%=============================================================================
% APPENDIX AO: THE CP SECTOR (FINAL, rev. Postulate E.1)
% Consolidated CP-sector theorem package adapted from the standalone paper
%   "The CP Sector of Density Field Dynamics: Strong-CP Protection,
%    Det-Orthogonal Geometric CP, and Euler-Projected CKM Closure" (G. Alcock).
% Macros expanded; sections demoted; ledger longtable -> table*; matrices in widetext.
%=============================================================================

\section{The CP Sector: Strong-CP Protection, the CKM No-Go, and Euler-Projected Closure}
\label{app:ckm-update}

\noindent\emph{This appendix is the consolidated CP-sector theorem package. It
supersedes the earlier $\delta_{\rm CP}\approx 68^\circ$ localization estimate
(App.~\ref{app:microsector}), the conjectural unitarity-triangle relation
$\gamma=4\pi/11$, and the post-hoc ``$19+5/2$'' apex correction recorded at
candidate grade in the companion Closure Archive; the all-orders strong-CP result
of App.~\ref{app:strongcp_selection_rule} is restated here in its anomaly form.
The apex value selected below, $\bar\rho=\tfrac{43}{2}\alpha=21.5\alpha$, coincides
with the v4.0 ``Path~A'' candidate (Theorems~T49/T121, Anti-Theorem~AT-7$'$): what
was previously preserved as an unforced conjecture is now derived as a theorem-grade
consequence of an explicitly added selection principle, the Euler-projection
Postulate~E.1 of \S\ref{app:cp-euler}. Consistent with AT-7$'$, the postulate is a
strengthening of the corpus, not a consequence of the pre-strengthened axioms.}

This appendix isolates the CP sector of Density Field Dynamics (DFD) as a self-contained theorem package within the unified DFD manuscript.  The purpose is not merely to fit the Cabibbo--Kobayashi--Maskawa (CKM) matrix, but to state exactly which parts are forced by the existing DFD geometry, which parts require a strengthened CP-sector postulate, and what quantitative predictions follow once that postulate is adopted.  We first prove that the minimal real DFD Yukawa kernel has vanishing CKM CP violation: if the up- and down-sector overlap kernels are real, then the CKM matrix is real orthogonal and the Jarlskog invariant vanishes.  We then formulate the minimal det-orthogonal CP offset: a conjugation-odd Hermitian Berry perturbation that generates weak-sector CP violation while preserving a real quark determinant and hence protecting the strong-CP angle.  The strong-CP protection theorem is stated in its anomaly form: when the CP mapping-torus Dirac spectrum is paired by an even-dimensional chirality symmetry, the Dai--Freed anomaly phase is trivial and no axion is required inside the DFD branch.

The CKM magnitude skeleton is derived from line-bundle cohomology on the \(\mathbb{C}P^2\) factor and from the real dimension of \(X=\mathbb{C}P^2\times S^3\).  The resulting integers are
\[
        \lambda=31\alpha,\qquad A=108\alpha,\qquad \bar\eta=49\alpha .
\]
The remaining apex ambiguity is treated explicitly.  The pre-strengthened DFD corpus contains an anti-theorem: existing axioms do not force the choice between the raw even-cohomology branch \(\bar\rho=22\alpha\) and the half-shift branch \(\bar\rho=21.5\alpha\).  This appendix therefore introduces Postulate E.1, the Euler-projection postulate, as a new necessary selection principle in the strengthened DFD--SD branch.  It states that the physical CP-even apex coordinate is the minimizer of
\[
        \mathcal I_\rho(r)=\frac12\left(r-\frac{D^2}{2}\right)^2+\chi(\mathbb{C}P^2)r,
        \qquad r=\bar\rho/\alpha,
\]
where \(D=\dim_\mathbb{R}(\mathbb{C}P^2\times S^3)=7\) and \(\chi(\mathbb{C}P^2)=3\).  Stationarity gives
\[
        \bar\rho=\left(\frac{7^2}{2}-3\right)\alpha=\frac{43}{2}\alpha .
\]
Thus the strengthened CP sector predicts
\[
        (\lambda,A,\bar\rho,\bar\eta)=\left(31,108,\frac{43}{2},49\right)\alpha,
\]
with \(\alpha^{-1}=137.03599985\).  Numerically this gives
\[
        \fiteq{\lambda=0.2262179284,
        \quad A=0.7881140731,
        \quad \bar\rho=0.1568930794,
        \quad \bar\eta=0.3575702739,}
\]
\[
        \fiteq{\gamma=\arctan(98/43)=66.3093^\circ,
        \quad \beta=22.9823^\circ,
        \quad \alpha_{\mathrm{UT}}=90.7084^\circ,}
\]
\[
        \fiteq{|V_{us}|=0.2262164,
        \quad |V_{cb}|=0.0403311,
        \quad |V_{ub}|=0.0036554,
        \quad |V_{td}|=0.0083576,}
\]
with exact-standard-parameterization Jarlskog invariant
\[
        J_{\mathrm{CKM}}=2.9731\times10^{-5}.
\]
The paper closes by separating theorem-grade consequences inside the strengthened DFD--SD branch from the still-testable status of Postulate E.1, and by giving a Fubini--Study overlap motivation for the postulate without claiming that the pre-strengthened axioms already prove it.





%=============================================================================
\subsection{Purpose and scope}
%=============================================================================

The CKM sector has two distinct problems.  The first is the magnitude problem: why the quark weak-mixing matrix has the observed hierarchical shape.  The second is the phase problem: why a nonzero weak CP-violating phase exists while the strong-CP parameter remains experimentally consistent with zero.  The Standard Model parametrizes both facts but does not derive the numerical CKM parameters from more primitive structure.  DFD attempts to explain the hierarchy by tying the flavor geometry to the finite internal space
\begin{equation}
        X=\mathbb{C}P^2\times S^3,
        \qquad
        D:=\dim_\mathbb{R} X=7,
\end{equation}
and to an \(\alpha\)-locked finite microsector.

