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% Appendix AD: Regular-Module Forcing from Finite Real-Bimodule Consistency
%
% PURPOSE: Promote the microsector Hilbert-space choice H_F = M_d(C) (Branch A)
% from "locked under no-knobs policy" to "forced by the minimal faithful real-
% bimodule theorem." Branch B is excluded structurally, not numerically.
%=============================================================================

\section{Regular-Module Forcing from Finite Real-Bimodule Consistency}
\label{app:regular-module-forcing}

\subsection{AD.1\quad Purpose}

The microsector derivation of $\alpha^{-1} = 137.036$ contains a forced binary fork: the finite spectral trace may be carried either by the regular module $\mathcal{H}_F = A = M_d(\mathbb{C})$ or by a $d$-dimensional fermion-representation module $\mathcal{H}_F \simeq \mathbb{C}^d$. The numerical comparison (Table~\ref{tab:microsector-fork}) selects the regular module by 43 ppm. This appendix proves that the regular module is also the \emph{minimal} finite Hilbert space compatible with the DFD microsector's real-bimodule structure, upgrading the lock from a no-knobs convention to a structural theorem.

\subsection{AD.2\quad The Finite Algebra and Its Role}

Let
\begin{equation}
\fiteq{A = M_d(\mathbb{C}).}
\end{equation}
The DFD microsector is \emph{not} the observed fermion-representation space. It is the finite carrier of the vacuum gauge-frame algebra whose spectral trace normalizes the emergent gauge kinetic term. Gauge variations are represented by inner derivations,
\begin{equation}
\fiteq{\mathrm{ad}_a(X) = [a, X],
\qquad a \in A,\ X \in \mathcal{H}_F.}
\end{equation}

\subsection{AD.3\quad Minimal Faithful Real-Bimodule Theorem}

\begin{theorem}[Minimal faithful real-bimodule]
\label{thm:minimal-bimodule}
Let $\mathcal{H}_F$ be a finite-dimensional Hilbert space carrying a faithful left action $\pi_L: A \to B(\mathcal{H}_F)$ and a faithful right (opposite) action $\pi_R: A^{op} \to B(\mathcal{H}_F)$ satisfying
\begin{enumerate}
\item \textbf{Order-zero condition:} $[\pi_L(a), \pi_R(b)] = 0$ for all $a, b \in A$.
\item \textbf{Mutual commutant closure} (finite-factor standard form):
\begin{equation}
\fiteq{\pi_L(A)' = \pi_R(A^{op}),
\qquad
\pi_R(A^{op})' = \pi_L(A).}
\end{equation}
\end{enumerate}
Then the minimal such Hilbert space is
\begin{equation}
\fiteq{\boxed{\;
\mathcal{H}_F \simeq \mathbb{C}^d \otimes \overline{\mathbb{C}^d} \simeq M_d(\mathbb{C}),
\;}}
\end{equation}
and therefore
\begin{equation}
\fiteq{\boxed{\;
\dim \mathcal{H}_F = d^2.
\;}}
\end{equation}
\end{theorem}

\begin{proof}
Every finite-dimensional faithful representation of the simple algebra $M_d(\mathbb{C})$ is a multiple of the defining representation. Hence
\begin{equation}
\fiteq{\mathcal{H}_F \simeq \mathbb{C}^d \otimes \mathbb{C}^m,
\qquad
\pi_L(a) = a \otimes I_m.}
\end{equation}
Its commutant is
\begin{equation}
\fiteq{\pi_L(A)' = I_d \otimes M_m(\mathbb{C}).}
\end{equation}
For $\pi_R(A^{op})$ to be faithful and equal to the full commutant, the multiplicity space must contain a full copy of $M_d(\mathbb{C})^{op}$. This requires
\begin{equation}
\fiteq{m \ge d.}
\end{equation}
Minimality gives $m = d$. Therefore
\begin{equation}
\fiteq{\mathcal{H}_F \simeq \mathbb{C}^d \otimes \mathbb{C}^d \simeq \mathrm{End}(\mathbb{C}^d) \simeq M_d(\mathbb{C}).}
\end{equation}
Under this identification, $A$ acts by left multiplication and $A^{op}$ acts by right multiplication. The adjoint gauge action is
\begin{equation}
\fiteq{\mathrm{ad}_a(X) = aX - Xa = [a, X],}
\end{equation}
which is exactly the DFD regular-module microsector.
\end{proof}

