%=============================================================================
% APPENDIX AB: UNIQUENESS OF THE INTERNAL MANIFOLD
% Deriving CP^2 x S^3 from vacuum axioms
% Integrated from: Alcock (2026), "Uniqueness of the Internal Manifold"
%=============================================================================

\section{Uniqueness of the Internal Manifold $\mathbb{C}P^2\times S^3$}\label{app:uniqueness-manifold}

The preceding appendices have used the internal manifold $X = \mathbb{C}P^2 \times S^3$ to derive Standard Model gauge structure, the generation count, the fine-structure constant, the CKM pattern, the Strong-CP vanishing, and the $G$--$H_0$ invariant. A natural question is whether this manifold is an ansatz or whether it is uniquely forced by first principles. This appendix closes that gap: we show that $X = \mathbb{C}P^2 \times S^3$ is the unique compact internal manifold satisfying a finite list of axioms motivated by the optical structure of DFD, the observed Standard Model gauge group, the observed generation count, and proton stability.

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\subsection{Vacuum Axioms}\label{app:uniq-axioms}
%-----------------------------------------------------------------------------

We seek a compact Riemannian manifold $X$ ---the internal fiber of the vacuum on which the refractive field $\psi$ lives--- satisfying:

\begin{itemize}\itemsep2pt
\item[\textbf{(V1)}] \textbf{Compactness and smoothness.} $X$ is a compact, connected, smooth, orientable Riemannian manifold without boundary.
\item[\textbf{(V2)}] \textbf{Optical dimension.} $\dim X = 7$. This is the minimal internal dimension that admits the Standard Model gauge group via an isometric partition (Lemma~\ref{lem:uniq-dim}) and is consistent with the spectral-action structure of the $\alpha^{57}$ invariant (Appendix~\ref{app:O_alpha57_clean}).
\item[\textbf{(V3)}] \textbf{Spin$^c$.} $X$ admits a Spin$^c$ structure. This is required for the Dirac operator that supplies fermion zero modes and for the Spin$^c$ Hirzebruch--Riemann--Roch index computation of Appendix~\ref{app:bridge}.
\item[\textbf{(V4)}] \textbf{Product structure.} $X = M_c \times M_g$ where $M_c$ is complex (K\"ahler) and $M_g$ is a compact Lie group or group quotient. The complex factor supplies the chiral (holomorphic) Yukawa structure, and the group factor supplies the topological protection mechanisms (proton stability, generation count, Strong-CP).
\item[\textbf{(V5)}] \textbf{Three generations.} $M_c$ has complex dimension $\dim_{\mathbb{C}} M_c \geq 2$ with $\chi_{\mathrm{top}}(M_c) = 3$, supplying the generation count via the index theorem.
\item[\textbf{(V6)}] \textbf{Gauge partition $(3,2,1)$.} The tangent bundle $TX$ admits an isometric partition $TX = V_3 \oplus V_2 \oplus V_1 \oplus V_s$ with $\dim V_i = i$, supporting the gauge group $\mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1)$ by the mechanism of Appendix~\ref{app:gauge-emergence} and Sec.~\ref{sec:gauge-coupling}.
\item[\textbf{(V7)}] \textbf{Proton stability.} The group factor $M_g$ has $\pi_3(M_g) = \mathbb{Z}$, supplying the topological winding number that defines baryon number (Theorem~\ref{thm:proton-bombproof}).
\item[\textbf{(V8)}] \textbf{Strong-CP closure.} The mapping torus $T_{\mathrm{CP}}$ of the CP involution on $X$ has even dimension, ensuring $\eta(D_{T_{\mathrm{CP}}}) = 0$ by the spectral-symmetry theorem of Appendix~\ref{app:strongcp_selection_rule}.
\item[\textbf{(V9)}] \textbf{Finite spectral rank.} The Dirac spectrum on $X$ admits a finite Toeplitz truncation at rank $k_{\max} = 60$ consistent with the Chern--Simons level quantization of the $\alpha$-derivation (Appendix~\ref{app:alpha-cs}).
\end{itemize}

\paragraph{Remark on axiom independence.}
Not all of (V1)--(V9) are logically independent. Several are consequences of the others once a candidate $X$ is chosen; the axioms are listed as a convenient \emph{sufficient} set rather than a minimal independent one. In particular, (V9) is implied by the combination (V2)+(V3)+(V4)+(V6) once the Toeplitz truncation of Appendix~\ref{app:alpha-cs} is applied, and (V8) is implied by (V2).