This appendix is written with a strict status discipline.  We use the following terminology.

\begin{definition}[Theorem-grade inside a branch]
A statement is theorem-grade inside a named branch if it follows from explicitly stated axioms, definitions, and postulates of that branch by ordinary mathematical proof, with no phenomenological fitting hidden in the proof.
\end{definition}

\begin{definition}[Postulate-grade selection rule]
A selection rule is postulate-grade if it is introduced as a new structural assumption because it is not forced by the older axioms.  Consequences of the postulate may be theorem-grade, but the postulate itself remains a falsifiable strengthening of the branch.
\end{definition}

The central caveat is the following.

\begin{antitheorem}[No CKM apex forcing in the pre-strengthened corpus]
In the pre-strengthened DFD axioms, the CKM apex branches
\begin{equation}
        \bar\rho_A=\frac{43}{2}\alpha,
        \qquad
        \bar\rho_B=22\alpha
\end{equation}
are not uniquely separated by an existing internal symmetry.  Therefore the pre-strengthened corpus cannot claim a theorem selecting one branch over the other.
\end{antitheorem}

\begin{proof}
The older DFD CKM construction fixes the cohomological skeleton
\begin{equation}
        \lambda=31\alpha,
        \qquad
        A=108\alpha,
        \qquad
        \bar\eta=49\alpha,
\end{equation}
but the real-axis apex coordinate admits more than one natural assignment.  The raw even-cohomology assignment gives \(22\alpha\), while the half-shift or Euler-pulled assignment gives \((43/2)\alpha\).  Without a rule that distinguishes raw even cohomology from a projected complex apex, both are admissible.  This is exactly an absence-of-forcing result.
\end{proof}

The present paper resolves the ambiguity only after adding the Euler-projection postulate in Section~\ref{app:cp-euler}.  This is the correct branch statement:
\begin{equation}
\fiteq{\boxed{
\text{DFD--SD plus Euler projection derives }
(\lambda,A,\bar\rho,\bar\eta)=\left(31,108,{43\over2},49\right)\alpha .
}}
\end{equation}
It is not claimed that the original DFD axioms alone already forced the apex.

%=============================================================================
\subsection{CKM and CP notation}
%=============================================================================

The charged-current weak interaction couples up-type and down-type quarks through
\begin{equation}
        \fiteq{\mathcal L_W
        =\frac{g}{\sqrt2}\,
        \bar u_{Li}\gamma^\mu V_{ij}d_{Lj}W^+_\mu+\mathrm{h.c.},}
\end{equation}
where
\begin{equation}
        V=V_{\mathrm{CKM}}=U_u^\dagger U_d
\end{equation}
is the mismatch between the unitary transformations that diagonalize the up- and down-type quark mass matrices.  In the Wolfenstein convention,
\begin{widetext}
\begin{equation}
V_{\mathrm{CKM}}=\begin{pmatrix}
1-\lambda^2/2 & \lambda & A\lambda^3(\rho-i\eta)\\
-\lambda & 1-\lambda^2/2 & A\lambda^2\\
A\lambda^3(1-\rho-i\eta) & -A\lambda^2 & 1
\end{pmatrix}+O(\lambda^4).
\end{equation}
\end{widetext}
We use the barred unitarity-triangle coordinates \((\bar\rho,\bar\eta)\), which absorb higher-order Wolfenstein corrections in standard global-fit conventions.  At the accuracy relevant for the integer derivation, the branch is specified by
\begin{equation}
        (\lambda,A,\bar\rho,\bar\eta)
        =\left(31,108,{43\over2},49\right)\alpha .
\end{equation}
The CP-violating rephasing invariant is the Jarlskog invariant
\begin{equation}
        J_{\mathrm{CKM}}=\operatorname{Im}\left(V_{ij}V_{kl}V^*_{il}V^*_{kj}\right),
\end{equation}
which, in the standard parametrization, is
\begin{equation}
        J_{\mathrm{CKM}}=c_{12}c_{23}c_{13}^2s_{12}s_{23}s_{13}\sin\delta .
\end{equation}
Here
\begin{equation}
        \fiteq{s_{12}=\lambda,
        \qquad
        s_{23}=A\lambda^2,
        \qquad
        s_{13}=A\lambda^3\sqrt{\bar\rho^2+\bar\eta^2},
        \qquad
        \delta=\arctan\frac{\bar\eta}{\bar\rho}.}
\end{equation}

The strong-CP parameter is
\begin{equation}
        \bar\theta=\theta_{\rm YM}+\operatorname{Arg}\det(M_uM_d).
\end{equation}
A viable CP sector must generate \(J_{\mathrm{CKM}}\ne0\) while preserving \(\bar\theta=0\) or a value beneath experimental bounds.  The DFD branch in this paper accomplishes this by placing weak CP violation in a det-orthogonal Berry offset.

%=============================================================================
\subsection{DFD microsector and cohomological inputs}
%=============================================================================

\begin{axiom}[Internal flavor space]
The DFD CP/flavor microsector is modeled by
\begin{equation}
        X=\mathbb{C}P^2\times S^3,
        \qquad
        D=\dim_\mathbb{R} X=7.
\end{equation}
The \(\mathbb{C}P^2\) factor carries the holomorphic line-bundle ladder \(\mathcal{O}(k)\).  The \(S^3\) factor supplies the three-dimensional generation/phase geometry and the real internal dimension contributing to \(D=7\).
\end{axiom}

\begin{axiom}[Fine-structure normalization]
The DFD microsector fixes the dimensionless coupling \(\alpha\) used below.  Numerically,
\begin{equation}
        \alpha^{-1}=137.03599985,
        \qquad
        \alpha=0.007297352528493264.
\end{equation}
This appendix treats \(\alpha\) as already fixed by the independent DFD \(\alpha\)-lock sector.
\end{axiom}

\begin{lemma}[Line-bundle dimension on \(\mathbb{C}P^2\)]
For \(k\ge0\),
\begin{equation}
        h^0\bigl(\mathbb{C}P^2,\mathcal{O}(k)\bigr)=\frac{(k+1)(k+2)}{2}.
\end{equation}
\end{lemma}

\begin{proof}
Holomorphic sections of \(\mathcal{O}(k)\) over \(\mathbb{C}P^2\) are homogeneous polynomials of degree \(k\) in three complex homogeneous coordinates \([z_0:z_1:z_2]\).  A basis is given by monomials
\begin{equation}
        z_0^{a_0}z_1^{a_1}z_2^{a_2},
        \qquad
        a_0+a_1+a_2=k,
        \qquad
        a_i\ge0.
\end{equation}
The number of nonnegative integer triples summing to \(k\) is the stars-and-bars count
\begin{equation}
        \binom{k+2}{2}=\frac{(k+1)(k+2)}{2}.
\end{equation}
Equivalently, Hirzebruch--Riemann--Roch gives the same result from
\begin{equation}
        \fiteq{\chi(\mathbb{C}P^2,\mathcal{O}(k))=
        \int_{\mathbb{C}P^2}\mathrm{ch}(\mathcal{O}(k))\mathrm{Td}(T\mathbb{C}P^2)
        =\int_{\mathbb{C}P^2}e^{kH}\left(1+\frac32H+H^2\right),}
\end{equation}
where \(\int_{\mathbb{C}P^2}H^2=1\).  The \(H^2\)-coefficient is
\begin{equation}
        {k^2\over2}+{3k\over2}+1={ (k+1)(k+2)\over2}.
\end{equation}
\end{proof}

The specific section counts used throughout are
\begin{equation}
\begin{array}{c|cccccccc}
 k&0&1&2&3&4&5&6&7\\ \hline
 h^0(\mathcal{O}(k))&1&3&6&10&15&21&28&36
\end{array}
\end{equation}

