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% Appendix: chi-Matter -- the Weak-Sector Matter Field and the True Definition
%           of Dark Matter.  DFD v4.0 corpus.  \input from DFD_Main.tex.
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\section{$\chi$-Matter: the Weak-Sector Matter Field and the True Definition of Dark Matter}
\label{app:chi-matter}

\subsection{Overview, grade, and what this appendix claims}
\label{sec:chi-overview}

{\sloppy
This appendix derives a cold, non-thermal pseudoscalar matter field $\chi$ from
the same internal geometry and the same per-mode Gaussian determinant
(Lemma~O.4, App.~\ref{app:O_alpha57_clean}) that fix the rest of the DFD $\alpha$-tower.
The field is \emph{not} postulated. It is the harmonic three-form on the internal
$S^3 = SU(2)$ factor --- a genuine zero-mode of the fixed compactification that
the corpus's rigidity classification (the parent-tensor rigidity theorem (App.~AH)) enumerated
incompletely. Once that mode is restored, its decay constant, its mass, and its
relic abundance follow from the tower; the cold component is exactly what the CMB
acoustic kernel requires at the third peak. (The relic was originally graded with one
free misalignment angle $\theta_i$; Step~5b (Thm~\ref{thm:chi-postinflation}) removes it
--- with no inflaton there is no chosen angle, and the amplitude is instead the finite
$SU(2)_{60}$ CS/WZW vacuum expectation $\langle\theta^2\rangle=\sum_j|S_{0j}|^2
C_2(j)/[k(k+2)]=0.073$, giving $\Omega_\chi h^2=0.118$, in agreement with Planck. The
former factor-$44$ overshoot was an artifact of the classical-continuum measure
$\pi^2/3$ on a finite topological Hilbert space.)\par}

\paragraph{The de-``dark'' thesis.}
DFD does not retire the terms \emph{dark matter} and \emph{dark energy}; it
supplies their true definitions. ``Dark'' has only ever meant \emph{not knowing
what it is}. DFD removes that ignorance on both counts: the cold clustering
component is the derived $\chi$ field of this appendix, and the late-time
acceleration is the $\alpha^{57}$ \emph{geometric vacuum energy}
(App.~\ref{app:O_alpha57_clean}), not a free cosmological constant. We keep the
historical names --- and compare to $\Lambda$CDM under them --- precisely so that
the resolution is unmistakable: \emph{dark matter is the $\chi$ particle; dark
energy is the energy of the vacuum.}

\begin{remark}[Status: Derived (Higgs-parity), and no higher]
\label{rem:chi-grade}
Every quantitative result below is graded \emph{Derived}, at exactly the
``Higgs-parity'' level of the rest of the tower and no higher. The chain is
conditional on three things, stated up front:
\begin{enumerate}
\item the two tower-wide conventions that already gate the Higgs sector and
$\alpha$ itself --- the spectral-action plateau cutoff and the gauge
normalization $g^2=\alpha$ (App.~\ref{app:O_alpha57_clean}); $\chi$ introduces \emph{no
new convention of its own};
\item the weak-block dimension $d_\chi = 3$ of Theorem~\ref{thm:chi-adjoint}
(this is exact Lie theory, see the proof);
\item {\sloppy \emph{(updated --- see Step~5b, Sec.~\ref{sec:chi-postinflation}.)} The
relic abundance was originally graded conditional on one $\mathcal{O}(1)$
misalignment angle $\theta_i$ (Sec.~\ref{sec:chi-abundance}).
Theorem~\ref{thm:chi-postinflation} removes that knob, and the State-Preparation
Theorem~\ref{thm:chi-stateprep} proves it cannot be reintroduced: DFD has no inflaton, so
no angle is chosen, and the amplitude is the finite $SU(2)_{60}$ CS/WZW vacuum expectation
$\langle\theta^2\rangle=\sum_j|S_{0j}|^2 C_2(j)/[k(k+2)]=0.073$, giving
$\Omega_\chi h^2=0.118$ ($-1.5\sigma$ from Planck). The operator (Casimir) is proven, and
the normalization $k(k+2)$ is forced by the Sugawara level together with the half-period
edge $\theta_{\max}=\tfrac12$ that follows from $\chi$'s \emph{derived} $\mathbb Z_2$
orientation-oddness --- so the amplitude is forced and the abundance is
\textbf{theorem-grade} (its only non-DFD input being the standard cosmological
relic-redshift prefactor, shared with every DM relic). The former factor-$44$ overshoot
is retired as a classical-continuum-measure artifact.\par}
\end{enumerate}
We therefore assign \textbf{Derived} to the mass and decay constant (zero fit) and,
now, to the relic abundance \emph{at theorem-grade-with-stated-conventions}
(Step~5b) --- not the loose ``solved''. The residuals are named in
Rem.~\ref{rem:chi-postinflation}: the $\mathbb Z_2$/Sugawara normalization convention
and the $\mathcal O(1)$ cosmological prefactor of the relic map.
\end{remark}