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\subsection{Uniqueness Theorem}\label{app:uniq-theorem}
%-----------------------------------------------------------------------------

\begin{theorem}[Uniqueness of $\mathbb{C}P^2\times S^3$]\label{thm:uniqueness}
Let $X$ be a compact manifold satisfying axioms (V1)--(V9). Then, up to isometry,
\begin{equation}\label{eq:AB-unique}
\fiteq{\boxed{X = \mathbb{C}P^2 \times S^3.}}
\end{equation}
\end{theorem}

The proof proceeds by reducing each axiom to a structural constraint and showing that the intersection of these constraints admits a unique solution.

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\subsection{Reduction of the Axioms}\label{app:uniq-reduction}
%-----------------------------------------------------------------------------

\subsubsection{Dimension, product structure, and the split $7 = 4 + 3$}

\begin{lemma}[Forced split]\label{lem:uniq-split}
Under (V1)--(V4), the factorization of $X$ is forced to be $X = M_c^{(4)} \times M_g^{(3)}$, i.e.\ $M_c$ is 4-dimensional (complex dimension 2) and $M_g$ is 3-dimensional.
\end{lemma}

\begin{proof}
(V2) requires $\dim X = 7$. (V4) requires $X = M_c \times M_g$ with $M_c$ K\"ahler (even real dimension) and $M_g$ a compact Lie group or homogeneous quotient. The possible splittings of 7 into (even) + (odd) are $2+5$, $4+3$, and $6+1$.

The splitting $6+1$ is excluded: $M_g = S^1$ has $\pi_3(S^1) = 0$, violating (V7). The splitting $2+5$ is excluded: $M_c$ would be a Riemann surface, with $\chi_{\mathrm{top}}(M_c) \in 2\mathbb{Z}$ (the genus-$g$ surface has $\chi = 2 - 2g$), contradicting (V5) which requires $\chi = 3$. The only remaining splitting is $4+3$.
\end{proof}

\subsubsection{Identification of $M_c^{(4)}$}

\begin{lemma}[K\"ahler surface with $\chi = 3$]\label{lem:uniq-Mc}
A compact K\"ahler surface with $\chi_{\mathrm{top}} = 3$ admitting a Spin$^c$ structure is isomorphic to $\mathbb{C}P^2$.
\end{lemma}

\begin{proof}
Compact K\"ahler surfaces with positive first Chern class (Fano) are classified~\cite{Hirzebruch1958}: they are $\mathbb{C}P^2$, $\mathbb{C}P^1\times\mathbb{C}P^1$, and the del Pezzo surfaces $\mathrm{dP}_k$ ($1\leq k\leq 8$). Their Euler characteristics are:
\begin{center}
\fittab{%
\begin{tabular}{lc}
Surface & $\chi_{\mathrm{top}}$ \\
\hline
$\mathbb{C}P^2$ & $3$ \\
$\mathbb{C}P^1\times\mathbb{C}P^1$ & $4$ \\
$\mathrm{dP}_k$ & $3+k$ \\
\end{tabular}%
}
\end{center}
Only $\mathbb{C}P^2$ satisfies $\chi = 3$.

Non-Fano K\"ahler surfaces (K3, Enriques, bi-elliptic, general type) have $\chi \in \{24, 12, 0, \geq 3 \text{ with } c_1^2 > 0\}$ by the Miyaoka--Yau inequality and the Noether formula. The only one with $\chi = 3$ would be a surface with $c_1^2 = 9$ and $K_X$ ample, which by the classification (Bogomolov--Miyaoka--Yau inequality saturation) would have to be the ball quotient $\mathbb{B}^2/\Gamma$; but such quotients are hyperbolic and do not admit positive Ricci curvature, inconsistent with the Fubini--Study metric required for the gauge-emergence Ricci-based stiffness derivation of Appendix~\ref{app:gauge-emergence}. Hence $M_c = \mathbb{C}P^2$.
\end{proof}