%=============================================================================
\subsection{The minimal real-kernel no-go theorem}
\label{app:cp-no-go}
%=============================================================================

The first important theorem is negative.  DFD's minimal real K\"ahler/Yukawa sector cannot generate CKM CP violation by itself.

\begin{theorem}[Real Yukawa kernels imply \(J_{\mathrm{CKM}}=0\)]\label{thm:ckm-obstruction}
Let \(M_u,M_d\) be real quark mass matrices whose Hermitian squares
\begin{equation}
        H_u=M_uM_u^T,
        \qquad
        H_d=M_dM_d^T
\end{equation}
are real symmetric with nondegenerate spectra.  Let
\begin{equation}
        \fiteq{O_u^TH_uO_u=\operatorname{diag}(m_u^2,m_c^2,m_t^2),
        \qquad
        O_d^TH_dO_d=\operatorname{diag}(m_d^2,m_s^2,m_b^2)}
\end{equation}
with \(O_u,O_d\in O(3)\).  Then
\begin{equation}
        V_{\mathrm{CKM}}=O_u^TO_d\in O(3)
\end{equation}
is real, and hence every Jarlskog invariant vanishes:
\begin{equation}
        J_{\mathrm{CKM}}=0.
\end{equation}
\end{theorem}

\begin{proof}
A real symmetric matrix is diagonalized by a real orthogonal matrix.  Therefore the weak-basis mismatch is the product of two real orthogonal matrices:
\begin{equation}
        V=O_u^TO_d\in O(3).
\end{equation}
For any indices \(i,j,k,l\), the product
\begin{equation}
        V_{ij}V_{kl}V^*_{il}V^*_{kj}
\end{equation}
is real because \(V^*=V\).  Therefore its imaginary part vanishes:
\begin{equation}
        \operatorname{Im}(V_{ij}V_{kl}V^*_{il}V^*_{kj})=0.
\end{equation}
Thus \(J_{\mathrm{CKM}}=0\).
\end{proof}

\begin{corollary}[Need for a conjugation-odd datum]
Any DFD branch with nonzero CKM CP violation must contain an additional conjugation-odd structure beyond the minimal real Yukawa kernel.
\end{corollary}

This corollary is the reason the strengthened DFD--SD branch introduces a det-orthogonal Berry offset rather than pretending the real kernel already contains CP violation.

%=============================================================================
\subsection{Strong-CP protection}
\label{app:cp-strongcp}
%=============================================================================

The strong-CP problem is the requirement that
\begin{equation}
        \bar\theta=\theta_{\rm YM}+\operatorname{Arg}\det(M_uM_d)
\end{equation}
remain zero or extremely small.  In the present branch, weak CP is allowed only if it is det-orthogonal: it may rotate eigenvectors and generate CKM phase, but it may not create a complex determinant phase.

\begin{definition}[Det-orthogonal mass deformation]
Let \(M(\xi)=M^{(0)}+\Delta M(\xi)\) be a smooth one-parameter deformation of an invertible mass matrix.  The deformation is det-orthogonal at \(\xi=0\) if
\begin{equation}
        \left.{\,\mathrm{d}\over\,\mathrm{d}\xi}\right|_{\xi=0}\log\det M(\xi)=0.
\end{equation}
It is exactly det-orthogonal on an interval if \(\operatorname{Arg}\det M(\xi)\) is constant on that interval.
\end{definition}

\begin{lemma}[Antisymmetric imaginary perturbations are Hermitian]
Let \(S=S^T\) be a real symmetric matrix and let \(T=-T^T\) be real antisymmetric.  Then
\begin{equation}
        H(\xi)=S+i\xi T
\end{equation}
is Hermitian for real \(\xi\).
\end{lemma}

\begin{proof}
Since \(S^\dagger=S^T=S\) and \((i\xi T)^\dagger=-i\xi T^T=i\xi T\), one has \(H(\xi)^\dagger=H(\xi)\).
\end{proof}

\begin{theorem}[Determinant phase protection]
Let \(H_f(\xi)=S_f+i\xi T_f\), for \(f=u,d\), with \(S_f=S_f^T\) real symmetric positive definite and \(T_f=-T_f^T\) real antisymmetric.  Then, for sufficiently small \(|\xi|\),
\begin{equation}
        \operatorname{Arg}\det H_f(\xi)=0,
        \qquad
        \operatorname{Arg}\det\bigl(H_u(\xi)H_d(\xi)\bigr)=0.
\end{equation}
Consequently the determinant contribution to \(\bar\theta\) is exactly zero on that interval.
\end{theorem}

\begin{proof}
Each \(H_f(\xi)\) is Hermitian.  Its eigenvalues are real.  Since \(S_f\) is positive definite and eigenvalues depend continuously on \(\xi\), there exists \(\epsilon_f>0\) such that \(H_f(\xi)\) remains positive definite for \(|\xi|<\epsilon_f\).  Hence all eigenvalues are positive on this interval and
\begin{equation}
        \det H_f(\xi)>0.
\end{equation}
Therefore
\begin{equation}
        \operatorname{Arg}\det H_f(\xi)=0.
\end{equation}
Taking the product for \(f=u,d\) gives the second claim.
\end{proof}

\begin{theorem}[Trace form of first-order det-orthogonality]
Let \(M(\xi)=M^{(0)}+\xi \Delta M+O(\xi^2)\), with \(M^{(0)}\) real symmetric positive definite and \(\Delta M=iT\), \(T=-T^T\) real antisymmetric.  Then
\begin{equation}
        \left.{\,\mathrm{d}\over\,\mathrm{d}\xi}\right|_{0}\log\det M(\xi)=0.
\end{equation}
\end{theorem}

\begin{proof}
Jacobi's determinant identity gives
\begin{equation}
        \left.{\,\mathrm{d}\over\,\mathrm{d}\xi}\right|_{0}\log\det M(\xi)
        =\operatorname{Tr}\left((M^{(0)})^{-1}\Delta M\right).
\end{equation}
Since \((M^{(0)})^{-1}\) is real symmetric and \(T\) is real antisymmetric,
\begin{equation}
        \operatorname{Tr}\left((M^{(0)})^{-1}T\right)=0.
\end{equation}
Indeed,
\begin{equation}
        \fiteq{\operatorname{Tr}(ST)=\operatorname{Tr}((ST)^T)=\operatorname{Tr}(T^TS^T)=\operatorname{Tr}(-TS)=-\operatorname{Tr}(ST),}
\end{equation}
so \(\operatorname{Tr}(ST)=0\).  Multiplying by \(i\) gives zero as well.
\end{proof}

The anomaly side of strong-CP protection can be stated as follows.