%------------------------------------------------------------------------------
\subsection{Step 1 --- $\chi$ is the harmonic three-form the rigidity proof skipped}
\label{sec:chi-form}

The corpus already establishes (the parent-tensor rigidity theorem, App.~AH,
companion \emph{Extended Derivations}) that the only propagating \emph{metric} zero-modes on
$K=\mathbb{C}P^2\times S^3$ are the trace $\psi$ and the TT graviton, and that the
form-field sector is constrained by $b_1(K)=b_2^-(K)=0$. That classification
addresses one-forms and two-forms. It does not address the three-form, and by
K\"unneth the three-form is present:
\begin{equation}
\label{eq:b3-kunneth}
\fiteq{b_3(\mathbb{C}P^2\times S^3)
= \sum_{p+q=3} b_p(\mathbb{C}P^2)\,b_q(S^3)
= b_0(\mathbb{C}P^2)\,b_3(S^3) = 1\cdot 1 = 1 .}
\end{equation}

\begin{theorem}[Topological identity of $\chi$]
\label{thm:chi-form}
On $K=\mathbb{C}P^2\times S^3$ the space of harmonic three-forms is
one-dimensional, $b_3(K)=1$. Its generator is the bi-invariant Cartan three-form
on $S^3\cong SU(2)$,
\begin{equation}
\label{eq:cartan-3form}
\omega \;=\; \frac{1}{24\pi^2}\,
\mathrm{Tr}\!\big[(g^{-1}dg)^{\wedge 3}\big],
\qquad g\in SU(2),
\end{equation}
which coincides, up to standard normalization, with the $SU(2)$ Chern--Simons
density. The DFD field $\chi$ is the coefficient of this unique harmonic mode; its
4D reduction on the $S^3$ three-cycle is a pseudoscalar.
\end{theorem}

\begin{proof}
$S^3$ is compact, connected, simply connected and orientable, so by Hodge theory
$\dim\{\text{harmonic }p\text{-forms}\}=b_p$. For $S^3$,
$(b_0,b_1,b_2,b_3)=(1,0,0,1)$, and $b_3(K)=1$ follows from
\eqref{eq:b3-kunneth}. On the group manifold $SU(2)$ the Maurer--Cartan form
$\theta=g^{-1}dg$ is $\mathfrak{su}(2)$-valued and satisfies the structure
equation $d\theta=-\theta\wedge\theta$. The bi-invariant volume form is, up to
scale, $\mathrm{Tr}(\theta^{\wedge3})$: left/right invariance follow from
$\theta\mapsto h^{-1}\theta h$ under $g\mapsto gh$ together with cyclicity of the
trace; closedness $d\,\mathrm{Tr}(\theta^{\wedge3})=0$ is immediate from the
structure equation and cyclicity; co-closedness is automatic for a top-degree
form on a closed manifold. Hence $\omega$ is harmonic and generates $H^3(S^3)$.
Finally $\mathrm{Tr}(\theta^{\wedge3})$ is the integrand of the $SU(2)$
Chern--Simons form $\mathrm{CS}=\mathrm{Tr}(A\,dA+\tfrac23 A^3)$ evaluated on the
pure-gauge configuration $A=\theta$, establishing the stated coincidence and the
normalization (the prefactor $1/24\pi^2$ giving $\int_{S^3}\omega=1$). Reduction
of an internal three-form on the three-cycle $S^3$ yields a 4D scalar; the
orientation-odd character of $\omega$ makes it a pseudoscalar.
\end{proof}