\subsubsection{Identification of $M_g^{(3)}$}

\begin{lemma}[Group factor with $\pi_3 = \mathbb{Z}$]\label{lem:uniq-Mg}
A compact, connected, simply-connected 3-dimensional Lie group with $\pi_3 = \mathbb{Z}$ is isomorphic to $S^3 = \mathrm{SU}(2)$.
\end{lemma}

\begin{proof}
The classification of compact connected 3-dimensional Lie groups gives: $T^3$, $S^1\times T^2$, $S^1\times\mathrm{SU}(2)$, and $\mathrm{SU}(2)\cong S^3$. Their homotopy:
\begin{center}
\fittab{%
\begin{tabular}{lc}
Group & $\pi_3$ \\
\hline
$T^3 = S^1\times S^1\times S^1$ & $0$ \\
$S^1\times\mathrm{SU}(2)$ & $\mathbb{Z}$ (but not simply connected) \\
$\mathrm{SU}(2) \cong S^3$ & $\mathbb{Z}$ \\
\end{tabular}%
}
\end{center}
Only $S^3$ is simply connected and has $\pi_3 = \mathbb{Z}$. The quotient $\mathrm{SO}(3) = S^3/\mathbb{Z}_2$ --- and more generally the lens family $L(p,1) = S^3/\mathbb{Z}_p$ --- also has $\pi_3 = \mathbb{Z}$ but is not simply connected. These quotients are excluded here only by the simple-connectedness \emph{hypothesis} of the lemma, which is a discrete selection, not a consequence of axioms (V1)--(V9) as written (Remark~\ref{rem:uniq-lens-selection}). Hence $M_g = S^3$ among simply connected candidates.
\end{proof}

\begin{remark}[Status: the lens-quotient family is excluded by selection, not derivation --- v4.0]\label{rem:uniq-lens-selection}
Axiom (V4) as stated admits ``a compact Lie group \emph{or group quotient},'' so the lemma above does \emph{not} close the quotient branch of (V4): every lens space $L(p,1) = S^3/\mathbb{Z}_p$ shares the universal cover $S^3$, hence has $\pi_3 = \mathbb{Z}$, and satisfies (V1)--(V8) as written ((V5) constrains only $M_c$). The level-$60$ Chern--Simons quantization underlying (V9) truncates the viable quotient family to $p \,|\, 60$, i.e.\ $p \in \{1,2,3,4,5,6,10,12,15,20,30,60\}$ --- a $12$-member discrete menu, with $p=1$ being $S^3$ itself; the countable tail $p \nmid 60$ breaks level integrality and is excluded. The corpus \emph{selects} the simply connected member $p=1$. The physical motivation --- $\pi_1$ torsion would introduce discrete hair on the baryon-number winding sectors of Theorem~\ref{thm:proton-bombproof} --- is a plausibility argument adopted as a discrete axiom-bit, not a proven no-go. Theorem~\ref{thm:uniqueness} is accordingly conditional on this selection, and the $\times 12$ menu enlargement on the $M_g$ coordinate is priced in the trials-factor audit of App.~CE (Sec.~\ref{subsec:ce-trials-audit}).
\end{remark}

\subsubsection{Consistency with remaining axioms}

\begin{lemma}[Remaining axioms satisfied]\label{lem:uniq-check}
The manifold $X = \mathbb{C}P^2 \times S^3$ satisfies (V3), (V6), (V8), (V9).
\end{lemma}

\begin{proof}
(V3) $\mathbb{C}P^2$ has Spin$^c$ structure from $c_1(\mathbb{C}P^2) = 3H$ (determinant line $\mathcal{O}(3)$); $S^3$ has Spin structure (hence Spin$^c$ a fortiori) from parallelizability. The product carries a product Spin$^c$ structure.

(V6) $T\mathbb{C}P^2 \oplus TS^3$ has rank $4 + 3 = 7$. The isometric partition is $V_3 = \mathrm{span}\{e_1,e_2,e_3\}_{S^3}$ (the full tangent of $S^3$), $V_2 = \mathrm{span}\{f_1,f_2\}_{\mathbb{C}P^1\subset\mathbb{C}P^2}$ (a linearly embedded projective line), $V_1 = \mathrm{normal}(\mathbb{C}P^1\hookrightarrow\mathbb{C}P^2)$ (complex dim 1), and $V_s$ is absent since the partition already saturates $3+2+1=6$ plus the remaining $1$ direction can be taken as $V_s$ or absorbed into the Ricci-based stiffness assignment of Lemma~\ref{lem:uniq-Mc}.

More carefully: the $7 = 3+2+1+1$ decomposition has the singlet direction given by the orientation line in the Fubini--Study form; see Appendix~\ref{app:gauge-emergence} for the explicit construction.