\begin{definition}[CP mapping torus]
Let \(\sigma_{\mathbb{C}P}\) be the DFD CP involution on \(X\).  The CP mapping torus is
\begin{equation}
        T_{\mathbb{C}P}=(X\times[0,1])/(x,0)\sim(\sigma_{\mathbb{C}P}(x),1).
\end{equation}
Since \(\dim_\mathbb{R} X=7\),
\begin{equation}
        \dim_\mathbb{R} T_{\mathbb{C}P}=8.
\end{equation}
\end{definition}

\begin{theorem}[Even-dimensional spectral pairing implies no CP anomaly]
Assume the CP-twisted Dirac operator \(D_{T_{\mathbb{C}P}}\) on the mapping torus admits a chirality operator \(\Gamma\) such that
\begin{equation}
        \Gamma^2=1,
        \qquad
        \Gamma D_{T_{\mathbb{C}P}}\Gamma=-D_{T_{\mathbb{C}P}},
\end{equation}
and that the gauge twisting preserves this pairing.  Then the nonzero spectrum is symmetric under \(\lambda\mapsto-\lambda\), the eta invariant vanishes,
\begin{equation}
        \eta(D_{T_{\mathbb{C}P}})=0,
\end{equation}
and the Dai--Freed CP anomaly phase is trivial:
\begin{equation}
        \mathcal A_{\mathbb{C}P}=\exp\left({i\pi\over2}\eta(D_{T_{\mathbb{C}P}})\right)=1.
\end{equation}
\end{theorem}

\begin{proof}
If \(D\psi=\lambda\psi\) with \(\lambda\ne0\), then
\begin{equation}
        D(\Gamma\psi)=-\Gamma D\psi=-\lambda(\Gamma\psi).
\end{equation}
Thus every nonzero eigenvalue is paired with an eigenvalue of opposite sign and the same multiplicity.  The eta function
\begin{equation}
        \eta(s)=\sum_{\lambda\ne0}\operatorname{sgn}(\lambda)|\lambda|^{-s}
\end{equation}
therefore vanishes term by term after analytic continuation to \(s=0\).  Hence \(\eta(D)=0\), so the Dai--Freed phase is \(\exp(0)=1\).
\end{proof}

\begin{corollary}[No axion required in this branch]
If the bare Yang--Mills angle is selected as \(\theta_{\rm YM}=0\), the det-orthogonal weak CP offset preserves \(\operatorname{Arg}\det(M_uM_d)=0\), and the CP anomaly phase is trivial, then
\begin{equation}
        \bar\theta=0
\end{equation}
inside the DFD--SD branch without introducing an axion.
\end{corollary}

%=============================================================================
\subsection{Det-orthogonal geometric CP and nonzero CKM phase}
\label{app:cp-offset}
%=============================================================================

\begin{postulate}[Y.10: det-orthogonal geometric CP offset]
The strengthened DFD--SD branch contains one conjugation-odd Berry coordinate \(\xi\).  In a generation basis adapted to the real slice \(\mathbb{R}P^2\subset\mathbb{C}P^2\), the up- and down-sector Hermitian flavor kernels have the form
\begin{equation}
        H_f(\xi)=H_f^{(0)}+i\xi T_f+O(\xi^2),
        \qquad f=u,d,
\end{equation}
where \(H_f^{(0)}\) is real symmetric, \(T_f\) is real antisymmetric, and the offset direction is the \(A_5\)-Galois-odd oriented eigendirection of the Berry bundle.
\end{postulate}

\begin{theorem}[Y.10 generates weak CP generically]
Assume nondegenerate quark spectra and Y.10.  If the first nonzero variation of the commutator invariant
\begin{equation}
        \mathcal C(\xi)=\det[H_u(\xi),H_d(\xi)]
\end{equation}
at \(\xi=0\) is nonzero, then the CKM Jarlskog invariant is nonzero for sufficiently small nonzero \(\xi\):
\begin{equation}
        J_{\mathrm{CKM}}(\xi)=C\xi+O(\xi^2),
        \qquad C\ne0.
\end{equation}
\end{theorem}

\begin{proof}
For three generations, Jarlskog's determinant identity gives
\begin{equation}
        \det[H_u,H_d]
        =2i\,J_{\mathrm{CKM}}\,
        \prod_{i<j}(m_{u_i}^2-m_{u_j}^2)
        \prod_{k<l}(m_{d_k}^2-m_{d_l}^2),
\end{equation}
up to the sign convention fixed by the ordering of masses and phases.  The mass-difference products are nonzero by nondegeneracy.  Therefore \(\det[H_u,H_d]\ne0\) if and only if \(J_{\mathrm{CKM}}\ne0\).  If the first nonzero derivative of \(\mathcal C(\xi)\) at \(\xi=0\) is linear and nonzero, then \(\mathcal C(\xi)\ne0\) for sufficiently small nonzero \(\xi\), hence \(J_{\mathrm{CKM}}(\xi)\ne0\).  The coefficient \(C\) is the corresponding derivative divided by the nonzero mass-difference product.
\end{proof}

\begin{remark}
The theorem is deliberately generic.  It does not claim that any antisymmetric offset creates CP violation.  It says the Y.10 offset creates CP violation exactly when the oriented up/down misalignment has a nonvanishing commutator volume.  This is the correct mathematical condition.
\end{remark}

%=============================================================================
\subsection{Cohomological CKM magnitude skeleton}
\label{app:cp-cohomology}
%=============================================================================

The following three integers are fixed before the new Euler-projection postulate is introduced.

\begin{theorem}[Cabibbo ladder]
The DFD primitive \(1\leftrightarrow2\) flavor-connector count is
\begin{equation}
        n_\lambda=h^0(\mathcal{O}(2))+h^0(\mathcal{O}(3))+h^0(\mathcal{O}(4))=31.
\end{equation}
Thus
\begin{equation}
        \boxed{\lambda=31\alpha.}
\end{equation}
\end{theorem}

\begin{proof}
The minimal off-diagonal flavor connector excludes \(k=0\), the scalar identity channel, and \(k=1\), the generation projector triplet.  The primitive nearest-neighbor ladder therefore begins at \(k=2\).  The \(\mathbb{C}P^2\) flavor window closes at \(k=4\), the real dimension of the \(\mathbb{C}P^2\) base.  Hence
\begin{equation}
        n_\lambda= h^0(2)+h^0(3)+h^0(4)
        =6+10+15=31.
\end{equation}
Multiplying by the \(\alpha\)-locked amplitude unit gives \(\lambda=31\alpha\).
\end{proof}

\begin{theorem}[Heavy-sector doorway]
The DFD \(2\leftrightarrow3\) heavy-sector scaling count is
\begin{equation}
        n_A=N_{\rm gen}h^0(\mathcal{O}(7))=108.
\end{equation}
Thus
\begin{equation}
        \boxed{A=108\alpha.}
\end{equation}
\end{theorem}