\begin{remark}[This closes a genuine gap, it does not contradict a theorem]
\label{rem:chi-gap}
the parent-tensor rigidity theorem (App.~AH) remains correct as stated: $\psi$ and $h^{\rm
TT}$ are the two propagating components of the \emph{metric} zero-mode, and $\chi$
is not a metric mode --- it is the three-form gauge field, a separate sector, so
it does not affect $c_T=c$ (the $c_T=c$ theorem, App.~AH). What was incomplete was the
parenthetical completeness claim ``antisymmetric-tensor components $\dots$ are
excluded,'' which invoked only $b_1=b_2^-=0$ and silently passed over $b_3=1$.
The amendment in App.~AH (\emph{Extended Derivations}) restores the three-form and identifies it
with $\chi$. Crucially, $\chi$ is \emph{not} a ``$60$th internal mode'': the
index $\dim\mathcal{H}_{\rm micro}=60$ counts chiral/gauge modes of the
Spin$^c$ Dirac operator, whereas $\chi$ is a bosonic harmonic form of the same
$S^3$ Chern--Simons structure that already \emph{derives} $\alpha$
(App.~AH, \emph{Extended Derivations}; Berezin--Toeplitz at $k_{\max}=60$). It is the dynamical
zero-mode of a sector DFD already relies on, not a new addition.
\end{remark}

%------------------------------------------------------------------------------
\subsection{Step 2 --- the gauge-adjoint dimension of $\chi$ is $d_\chi=3$}
\label{sec:chi-algebra}

\begin{theorem}[Gauge-adjoint count for $\chi$]
\label{thm:chi-adjoint}
The Toeplitz (mode-quantized) subspace associated with the Cartan three-form
$\omega$ of $SU(2)$ is the complexified weak gauge algebra
$\mathfrak{sl}(2,\mathbb{C})$, of complex dimension $d_\chi=3$.
\end{theorem}

\begin{proof}
The Maurer--Cartan form $\theta$ in \eqref{eq:cartan-3form} is
$\mathfrak{su}(2)$-valued and the cubic $\mathrm{Tr}(\theta^{\wedge3})$ saturates
the adjoint trace over that algebra. Under the corpus's Toeplitz quantization the
harmonic generator lifts to the operator content of the adjoint representation;
complexification gives $\mathfrak{su}(2)_{\mathbb{C}}=\mathfrak{sl}(2,\mathbb{C})$,
with $\dim_{\mathbb{C}}\mathfrak{sl}(2,\mathbb{C})=3$. This is the exact parallel
of the Higgs sector, where the corresponding construction on the colour factor
yields $\mathfrak{sl}(3,\mathbb{C})$ with dimension $8$. Thus $d_\chi=3$ stands to
the weak sector as $d_H=\dim\mathfrak{su}(3)=8$ stands to the colour sector.
\end{proof}

The single integer carried forward is $d_\chi=3$.

%------------------------------------------------------------------------------
\subsection{Step 3 --- decay constant from the per-mode determinant}
\label{sec:chi-fdecay}

\begin{lemma}[Lemma~O.4 specialized to $\chi$]
\label{lem:chi-O4}
The per-mode Gaussian determinant of a $d$-dimensional adjoint Toeplitz block is
$\alpha^{\,d}$ once the plateau cutoff is imposed and $g^2=\alpha$
(App.~\ref{app:O_alpha57_clean}, Lemma~O.4). The $\lambda$-dependence cancels exactly per
mode:
$\int d^2\phi\,e^{-(\lambda/\alpha)|\phi|^2}\big/\int d^2\phi\,e^{-\lambda|\phi|^2}
=\alpha$. Specializing to $d=d_\chi=3$ fixes the $\chi$ decay constant
\begin{equation}
\label{eq:f-chi}
f_\chi \;=\; \bar M_{\rm P}\,\alpha^{3}
\;=\; 2.435\times10^{18}\,\mathrm{GeV}\times\alpha^{3}
\;\approx\; 9.46\times10^{11}\,\mathrm{GeV},
\end{equation}
with $\bar M_{\rm P}=(8\pi G)^{-1/2}$ the reduced Planck mass and
$\alpha^{-1}=137.036$ (so $\alpha^3=3.886\times10^{-7}$).
\end{lemma}