\subsection{AD.4\quad Branch B Exclusion}

\begin{corollary}[Branch B is structurally invalid]
\label{cor:branch-B-excluded}
The fermion-representation branch takes
\begin{equation}
\fiteq{\mathcal{H}_F \simeq \mathbb{C}^d.}
\end{equation}
The commutant of the defining representation of $M_d(\mathbb{C})$ is only
\begin{equation}
\fiteq{\pi_L(M_d)' = \mathbb{C}.}
\end{equation}
Therefore $\mathcal{H}_F \simeq \mathbb{C}^d$ cannot carry a faithful right $M_d(\mathbb{C})^{op}$ action satisfying the standard-form real-bimodule condition. Branch B is therefore not a valid DFD vacuum microsector trace carrier.
\end{corollary}

\begin{remark}[Scope of the exclusion]
Corollary~\ref{cor:branch-B-excluded} excludes $\mathcal{H}_F \simeq \mathbb{C}^d$ \emph{only as the DFD vacuum gauge-frame trace carrier}. It does not imply that $\mathbb{C}^d$ is invalid for ordinary fermion representations elsewhere in the framework. Physical fermion fields transform in the defining representation; the vacuum trace carrier is a separate structure responsible for gauge-frame normalization.
\end{remark}

\subsection{AD.5\quad Finite-$d$ Consequence}

For the DFD Toeplitz cutoff $k_{\max} = 60$ (Lemma~\ref{lem:E_bundle_kmax60}),
\begin{equation}
\fiteq{d = k_{\max} + 4 = 64.}
\end{equation}
Hence
\begin{equation}
\fiteq{\dim \mathcal{H}_F = d^2 = 4096.}
\end{equation}

\begin{corollary}[Forced trace conversion factor]
\label{cor:branch-A-factor}
The democratic trace over $\mathcal{H}_F = M_d(\mathbb{C})$ and conversion to canonical $\mathfrak{su}(d)$ generator normalization gives
\begin{equation}
\fiteq{\boxed{\;
\varepsilon_{\rm adj}^{(A)}
=
\frac{d^2}{d^2 - 1}
=
\frac{4096}{4095}.
\;}}
\end{equation}
This is the regular-module factor used in the successful branch of the fine-structure constant derivation. The alternative factor $(d^2 - 1)/d^2 = 4095/4096$ belongs to a module that fails the DFD finite real-bimodule condition.
\end{corollary}

\begin{proof}
The regular module $M_d(\mathbb{C})$ decomposes as $M_d(\mathbb{C}) = \mathbb{C}\cdot 1 \oplus \mathfrak{su}(d)$ as a representation of the inner automorphism group, where the singlet corresponds to the identity matrix and $\mathfrak{su}(d)$ is the $(d^2 - 1)$-dimensional traceless adjoint. The democratic UV trace runs over all $d^2$ basis elements; the gauge kinetic term in canonical normalization runs only over the $\mathfrak{su}(d)$ factor. The conversion is the inverse ratio of dimensions, $d^2/(d^2 - 1)$.
\end{proof}

\subsection{AD.6\quad Status Upgrade}

\noindent
\begin{tcolorbox}[colback=green!5!white, colframe=green!60!black, title=Microsector Lock --- Theorem-Grade Status]
\textbf{Previous (v4.0 and earlier):} Branch A ($\mathcal{H}_F = M_d(\mathbb{C})$, BOOST $= 4096/4095$) adopted under the no-knobs policy because Branch B ($\mathcal{H}_F \simeq \mathbb{C}^d$, drop $= 4095/4096$) misses $\alpha^{-1}$ by 43 ppm and cannot be rescued without violating no-knobs.

\textbf{Current (this appendix, v4.0):} Branch A is \emph{forced} by the minimal faithful real-bimodule theorem (Theorem~\ref{thm:minimal-bimodule}). Branch B fails the bimodule consistency requirement (Corollary~\ref{cor:branch-B-excluded}); the 43 ppm numerical mismatch becomes a redundant consistency check rather than the basis for selection.

\textbf{The $\alpha$ derivation is now structurally locked at the module level:}
\begin{itemize}
\item Bimodule structure: $\mathcal{H}_F = M_d(\mathbb{C})$ (forced minimal faithful real-bimodule)
\item Dimension: $d^2 = 4096$
\item Trace conversion: $\varepsilon_{\rm adj}^{(A)} = 4096/4095$ (Corollary~\ref{cor:branch-A-factor})
\item Gauge action: inner derivations $\mathrm{ad}_a(X) = [a, X]$
\item $\alpha^{-1} = 137.03599985$ ($0.0056$~ppm match to experiment)
\end{itemize}

\textbf{Theorem dependencies:} order-zero $+$ faithful left/right actions $+$ mutual commutant closure $+$ minimality. These constitute a legitimate spectral-triple consistency requirement for the DFD vacuum trace carrier.

\textbf{Falsifier:} If a finite Hilbert space $\mathcal{H}_F \simeq \mathbb{C}^d$ can be shown to satisfy the faithful real-bimodule conditions (faithful left and right actions of $M_d(\mathbb{C})$ and $M_d(\mathbb{C})^{op}$ with mutual commutant closure), then Theorem~\ref{thm:minimal-bimodule} is wrong. No such construction exists in the current literature; the commutant of the defining representation is provably $\mathbb{C}$ by Schur's lemma.
\end{tcolorbox}