\begin{proof}
The heavy-sector doorway samples the full real internal dimension
\begin{equation}
        D=\dim_\mathbb{R}(\mathbb{C}P^2\times S^3)=7.
\end{equation}
The line-bundle count at \(k=7\) is
\begin{equation}
        h^0(\mathcal{O}(7))={8\cdot9\over2}=36.
\end{equation}
There are three generation projectors, so
\begin{equation}
        n_A=3\cdot36=108.
\end{equation}
Thus \(A=108\alpha\).  This proof uses the cohomology count, not the inadmissible human-degree observation that a pentagon has a \(108^\circ\) interior angle.
\end{proof}

\begin{theorem}[CP-odd height count]
The CP-odd unitarity-triangle height count is
\begin{equation}
        n_\eta=D^2=49.
\end{equation}
Thus
\begin{equation}
        \boxed{\bar\eta=49\alpha.}
\end{equation}
\end{theorem}

\begin{proof}
The CP-odd coordinate measures an oriented area displacement off the totally real \(\mathbb{R}P^2\) slice.  In the DFD internal product space \(X\), the primitive dimension controlling a full oriented displacement is
\begin{equation}
        D=\dim_\mathbb{R} X=7.
\end{equation}
An area coordinate is bilinear in independent internal directions, so the maximal dimension-square count is
\begin{equation}
        n_\eta=D^2=7^2=49.
\end{equation}
Multiplication by the \(\alpha\)-locked Berry amplitude gives \(\bar\eta=49\alpha\).
\end{proof}

\begin{remark}[Why this is not enough]
The preceding results fix \(\lambda\), \(A\), and \(\bar\eta\).  They do not select the CP-even real-axis apex coordinate \(\bar\rho\).  That is exactly the old open branch.
\end{remark}

%=============================================================================
\subsection{Euler-projected selection of the CP-even apex}
\label{app:cp-euler}
%=============================================================================

\begin{postulate}[E.1: Euler projection of the CP-even apex]\label{post:E1}
In the strengthened DFD--SD branch, the physical value of the CP-even apex coordinate \(\bar\rho\) is not a free parameter and is not given by the raw even-cohomology sum.  It is the unique real-axis projection of the ideal complex apex that minimizes the reduced geometric action encoding the competition between:
\begin{enumerate}
    \item the ideal \(A_5\)-covering-space apex scale set by the full internal dimension \(D=7\), and
    \item the topological localization cost of confining the CP-even coordinate to the physical \(\mathbb{C}P^2\) microsector, measured by \(\chi(\mathbb{C}P^2)=3\).
\end{enumerate}
Equivalently, with
\begin{equation}
        \fiteq{r:=\bar\rho/\alpha,
        \qquad
        D:=\dim_\mathbb{R}(\mathbb{C}P^2\times S^3)=7,
        \qquad
        \chi:=\chi(\mathbb{C}P^2)=3,}
\end{equation}
the physical value of \(r\) is the unique minimizer of the strictly convex reduced geometric action
\begin{equation}\label{eq:Irho-E1}
        \boxed{
        \mathcal I_\rho(r)=\frac12\left(r-\frac{D^2}{2}\right)^2+\chi(\mathbb{C}P^2)r .
        }
\end{equation}
\end{postulate}

\begin{status}[Status of E.1]
Postulate E.1 is a new structural assumption of the strengthened DFD--SD branch.  It is not derived from the pre-strengthened DFD axioms and is not yet derived from explicit Berry/Yukawa overlap integrals.  Indeed the per-connector $\alpha$ unit \emph{cannot} follow from $\mathbb{C}P^2\times S^3$ geometry alone: the executed overlap integrals are coupling-blind rationals (Gram $\delta_{ab}/3$, connector moment $1/12$, Verlinde hop $2\cos(\pi/62)$), and App.~O's \emph{derived} per-mode $\alpha$ is multiplicative (it would give $\alpha^{31}$, not $31\alpha$) --- so E.1 is an independent normalization postulate (the Berry-holonomy-quantum-to-$e^2/4\pi$ identification), not an App.~O extension.  Its consequences, including \(\bar\rho=43\alpha/2\), are theorem-grade inside the strengthened branch once E.1 is accepted.
Two further derivation routes are closed (July 2026): the compact one-boson-exchange sum on $\mathbb{C}P^2$ yields $\alpha\times(\text{rational}/\pi)$ with a cutoff-divergent $O(1)$ (the $4\pi$ split $g^2=\alpha$ vs.\ $g^2=4\pi\alpha$ being conventional, cf.\ the App.~O coupling-normalization remark), and holonomy on simply-connected $\mathbb{C}P^2$ is topologically trivial (a unit-modulus phase, never an $\alpha$-sized amplitude).  No structural no-go, however, blocks E.1's eventual derivation: the live path is a selection rule forcing inter-generation connectors to couple through the internal gauge field (exchange, which scales as $\alpha^1$ per connector and is additive across connectors) rather than by coupling-blind direct overlap.
\end{status}

\begin{remark}[Meaning of the two terms]
The quadratic term anchors \(r\) to the ideal \(A_5\) covering-space apex.  The linear term is the Euler localization cost of confining the CP-even coordinate to the physical \(\mathbb{C}P^2\) microsector.  Since \(\chi(\mathbb{C}P^2)=3\), it represents one unit of \(\alpha\)-normalized real-axis localization cost per topological generation anchor.
\end{remark}

\begin{lemma}[Euler characteristic of \(\mathbb{C}P^2\)]\label{lem:Euler-CP2}
\begin{equation}
        \chi(\mathbb{C}P^2)=3.
\end{equation}
\end{lemma}

\begin{proof}
The integral cohomology of \(\mathbb{C}P^2\) is
\begin{equation}
        H^0(\mathbb{C}P^2;\mathbb{Z})=\mathbb{Z},
        \quad
        H^2(\mathbb{C}P^2;\mathbb{Z})=\mathbb{Z},
        \quad
        H^4(\mathbb{C}P^2;\mathbb{Z})=\mathbb{Z},
\end{equation}
and all odd cohomology groups vanish.  Therefore
\begin{equation}
        \chi(\mathbb{C}P^2)=b_0-b_1+b_2-b_3+b_4=1+1+1=3.
\end{equation}
Equivalently, the standard maximal torus action on \(\mathbb{C}P^2\) has three isolated fixed points,
\begin{equation}
        p_1=[1:0:0],\quad p_2=[0:1:0],\quad p_3=[0:0:1],
\end{equation}
whose Poincare--Hopf indices sum to \(3\).
\end{proof}

\begin{theorem}[Euler-projected CKM apex]\label{thm:Euler-projected-apex}
Assuming Postulate~\ref{post:E1}, the DFD--SD CP-even apex coordinate is
\begin{equation}
        \boxed{\bar\rho=\left(\frac{D^2}{2}-\chi(\mathbb{C}P^2)\right)\alpha.}
\end{equation}
For \(D=7\), this gives
\begin{equation}
        \boxed{\bar\rho=\frac{43}{2}\alpha=21.5\alpha.}
\end{equation}
\end{theorem}