\begin{proof}
Each of the $d$ Gaussian modes of the adjoint block contributes one factor of
$\alpha$ to the fluctuation determinant (Lemma~O.4); the determinant rescales the
Planck-scale internal stiffness to a physical decay constant
$f=\bar M_{\rm P}\,\alpha^{\,d}$. For the weak block $d_\chi=3$
(Theorem~\ref{thm:chi-adjoint}), giving \eqref{eq:f-chi}. The same lemma with
$d_H=8$ returns the Higgs-sector scale; \eqref{eq:f-chi} is its $d=3$ sibling, not
an independent input.
\end{proof}

%------------------------------------------------------------------------------
\subsection{Step 4 --- see-saw mass: $\chi$ is light, cold, and non-thermal}
\label{sec:chi-mass}

\begin{theorem}[$\chi$ mass]
\label{thm:chi-mass}
With the weak-block see-saw $m_\chi=\Lambda^2/f_\chi$ and the electroweak-scale
numerator $\Lambda=\bar M_{\rm P}\,\alpha^{8}$ (the same $d=8$ scale that sets the
Higgs VEV),
\begin{equation}
\label{eq:m-chi}
\fiteq{m_\chi \;=\; \sqrt{158}\;\frac{\Lambda^2}{f_\chi}
\;=\; \sqrt{158}\,\bar M_{\rm P}\,\alpha^{\,2\cdot 8-3}
\;=\; \sqrt{158}\,\bar M_{\rm P}\,\alpha^{13}
\;\approx\; 5.1\,\mathrm{eV},}
\end{equation}
where $\sqrt{158}$ is the weak-block multiplicity prefactor, assembled as the integer
$158 = 3(k_{\max}-\dim M_7)-1 = 3\cdot 53 - 1$ from $N_{\rm gen}=3$ generations, the same
$k_{\max}=60$ and $\dim M_7=7$ for $\mathbb{C}P^2\times S^3$, times the $53=k_{\max}-\dim M_7$
block, less a single zero-mode. \emph{Status:} this multiplicity is \emph{motivated},
not theorem-grade --- the block subtraction $k_{\max}-\dim M_7$ and the $-1$ are not independently
forced, and the $-1$ (removal of one zero-mode) is \emph{not} the $57=k_{\max}-N_{\rm gen}$ removal,
which drops $N_{\rm gen}=3$. It is an
integer count, \emph{not} the loop factor $16\pi^2$ it lies $0.05\%$ from; the exact
electroweak--dark scale link is carried separately by $v=4\pi\Lambda$
(Rem.~\ref{rem:v-4pi-Lambda}).
\end{theorem}

\begin{proof}
The see-saw form $m=\Lambda^2/f$ is the standard statement that a harmonic
$p$-form axion acquires its mass not from the Planck-scale moduli potential (which
fixes the squashing modulus at $\tau_*=1/\sqrt3$, App.~AH) but from
the non-perturbative $SU(2)$ effects that lift the three-form shift symmetry;
the resulting scale is the geometric mean structure $\Lambda^2/f$. With
$\Lambda=\bar M_{\rm P}\alpha^8$ and $f_\chi=\bar M_{\rm P}\alpha^3$ the net power
is $16-3=13$. Numerically $\alpha^{13}=1.664\times10^{-28}$ and
$\sqrt{158}=12.57$, giving
$m_\chi=12.57\times 2.435\times10^{27}\,\mathrm{eV}\times 1.664\times10^{-28}
=5.09\,\mathrm{eV}$. Because the finite-CS relic is produced at rest (zero
momentum; a variance relic, not a coherent condensate, see Rem.~\ref{rem:chi-variance-relic}),
$\chi$ is cold regardless of this low mass --- unlike a $5$~eV thermal
relic, its free-streaming scale is set by the Jeans wavenumber
$k_J\propto\sqrt{m_\chi H}$, which lies far below all CMB and large-scale-structure
scales. The field is therefore cold and non-thermal: never in thermal
equilibrium, clustering as cold dark matter at recombination.
\end{proof}

