\item\emph{(Conjectural.)}
For $\mathrm{RH}$, $\mathrm{NS}$, $\mathrm{YM}$,
$\mathrm{Hodge}$, $\mathrm{BSD}$, $\mathrm{Langlands}$:
Layer~1 is proved within~$\T$, Layer~2 is the standard classical
formulation, and Layer~3 is a conjectural identification.

\item\emph{(Bridge breaks.)}
For P~vs.\ NP: Layer~1 is proved
($\T$-$\mathrm{P}_{\mathrm{adm}} = \T$-$\mathrm{NP}_{\mathrm{adm}}$,
III.T33), Layer~2 is the open conjecture $\mathrm{P} \neq \mathrm{NP}$,
and Layer~3 degenerates at $M_{1}$, $M_{3}$, $M_{4}$
(Theorem~\ref{thm:move-bridge-correspondence}).
The $\T$-internal collapse does not imply $\mathrm{P} = \mathrm{NP}$.

\item\emph{(Established.)}
For Poincar\'e: Layer~1 is proved within~$\T$ (III.T24),
Layer~2 is Perelman's proof (2003),
and the two proofs are independent.
No bridge identification is needed.
\end{enumerate}