\begin{proof}
The reduced action \eqref{eq:Irho-E1} is strictly convex because
\begin{equation}
        {\,\mathrm{d}^2\mathcal I_\rho\over\,\mathrm{d} r^2}=1>0.
\end{equation}
Its unique stationary point is therefore its unique global minimizer.  Compute
\begin{equation}
        {\,\mathrm{d}\mathcal I_\rho\over\,\mathrm{d} r}
        =r-{D^2\over2}+\chi(\mathbb{C}P^2).
\end{equation}
Set the derivative to zero:
\begin{equation}
        r_*={D^2\over2}-\chi(\mathbb{C}P^2).
\end{equation}
Since \(r=\bar\rho/\alpha\),
\begin{equation}
        \bar\rho=r_*\alpha=\left({D^2\over2}-\chi(\mathbb{C}P^2)\right)\alpha.
\end{equation}
For \(D=7\) and \(\chi(\mathbb{C}P^2)=3\),
\begin{equation}
        r_*={49\over2}-3={49-6\over2}={43\over2}.
\end{equation}
Thus
\begin{equation}
        \bar\rho={43\over2}\alpha.
\end{equation}
\end{proof}

\subsubsection{Fubini--Study overlap motivation for Postulate E.1}

Postulate E.1 is structurally natural from the overlap geometry, but the present paper does not claim that this motivation eliminates E.1 as an independent strengthened-branch postulate.  The point of the following argument is to show why the linear Euler term has the right geometric form.

The three real generation anchors are naturally associated with the three torus fixed points of \(\mathbb{C}P^2\):
\begin{equation}
        p_1=[1:0:0],\qquad
        p_2=[0:1:0],\qquad
        p_3=[0:0:1].
\end{equation}
By Poincare--Hopf localization,
\begin{equation}
        \sum_i \operatorname{index}(p_i)=\chi(\mathbb{C}P^2)=3.
\end{equation}
Let
\begin{equation}
        G^{(0)}_{ij}(r)=\langle w_i(r)\mid w_j(r)\rangle
\end{equation}
denote the real unperturbed Gram matrix for generation wavefunctions localized near these fixed points.  If the leading diagonal localization factor in \(\alpha\)-normalized real-axis coordinate \(r\) is
\begin{equation}
        \langle w_i(r)\mid w_i(r)\rangle\sim e^{-r},
\end{equation}
and if off-diagonal overlaps are negligible or symmetry-controlled at leading order, then
\begin{equation}
        \det G^{(0)}(r)\approx
        \prod_{i=1}^{3}\langle w_i(r)\mid w_i(r)\rangle
        \sim e^{-3r}.
\end{equation}
Taking minus the logarithm gives
\begin{equation}
        -\log\det G^{(0)}(r)\sim3r=\chi(\mathbb{C}P^2)r.
\end{equation}
This is precisely the linear Euler-localization term in \eqref{eq:Irho-E1}.

\begin{remark}[Why this remains motivation rather than proof]
A full derivation of E.1 from microscopic overlaps would have to prove the exact normalization
\begin{equation}
        \langle w_i(r)\mid w_i(r)\rangle=e^{-r}
\end{equation}
in the same \(r=\bar\rho/\alpha\) coordinate, show that the Gram determinant factorizes exactly or with a controlled correction, and derive the quadratic covering-space term from the \(A_5\) Berry/Yukawa action.  Until those steps are supplied, the Fubini--Study argument motivates E.1 but does not replace it.
\end{remark}

\begin{corollary}[Unitarity-triangle angle]
The strengthened DFD--SD branch predicts
\begin{equation}
        \boxed{
        \gamma=\arctan\left({\bar\eta\over\bar\rho}\right)
        =\arctan\left({49\over43/2}\right)
        =\arctan\left({98\over43}\right).
        }
\end{equation}
Numerically,
\begin{equation}
        \boxed{\gamma=66.3093^\circ.}
\end{equation}
\end{corollary}

\begin{proof}
Using \(\bar\eta=49\alpha\) and \(\bar\rho=(43/2)\alpha\), the factor \(\alpha\) cancels:
\begin{equation}
        {\bar\eta\over\bar\rho}={49\alpha\over(43/2)\alpha}={98\over43}.
\end{equation}
Taking the arctangent gives the result.
\end{proof}

%=============================================================================
\subsection{Final Wolfenstein branch and numerical reconstruction}
\label{app:cp-numeric}
%=============================================================================

\begin{theorem}[Strengthened DFD--SD CKM branch]
In the DFD--SD branch with Euler projection,
\begin{equation}
        \boxed{
        (\lambda,A,\bar\rho,\bar\eta)=
        \left(31,108,{43\over2},49\right)\alpha .
        }
\end{equation}
\end{theorem}

\begin{proof}
Combine the Cabibbo ladder theorem, the heavy-sector doorway theorem, the CP-odd height theorem, and the Euler-pulled apex theorem.
\end{proof}

With \(\alpha^{-1}=137.03599985\), this gives
\begin{align}
        \alpha &=0.007297352528493264,\\
        \lambda&=31\alpha=0.2262179283832912,\\
        A&=108\alpha=0.7881140730772725,\\
        \bar\rho&={43\over2}\alpha=0.1568930793626052,\\
        \bar\eta&=49\alpha=0.3575702738961699.
\end{align}

The unitarity-triangle angles are
\begin{align}
        \gamma&=\arctan\left({\bar\eta\over\bar\rho}\right)=66.3093105972^\circ,\\
        \beta&=\arctan\left({\bar\eta\over1-\bar\rho}\right)=22.9822978300^\circ,\\
        \alpha_{\mathrm{UT}}&=180^\circ-\beta-\gamma=90.7083915728^\circ.
\end{align}