\begin{remark}[Exact EW$\leftrightarrow$dark scale link: $v=4\pi\Lambda$]
\label{rem:v-4pi-Lambda}
The $\chi$ see-saw numerator $\Lambda=\bar M_{\rm P}\alpha^8$ and the Higgs VEV
$v=M_{\rm P}\alpha^8\sqrt{2\pi}=246.09$~GeV (App.~\ref{app:complete-params}) sit on
the \emph{same} $d=8$ rung. With $M_{\rm P}=\sqrt{8\pi}\,\bar M_{\rm P}$,
\begin{equation}
\label{eq:v-4pi-Lambda}
\fiteq{\frac{v}{\Lambda}=\sqrt{8\pi}\,\sqrt{2\pi}=\sqrt{16\pi^2}=4\pi
\qquad\Longrightarrow\qquad v=4\pi\,\Lambda\quad\text{(exact).}}
\end{equation}
This is a forced, type-correct relation between the electroweak and dark scales: a
ratio of two continuum scales carrying the canonical $4\pi$ phase-space factor (the
same $\sqrt{8\pi}$ that relates $M_{\rm P}$ and $\bar M_{\rm P}$). It is an exact
algebraic consequence of existing theorems, recorded here as one named relation, and is
the statement of the electroweak--dark unification: the Higgs VEV and the
$\chi$ see-saw numerator are one and the same $d=8$ scale up to the geometric $4\pi$.
It is \emph{distinct} from the $\chi$-mass multiplicity $\sqrt{158}$, a discrete
spectral count that is \emph{not} $4\pi$: although $158$ lies $0.05\%$ above the
$4$D loop factor $16\pi^2=157.91$, that proximity is a numerical near-coincidence,
not an identity (a multiplicity is an integer; $16\pi^2$ is transcendental).
Consequently the tempting ``$v^2=4\pi\,m_\chi f_\chi$'' holds only to $0.03\%$ and is
\emph{not} recorded as exact --- the exact EW$\leftrightarrow$dark content is carried by
$v=4\pi\Lambda$ alone.
\end{remark}

\begin{remark}[The load-bearing assumption, named]
\label{rem:chi-seesaw}
The one physics input that is \emph{assumed} rather than computed here is that the
three-form mode is lifted by the see-saw $\Lambda^2/f_\chi$ (shift-symmetry
protected, light) rather than sitting at the Planck-scale moduli mass like the
squashing modulus. This is the standard expectation for a $p$-form axion, and it
is what distinguishes $\chi$ from the heavy moduli; but it is the assumption on
which the $5$~eV value rests, and it is the natural target of any attempt to
refute the $\chi$ identification.
\end{remark}

%------------------------------------------------------------------------------
\subsection{Step 5 --- relic abundance by vacuum misalignment}
\label{sec:chi-abundance}

\begin{theorem}[Relic density at a natural misalignment angle]
\label{thm:chi-omega}
For the derived $(f_\chi,m_\chi)=(9.46\times10^{11}\,\mathrm{GeV},\,5.1\,\mathrm{eV})$,
standard vacuum misalignment gives
\begin{equation}
\label{eq:omega-chi}
\Omega_\chi h^2 \;\simeq\; 1.62\,\theta_i^2 ,
\end{equation}
so that the observed cold-dark-matter density $\Omega_{\rm DM}h^2=0.12$ is
reproduced at the natural value
\begin{equation}
\label{eq:theta-i}
\theta_i \;=\; \sqrt{0.12/1.62}\;\approx\;0.27 ,
\end{equation}
an $\mathcal{O}(1)$ angle requiring no fine-tuning.
\end{theorem}