(V8) $\dim T_{\mathrm{CP}} = \dim X + 1 = 8$, which is even. The CP involution on $\mathbb{C}P^2$ is complex conjugation $[z_0:z_1:z_2]\mapsto[\bar z_0:\bar z_1:\bar z_2]$, an orientation-preserving isometry of the Fubini--Study metric; combined with the identity on $S^3$ it gives a smooth CP involution on $X$. By Theorem~\ref{thm:eta_even_vanishes}, $\eta(D_{T_{\mathrm{CP}}}) = 0$.

(V9) The Spin$^c$ Dirac index on $\mathbb{C}P^2$ twisted by $E = \mathcal{O}(9) \oplus \mathcal{O}^{\oplus 5}$ is $k_{\max} = 60$ by Hirzebruch--Riemann--Roch (Lemma~\ref{lem:E_bundle_kmax60}). The Toeplitz truncation of the $S^3$ Chern--Simons level at $k_{\max} = 60 = |A_5|$ closes the microsector spectral structure (Appendix~\ref{app:alpha-cs}).
\end{proof}

\subsubsection{Uniqueness assembly}

\begin{proof}[Proof of Theorem~\ref{thm:uniqueness}]
Combining Lemma~\ref{lem:uniq-split} with Lemmas~\ref{lem:uniq-Mc} and~\ref{lem:uniq-Mg} gives $X = \mathbb{C}P^2 \times S^3$ uniquely. Lemma~\ref{lem:uniq-check} then verifies that this manifold satisfies the remaining axioms (V3), (V6), (V8), (V9). Since no other manifold satisfies (V2)+(V4) and (V5)+(V7) together with the simple-connectedness selection on $M_g$ (Remark~\ref{rem:uniq-lens-selection}), the solution is unique up to isometry given that selection.
\end{proof}

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\subsection{Dimension Lemma}\label{app:uniq-dim-lemma}
%-----------------------------------------------------------------------------

\begin{lemma}[Minimal internal dimension]\label{lem:uniq-dim}
The minimal dimension for a compact internal manifold admitting the Standard Model gauge partition $(3,2,1)$ with a distinguished singlet direction is $\dim X = 7$.
\end{lemma}

\begin{proof}
The gauge partition $(3,2,1)$ requires a tangent-bundle splitting $V_3 \oplus V_2 \oplus V_1$ with $\dim V_i = i$. The minimal dimension is therefore $3+2+1 = 6$. However:
\begin{itemize}
\item $\dim = 6$ forces the partition to be the full tangent bundle with no residual direction. The singlet factor $V_1$ would then coincide with the K\"ahler angular direction of the $U(1)$ line, leaving no degree of freedom for the distinguished radial (orientation) direction needed to saturate the Higgs potential derivation (Appendix~\ref{app:higgs-yukawa}) and the orientation class that enters the CP involution of Appendix~\ref{app:strongcp_selection_rule}.
\item Moreover, any compact 6-dimensional manifold with this partition that admits the K\"ahler structure required by (V4) would have to be a Calabi--Yau threefold or a Fano threefold. The Fano threefolds are classified; none has $\chi_{\mathrm{top}} = 3$ consistent with (V5).
\item $\dim = 7$ is the minimal dimension admitting a Lie-group factor (required by (V4) and (V7)) with the required $\pi_3 = \mathbb{Z}$. The 3-dimensional $S^3$ is the unique simply connected such factor (Lemma~\ref{lem:uniq-Mg}; the quotient branch is closed by selection, Remark~\ref{rem:uniq-lens-selection}).
\end{itemize}
Hence $\dim X = 7$ is minimal.
\end{proof}

%-----------------------------------------------------------------------------
\subsection{Integer Cohomology and Standard Model Parameters}\label{app:uniq-integers}
%-----------------------------------------------------------------------------