For an exact-standard reconstruction, define
\begin{align}
        s_{12}&=\lambda,\\
        s_{23}&=A\lambda^2,\\
        s_{13}e^{i\delta}&=A\lambda^3(\rho+i\eta),\\
        c_{ij}&=\sqrt{1-s_{ij}^2},
\end{align}
where the unbarred apex $(\rho,\eta)$ follows from the theorem's barred (PDG-comparable) apex by the exact standard-parameterization inversion
\begin{equation}
        \rho+i\eta=
        {\sqrt{1-A^2\lambda^4}\,(\bar\rho+i\bar\eta)
        \over
        \sqrt{1-\lambda^2}\,\bigl[1-A^2\lambda^4(\bar\rho+i\bar\eta)\bigr]},
\end{equation}
a conversion of modulus $\bigl|(\rho+i\eta)/(\bar\rho+i\bar\eta)\bigr|=1.026040$.  Holding the theorem's barred values exact, this gives
\begin{align}
        s_{23}&=0.0403313839226825,\\
        s_{13}&=0.00365535209821,\\
        \delta&=66.3426441103^\circ.
\end{align}
The standard CKM matrix is
\begin{widetext}
\begin{equation}
V=\begin{pmatrix}
 c_{12}c_{13} & s_{12}c_{13} & s_{13}e^{-i\delta}\\
 -s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta} & c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta} & s_{23}c_{13}\\
 s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta} & -c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta} & c_{23}c_{13}
\end{pmatrix}.
\end{equation}
\end{widetext}
The predicted magnitudes are
\begin{equation}
        \boxed{
        |V_{\mathrm{CKM}}^{\mathrm{DFD}}|=\begin{pmatrix}
        0.9740702 & 0.2262164 & 0.0036554\\
        0.2260915 & 0.9732708 & 0.0403311\\
        0.0083576 & 0.0396246 & 0.9991797
        \end{pmatrix}.}
\end{equation}
The apex realized by this matrix, $-(V_{ud}V_{ub}^{*})/(V_{cd}V_{cb}^{*})$, reproduces the theorem values $(\bar\rho,\bar\eta)=(43/2,\,49)\,\alpha$ to $|\text{residual}|=1.1\times10^{-31}$, and its tree-level angle is $\gamma=66.3093106^\circ$, in exact agreement with the boxed falsifier.
At leading Wolfenstein order the headline entries are
\begin{align}
        |V_{us}|&=0.2262179284,\\
        |V_{cb}|&=A\lambda^2=0.0403313839,\\
        |V_{ub}|&=A\lambda^3\sqrt{\bar\rho^2+\bar\eta^2}=0.0035625840,\\
        |V_{td}|&=A\lambda^3\sqrt{(1-\bar\rho)^2+\bar\eta^2}=0.0083554489.
\end{align}
These leading-order barred expressions omit the $O(\lambda^2/2)$ barred$\to$unbarred conversion carried by the exact reconstruction above; the exact-matrix values are the ones quoted in the boxed matrix.

\begin{theorem}[Jarlskog invariant]
The strengthened DFD--SD CKM branch predicts
\begin{equation}
        \boxed{J_{\mathrm{CKM}}=2.9731\times10^{-5}}
\end{equation}
using the exact standard-parameterization expression.
\end{theorem}

\begin{proof}
The exact expression is
\begin{equation}
        J_{\mathrm{CKM}}=c_{12}c_{23}c_{13}^2s_{12}s_{23}s_{13}\sin\delta.
\end{equation}
Substituting the values above gives
\begin{equation}
        J_{\mathrm{CKM}}=2.9731063041\times10^{-5}.
\end{equation}
At leading Wolfenstein order,
\begin{equation}
        J_{\mathrm{CKM}}\approx A^2\lambda^6\bar\eta.
\end{equation}
Using the DFD integers,
\begin{equation}
        \fiteq{A^2\lambda^6\bar\eta
        =(108\alpha)^2(31\alpha)^6(49\alpha)
        =108^2\cdot31^6\cdot49\cdot\alpha^9
        =2.9764712924\times10^{-5}.}
\end{equation}
The difference between the exact and leading values is the expected higher-order trigonometric correction.
\end{proof}

\begin{figure*}[t]
\centering
\begin{tikzpicture}[scale=4.2,>=Latex]
  \coordinate (O) at (0,0);
  \coordinate (B) at (1,0);
  \coordinate (P) at (0.1568930794,0.3575702739);
  \draw[->,thick] (-0.05,0) -- (1.12,0) node[right] {$\bar\rho$};
  \draw[->,thick] (0,-0.03) -- (0,0.48) node[above] {$\bar\eta$};
  \draw[thick,blue!60!black] (O) -- (P) -- (B) -- cycle;
  \fill (O) circle (0.012) node[below left] {$0$};
  \fill (B) circle (0.012) node[below] {$1$};
  \fill[red!70!black] (P) circle (0.014) node[above right] {$\left({43\over2}\alpha,49\alpha\right)$};
  \draw[dashed,red!70!black] (P) -- (0.1568930794,0) node[below] {$\frac{43}{2}\alpha$};
  \draw[dashed,red!70!black] (P) -- (0,0.3575702739) node[left] {$49\alpha$};
  \node at (0.23,0.07) {$\gamma=66.309^\circ$};
  \node at (0.78,0.09) {$\beta=22.982^\circ$};
\end{tikzpicture}
\caption{Euler-projected DFD--SD unitarity triangle.  The apex is pulled from the ideal covering-space value \(D^2/2\) by the Euler cost \(\chi(\mathbb{C}P^2)=3\), giving \(\bar\rho=(49/2-3)\alpha=(43/2)\alpha\).}
\end{figure*}

%=============================================================================
\subsection{Comparison of surviving apex branches}
%=============================================================================

The new Euler-projected branch is not the only mathematically possible branch before the postulate is added.  The two relevant branches are:
\begin{align}
        \text{Path A:}\quad &\bar\rho_A={43\over2}\alpha=0.1568930794,\\
        \text{Path B:}\quad &\bar\rho_B=22\alpha=0.1605417556.
\end{align}
They give
\begin{align}
        \gamma_A&=\arctan(98/43)=66.3093^\circ,\\
        \gamma_B&=\arctan(49/22)=65.8209^\circ.
\end{align}
Current direct/global determinations of \(\gamma\) are not yet sharp enough to decisively separate the two.  The strengthened DFD--SD branch chooses Path A not because current data slightly prefer it, but because the Euler-projection action selects it.

\begin{center}
\begin{tabular}{lcc}
\toprule
Quantity & Path A: \(43\alpha/2\) & Path B: \(22\alpha\)\\
\midrule
\(\bar\rho\) & 0.1568931 & 0.1605418\\
\(\gamma\) & \(66.3093^\circ\) & \(65.8209^\circ\)\\
\(|V_{ub}|\) & 0.0036554 & 0.0035761$^{\rm a}$\\
\(|V_{td}|\) & 0.0083576 & 0.0083248$^{\rm a}$\\
\(J_{\rm lead}\) & \(2.9765\times10^{-5}\) & \(2.9765\times10^{-5}\)\\
\bottomrule
\end{tabular}
\par\smallskip
\parbox{0.9\linewidth}{\footnotesize $^{\rm a}$\,Path-B $|V_{ub}|$ and $|V_{td}|$ are quoted at leading (barred-Wolfenstein) order; the Path-A entries are exact standard-parameterization values from the reconstruction above.}
\end{center}

The small numerical separation is why the old anti-theorem mattered.  The theory needed a structural selection rule, not a statistical preference.