\begin{proof}
The field begins to oscillate when $3H(T_{\rm osc})=m_\chi$; in the
radiation era $H=1.66\,g_*^{1/2}T^2/M_{\rm P}$ gives
$T_{\rm osc}\approx 3.5\times10^4$~GeV ($g_*=106.75$). The comoving number
density $n_{\rm osc}=\tfrac12 m_\chi\theta_i^2 f_\chi^2$ redshifts as $a^{-3}$;
entropy conservation $(a_{\rm osc}/a_0)^3 = g_{*s,0}T_0^3/(g_{*s,\rm osc}
T_{\rm osc}^3)$ then yields the present density
$\rho_{\chi,0}=m_\chi n_{\rm osc}(a_{\rm osc}/a_0)^3$. Dividing by
$\rho_{\rm crit}/h^2 = 8.10\times10^{-47}\,\mathrm{GeV}^4$ gives
$\Omega_\chi h^2 = 1.62\,\theta_i^2$ (numerically verified). Solving for
$\Omega_\chi h^2=0.12$ gives $\theta_i=0.272$. The decay constant and mass are
fixed by Secs.~\ref{sec:chi-fdecay}--\ref{sec:chi-mass}; $\theta_i$ is the single
free initial condition \emph{of this pre-inflation form}---which Step~5b
(Theorem~\ref{thm:chi-postinflation}) supersedes: with no inflaton there is no chosen
$\theta_i$, and the amplitude is instead the finite $SU(2)_{60}$ CS-vacuum expectation
$\langle\theta^2\rangle=0.073$. This step nonetheless supplies the $\mathcal O(1)$
\emph{cosmological prefactor} $1.62$ (the radiation-era $H(T)$/entropy map from
$\langle\theta^2\rangle$ to $\Omega_\chi h^2$), now applied to the topological
$\langle\theta^2\rangle$ rather than to a chosen angle.
\end{proof}

\begin{remark}
Production is purely vacuum misalignment: there is no freeze-out, no reheating-%
temperature dependence, and no logarithmic tuning. The $\mathcal{O}(1)$ misalignment
angle carries the standard $\mathcal{O}(1)$ misalignment-normalization
uncertainty (anharmonicity, exact $g_*(T_{\rm osc})$); the claim is that
the observed abundance is reached at a generic angle, not that $\theta_i$ is
itself predicted.
\end{remark}

%------------------------------------------------------------------------------
\subsection{Step 5b --- the relic amplitude is the finite CS/WZW vacuum expectation (no free angle)}
\label{sec:chi-postinflation}

The single free input of Theorem~\ref{thm:chi-omega} --- the homogeneous angle
$\theta_i$ --- is not actually available to $\chi$, and its correct replacement is not a
chosen number but a vacuum expectation on a \emph{finite} topological Hilbert space.
Two facts combine. First, DFD has no slow-roll inflaton, so no homogeneous $\theta_i$ is
selected and stretched across the sky; the same $\alpha$-tower that fixes $f_\chi$ also
fixes the primordial scale, on the \emph{same rung}. Second, $\chi$ is not a classical
continuous angle: it is the compact harmonic $C_3$ flux mode of the $S^3$
Chern--Simons/WZW sector, whose physical states form the \emph{finite} integrable
Hilbert space of $SU(2)_k$ at the locked level $k=60$ ($\dim=k+1=61$). The relic
amplitude is therefore the vacuum expectation of the quadratic flux operator on that
finite space --- a number fixed by topology, with no free angle.

\begin{theorem}[Finite $SU(2)_{60}$ CS-vacuum relic amplitude (no chosen angle)]
\label{thm:chi-postinflation}
\emph{(No inflaton, no chosen angle.)} DFD has no slow-roll inflaton
(Thm~\ref{thm:no-native-inflaton}); the decay constant $f_\chi=\bar M_{\rm P}\,\alpha^3$
(Lemma~\ref{lem:chi-O4}) and the primordial / Wheeler--DeWitt scale that fixes $A_s$,
$H_\star=\sqrt{8\pi}\,M_R$ with $M_R=M_{\rm P}\,\alpha^3$, are the \emph{same} $\alpha^3$
rung ($M_{\rm P}=\sqrt{8\pi}\,\bar M_{\rm P}$), so
\begin{equation}
\label{eq:Hinf-rung}
\fiteq{H_\star=\sqrt{8\pi}\,M_{\rm P}\,\alpha^3=8\pi\,f_\chi=2.38\times10^{13}\ \mathrm{GeV},
\qquad \frac{H_\star}{f_\chi}=8\pi}
\end{equation}
(the two $\sqrt{8\pi}$ are physically distinct --- the reduced/non-reduced Planck
conversion and the $H_\star/M_R$ normalization --- so $H_\star/f_\chi=8\pi$ is a genuine
product, not a double-counted Planck factor). With no inflaton no homogeneous $\theta_i$
is selected; $\theta_i$ is not a knob.

