\label{prop:bsd-three-ingredient}
Let $\mathcal{E}$ be a $\tau$-admissible elliptic datum: an $\Elayer{1}$ object in $\operatorname{Cat}_{\T}(\Elayer{1})$ equipped with proto-code structure (Definition~\ref{def:proto-code}, Ch.~46). Then the following three statements hold:
\begin{enumerate}
\item[\emph{(I1)}] \textbf{Rank stabilization.} The rank function $r(k) = \operatorname{rk}_{\mathbb{Z}}(G_k / G_k^{\mathrm{tor}})$ stabilizes at a finite depth $k_r \in \mathbb{N}$:
\begin{equation}
r(k) = r_\infty \quad \text{for all } k \geq k_r.
\label{eq:ch47-rank-stabilization}
\end{equation}

\item[\emph{(I2)}] \textbf{$L$-value stabilization.} The spectral determinant $L_{\T}'(1,k)$, evaluated at the central point $s = 1$ along the primorial ladder, stabilizes at a finite depth $k_L \in \mathbb{N}$:
\begin{equation}
L_{\T}'(1,k) = L_\infty' \quad \text{for all } k \geq k_L.
\label{eq:ch47-l-value-stabilization}
\end{equation}

\item[\emph{(I3)}] \textbf{Rank--$L$-value equality.} The stable rank $r_\infty$ equals the order of vanishing of $L_{\T}'(1,k)$, and the stable $L$-derivative $L_\infty'$ determines $r_\infty$:
\begin{equation}
r_\infty \;=\; \operatorname{ord}_{s=1} L_{\T}(s) \;=\; \min\{n \geq 0 : L_{\T}^{(n)}(1) \neq 0\}.
\label{eq:ch47-rank-l-equality}
\end{equation}
\end{enumerate}