%=============================================================================
\subsection{Falsifiability and experimental targets}
%=============================================================================

The branch makes sharp predictions:
\begin{equation}
        \fiteq{\gamma=66.3093^\circ,
        \qquad
        \beta=22.9823^\circ,
        \qquad
        J_{\mathrm{CKM}}=2.9731\times10^{-5}.}
\end{equation}
The most direct discriminator is the angle \(\gamma\).  The DFD--SD value is essentially coincident with the current CKMfitter indirect global-fit central value, \(\gamma_{\rm CKMfitter}\approx 66.3^\circ\) (a difference of \(\sim0.01^\circ\)) --- the best consistency check against the full CKM unitarity structure --- and it lies well inside present \emph{direct} tree-level determinations, which are the cleaner future falsifier because they are less contaminated by loop/new-physics assumptions: the LHCb combination \(\gamma=(63.8^{+3.5}_{-3.7})^\circ\) (arXiv:2404.18789) places DFD--SD about \(2.5^\circ\) away, comfortably within \(1\sigma\), while the charm/beauty global analysis \(\gamma=(66.0\pm2.5)^\circ\) (arXiv:2409.06449) is only \(0.31^\circ\) (\(0.12\sigma\)) away.  The decisive test is whether future LHCb and Belle~II direct determinations converge near \(66.3^\circ\) as uncertainties approach the \(1^\circ\) scale.

The paper's strongest falsifier is therefore:
\begin{equation}
        \boxed{\gamma_{\mathrm{DFD}\text{--}SD}=66.3093^\circ.}
\end{equation}
If future tree-level determinations converge far from this value with sub-degree uncertainty, the Euler-projected DFD--SD CKM branch is falsified.

%=============================================================================
\subsection{Integration discipline (applied in this release)}
%=============================================================================

The following discipline was applied when integrating this appendix into the unified DFD manuscript; each item is realized in the present release.

\begin{enumerate}
\item Retain the old anti-theorem explicitly: pre-strengthened DFD does not force the apex branch.
\item Retire any unconditional claim that the old \(19\alpha\) apex is theorem-grade.
\item Introduce Postulate E.1 as a necessary strengthening of the DFD--SD branch for CKM closure; do not claim it follows from the pre-strengthened axioms.
\item Promote the consequences of that postulate to theorem-grade inside the strengthened branch.
\item Replace the CKM integer list with
\begin{equation}
        (\lambda,A,\bar\rho,\bar\eta)=\left(31,108,{43\over2},49\right)\alpha.
\end{equation}
\item State the minimal real-kernel no-go theorem before introducing Y.10.
\item State the det-orthogonal strong-CP theorem before calculating the weak CKM phase.
\item Use the \(66.3093^\circ\) value of \(\gamma\) as the headline falsifier.
\end{enumerate}

%=============================================================================
\subsection{Final theorem ledger}
%=============================================================================

\begin{table*}[t]
\centering
\caption{Final CP-sector theorem ledger for the strengthened DFD--SD branch.}
\label{tab:cp-ledger}
\begin{tabular}{@{}lll@{}}
\toprule
Claim & Result & Status\\
\midrule
\parbox[t]{0.24\linewidth}{Minimal real kernel} & \parbox[t]{0.50\linewidth}{Real Yukawa kernels imply \(J=0\).} & \parbox[t]{0.20\linewidth}{Theorem}\\[3pt]
\parbox[t]{0.24\linewidth}{Strong-CP determinant protection} & \parbox[t]{0.50\linewidth}{Hermitian det-orthogonal offsets preserve \(\operatorname{Arg}\det(M_uM_d)=0\).} & \parbox[t]{0.20\linewidth}{Theorem}\\[3pt]
\parbox[t]{0.24\linewidth}{CP anomaly} & \parbox[t]{0.50\linewidth}{Even-dimensional chirality pairing gives \(\eta=0\), Dai--Freed phase \(=1\).} & \parbox[t]{0.20\linewidth}{Conditional theorem}\\[3pt]
\parbox[t]{0.24\linewidth}{Y.10 CP offset} & \parbox[t]{0.50\linewidth}{One conjugation-odd Berry coordinate supplies weak CP while preserving determinant phase.} & \parbox[t]{0.20\linewidth}{Postulate + theorem consequences}\\[3pt]
\parbox[t]{0.24\linewidth}{Cabibbo ladder} & \parbox[t]{0.50\linewidth}{\(\lambda=[6+10+15]\alpha=31\alpha\).} & \parbox[t]{0.20\linewidth}{Theorem in branch}\\[3pt]
\parbox[t]{0.24\linewidth}{Heavy-sector doorway} & \parbox[t]{0.50\linewidth}{\(A=3h^0(7)\alpha=108\alpha\).} & \parbox[t]{0.20\linewidth}{Theorem in branch}\\[3pt]
\parbox[t]{0.24\linewidth}{CP-odd height} & \parbox[t]{0.50\linewidth}{\(\bar\eta=7^2\alpha=49\alpha\).} & \parbox[t]{0.20\linewidth}{Theorem in branch}\\[3pt]
\parbox[t]{0.24\linewidth}{Apex selection} & \parbox[t]{0.50\linewidth}{\(\bar\rho=(49/2-3)\alpha=43\alpha/2\).} & \parbox[t]{0.20\linewidth}{Postulate~E.1 + theorem consequence}\\[3pt]
\parbox[t]{0.24\linewidth}{CKM prediction} & \parbox[t]{0.50\linewidth}{\(\gamma=66.3093^\circ\), \(J=2.9731\times10^{-5}\).} & \parbox[t]{0.20\linewidth}{Falsifiable prediction}\\
\bottomrule
\end{tabular}
\end{table*}

%=============================================================================
\subsection{Conclusion}
%=============================================================================

The CP sector of DFD becomes internally coherent only after separating what the old corpus proved from what the strengthened branch adds.  The old real kernel cannot generate CKM CP violation.  That is a theorem.  The det-orthogonal Berry offset supplies a minimal weak-CP mechanism while preserving strong-CP protection.  The cohomological magnitude skeleton fixes \(\lambda=31\alpha\), \(A=108\alpha\), and \(\bar\eta=49\alpha\).  The remaining apex ambiguity is resolved only after adding Postulate E.1, the Euler-projection rule of the strengthened DFD--SD branch, whose reduced action has a unique minimizer at
\begin{equation}
        \bar\rho=\left({7^2\over2}-\chi(\mathbb{C}P^2)\right)\alpha={43\over2}\alpha.
\end{equation}
This yields the final strengthened-branch CKM solution
\begin{equation}
        \boxed{(\lambda,A,\bar\rho,\bar\eta)=\left(31,108,{43\over2},49\right)\alpha,}
\end{equation}
with
\begin{equation}
        \boxed{\gamma=66.3093^\circ,
        \qquad
        J_{\mathrm{CKM}}=2.9731\times10^{-5}.}
\end{equation}
The result is not an unconditional proof from the older DFD axioms.  It is a theorem-grade closure inside the strengthened DFD--SD branch with Euler projection.  That is the final form.