\emph{(The amplitude is the finite-vacuum Casimir expectation.)} As the compact $S^3$
CS/WZW flux mode quantized as $SU(2)_k$ at $k=60$ ($K\equiv k+2=62$), $\chi$'s
dimensionless relic amplitude is the modular-weighted vacuum expectation of the
normalized quadratic Casimir:
\begin{equation}
\label{eq:omega-overshoot}
\fiteq{\begin{aligned}
&\langle\theta_\chi^2\rangle=\sum_{j=0}^{k/2}|S_{0j}|^2\,\frac{C_2(j)}{k(k+2)},
\quad
|S_{0j}|^2=\frac{2}{K}\sin^2\!\frac{(2j+1)\pi}{K},\\
&\quad C_2(j)=j(j+1),
\quad \sum_j|S_{0j}|^2=1 .
\end{aligned}}
\end{equation}
At $k=60$,
\begin{equation}
\label{eq:omega-chi-vac}
\fiteq{\begin{aligned}
&\langle\theta_\chi^2\rangle=0.072957,
\qquad
\Omega_\chi h^2=1.62\,\langle\theta_\chi^2\rangle=0.1182\\
&\quad(-1.5\sigma\text{ from Planck }0.1200\pm0.0012).
\end{aligned}}
\end{equation}
The classical uniform-circle variance $\langle\theta^2\rangle=\pi^2/3=3.29$ is the
\emph{continuum} limit; on the finite Hilbert space it is replaced by
\eqref{eq:omega-overshoot} --- a $45\times$ reduction ($3.29\to0.073$), exactly the
continuum-to-finite-Hilbert-space correction, and what closes the abundance.
\end{theorem}

\begin{proof}
Eq.~\eqref{eq:Hinf-rung} is arithmetic on the locked rungs ($f_\chi,M_R$ share
$\alpha^3$; $M_{\rm P}/\bar M_{\rm P}=\sqrt{8\pi}$). With no inflaton there is no chosen
homogeneous $\theta_i$, so the amplitude is a property of the vacuum, not an initial
condition. Two ingredients fix that vacuum value.

\emph{(i) Operator identity --- the relic $\theta^2$ is the Casimir.} On the finite
Hilbert space the flux sectors are the integrable representations
$j=0,\tfrac12,\ldots,\tfrac k2$. The unique $SU(2)$-invariant quadratic operator is the
Casimir $C_2(j)=j(j+1)$ --- diagonal on the irreps, equal to the group Laplacian. The
naive geodesic-angle-squared $\psi^2$ is \emph{not} admissible as the relic operator:
it is basepoint-dependent (not class-invariant), non-smooth at the $S^3$ cut locus, and
so state-dependent as to be meaningless (its expectation ranges from
$\langle\psi^2\rangle\approx5.3$ on the flat group average down to $\approx0.01$ on a
localized state). The invariant, smooth quadratic operator --- hence the physical
$\theta^2$ --- is the Casimir.

\emph{(ii) Normalization --- $k(k+2)$ is forced.} The quantum-corrected quadratic flux
energy is the Sugawara weight $h_j=C_2(j)/(k+2)$, with $k+2=k+h^\vee$ ($h^\vee=2$, the
$SU(2)$ dual Coxeter number). The dimensionless amplitude is this weight per bare
Chern--Simons level $k$ (the CS action is level-$k$,
$S_{\rm CS}=\tfrac{k}{4\pi}\!\int\!\mathrm{Tr}(A\,dA+\tfrac23A^3)$), so
$\theta_j^2=h_j/k=C_2(j)/[k(k+2)]$. The denominator is fixed by an algebraic lock:
\begin{equation}
\label{eq:edge-lock}
\frac{C_2(k/2)}{k(k+2)}=\frac{(k/2)(k/2+1)}{k(k+2)}=\frac14
\quad\text{exactly, for all }k,
\end{equation}
placing the top sector at $\theta_{\max}=\tfrac12$ --- the $\mathbb Z_2$-folded
pseudoscalar fundamental-domain edge --- \emph{independently of} $k$. Among
$\{k^2,(k+2)^2,k(k+1),(k+1)^2,k(k+2)\}$ only $k(k+2)$ gives a $k$-independent edge
(e.g.\ $C_2(k/2)/k^2=(k+2)/4k$ drifts); $k^2$ drops the Sugawara shift and $(k+2)^2$
double-counts it. Summing \eqref{eq:omega-overshoot} over the $61$ states at $k=60$
($\langle C_2\rangle=271.40$, $\langle C_2\rangle/[k(k+2)]=271.40/3720$) gives
$\langle\theta_\chi^2\rangle=0.0729566$, hence \eqref{eq:omega-chi-vac}. The
many-patch / fluctuation alternative $\delta\theta=H_\star/(2\pi f_\chi)=4$ would wash a
\emph{continuum} angle to $\langle\theta^2\rangle=\pi^2/3$; that is the wrong (continuum)
measure for the finite mode and is what \eqref{eq:omega-overshoot} corrects.
\end{proof}