Once $X = \mathbb{C}P^2 \times S^3$ is established, the integers appearing throughout the Standard Model parameter derivations acquire a uniform topological interpretation:

\begin{widetext}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\fittab{%
\begin{tabular}{lcl}
\toprule
Integer & Origin & Role \\
\midrule
$3$ & $\dim S^3 = \chi_{\mathrm{top}}(\mathbb{C}P^2)$ & generations, CKM integer $\times\alpha^3$ \\
$7$ & $\dim X$ & internal dimension, gauge partition \\
$8$ & $\dim T_{\mathrm{CP}}$ & Strong-CP vanishing, Higgs quartic denominator \\
$13$ & $3 + 10$ & Weinberg angle denominator ($3/13$) \\
$19$ & $h^0(\mathcal{O}(1))+h^0(\mathcal{O}(2))+h^0(\mathcal{O}(3))$ & legacy CKM $\bar\rho$ integer (superseded by $\tfrac{43}{2}$, App.~\ref{app:ckm-update}) \\
$31$ & line-bundle section count & CKM $\lambda$ integer \\
$49 = 7^2$ & $(\dim X)^2$ & CKM $\bar\eta$ integer \\
$57 = k_{\max}-N_{\mathrm{gen}}$ & Toeplitz rank minus generations & $G\hbar H_0^2/c^5$ exponent \\
$60$ & $\chi(\mathbb{C}P^2, E) = |A_5|$ & Toeplitz truncation rank \\
$108 = 3 \cdot 36$ & $N_{\mathrm{gen}} \cdot h^0$ & CKM $A$ integer \\
$137$ & spectral weighted sum & $\alpha^{-1}$ \\
\bottomrule
\end{tabular}%
}
\end{center}
\end{widetext}

These are not free parameters: they are cohomological invariants of the unique manifold \eqref{eq:AB-unique}.

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\subsection{Falsifiability}\label{app:uniq-falsify}
%-----------------------------------------------------------------------------

Theorem~\ref{thm:uniqueness} supports a sharp falsification criterion:

\begin{itemize}
\item Observation of proton decay at any rate (violating axiom (V7)) would invalidate $\pi_3(S^3) = \mathbb{Z}$ as the baryon-winding invariant.
\item Discovery of a fourth generation (violating (V5)) would invalidate $\chi_{\mathrm{top}}(\mathbb{C}P^2) = 3$.
\item A confirmed deviation of $\sin^2\theta_W$ from $3/13$ after radiative corrections, or of $\alpha^{-1}$ from 137.036 beyond systematic uncertainty, would challenge the partition (V6) or the Toeplitz closure (V9).
\item A robust detection of $\bar\theta \neq 0$ in neutron or atomic EDM experiments would falsify the even-dimensionality argument of (V8).
\item Any coherent alternative internal manifold satisfying (V1)--(V9) would falsify Theorem~\ref{thm:uniqueness} by counterexample.
\end{itemize}

\paragraph{Status.}
The uniqueness theorem is theorem-grade conditional on the axiom list (V1)--(V9) \emph{plus one discrete selection}: simple connectedness of $M_g$, which removes the $12$-member $p\,|\,60$ lens-quotient menu by choice rather than by proof (Remark~\ref{rem:uniq-lens-selection}). Each axiom is physically motivated by an observed feature of nature or a structural feature of DFD. Given the axioms and that selection, the theorem collapses what would otherwise appear to be a choice of internal manifold into a forced consequence.

%-----------------------------------------------------------------------------
\subsection{Summary}\label{app:uniq-summary}
%-----------------------------------------------------------------------------

\noindent
\begin{tcolorbox}[colback=green!5!white, colframe=green!60!black, title=Uniqueness of the Internal Manifold]
\textbf{Statement (Theorem~\ref{thm:uniqueness}):} The unique compact internal manifold satisfying DFD's optical axioms together with the observed Standard Model gauge group, generation count, and proton stability --- and the simple-connectedness selection on $M_g$ (Remark~\ref{rem:uniq-lens-selection}) --- is $X = \mathbb{C}P^2 \times S^3$.

\textbf{Implication:} The integers $\{3, 7, 8, 13, 19, 31, 49, 57, 60, 108, 137\}$ appearing throughout the Standard Model and cosmology are not free: they are cohomological invariants of this unique manifold.

\textbf{Falsifier:} Any exhibited compact manifold $X' \neq \mathbb{C}P^2\times S^3$ satisfying the same axioms, or any observation violating (V5), (V7), (V8), or (V9).

\textbf{Consequence:} Combined with the $\alpha^{57}$ dimensionless constraint (Appendix~\ref{app:O_alpha57_clean}), DFD has \emph{no continuous free parameters}; the internal geometry is fixed up to the discrete selections booked in Remark~\ref{rem:uniq-lens-selection} and priced in the trials-factor audit (Sec.~\ref{subsec:ce-trials-audit}).
\end{tcolorbox}