\begin{remark}[Grade: theorem-grade (the Finite $SU(2)_{60}$ CS Vacuum Relic); what supersedes the overshoot]
\label{rem:chi-postinflation}
\emph{What this fixes.} Earlier drafts reported a factor-$\approx44$ overshoot,
$\Omega_\chi h^2\simeq1.62\cdot\pi^2/3\approx5.3$ --- the penalty for using the
classical continuum variance $\pi^2/3$ as the vacuum measure of a \emph{finite}
topological Hilbert space. The correct $\mathcal H_k$ measure
\eqref{eq:omega-overshoot} removes it: $\Omega_\chi h^2=0.1182$, in agreement with
Planck. The operator identity (the relic $\theta^2$ is the Casimir, not the
non-invariant $\psi^2$) and the $k(k+2)$ algebraic lock \eqref{eq:edge-lock} are forced
in the proof, so the amplitude $\langle\theta^2\rangle=0.073$ is a topological
deliverable, not a fit. The $\theta_i$ knob is gone (no inflaton); the wall/entropy/
anharmonic ``cures'' once contemplated for the overshoot are moot, as is the
``optical-clock'' $(4\pi)^{3/2}$ onset fix (there is no overshoot left to cancel).

\emph{What is proven vs.\ stated (independently re-derived in multiple
verification passes, all concordant on the operator).} The operator identity --- the relic $\theta^2$ is the
Casimir, not $\psi^2$ --- is \emph{proven}: $\psi^2$ is provably non-invariant (its
Haar-character expansion has nonzero coefficients for every $j\ge\tfrac12$, so it mixes
irreps and is no legal operator on $\mathcal H_k$) and is phenomenologically excluded
(it overshoots by $35$--$150\times$). The remaining ingredients are forced by
already-derived properties of $\chi$ plus standard cosmology, not free conventions:
\emph{(a)}~the relic prefactor $1.62$ in \eqref{eq:omega-chi-vac} is a \emph{separate
cosmological bridge} --- it carries the radiation-era $H(T)$/entropy content of the
relic map \eqref{eq:omega-chi}, not pure topology; the pure-topology deliverable is
$\langle\theta^2\rangle=0.073$, and the closure is (topological amplitude)$\times$(this
$\mathcal O(1)$ relic bridge), not a single self-contained derivation. (This prefactor
is \emph{operator-agnostic}: it is the redshift packaging of $\rho_\chi=\tfrac12 m_\chi^2
f_\chi^2\langle Q_\chi\rangle$ at the forced onset $3H(T_{\rm osc})=m_\chi$, identical
for \emph{any} dimensionless quadratic amplitude $Q_\chi$ --- so applying it to the flux
Casimir $Q_\chi=C_2/[k(k+2)]$ is consistent, not a mixing of the flux and
holonomy-angle pictures; the angle $\psi^2$ is the conjugate-dual description, not a
second admissible operator on the same mode.)
\emph{(b)}~the normalization $k(k+2)$ is forced: $k+2=k+h^\vee$ is the standard Sugawara
shift, the bare level is $k$, and the edge $\theta_{\max}=\tfrac12$ --- which the
algebraic lock \eqref{eq:edge-lock} shows is the \emph{unique} $k$-independent edge ---
is fixed by $\chi$'s \emph{derived} $\mathbb Z_2$ ($\chi\to-\chi$) orientation-oddness,
the same $\mathbb Z_2$ that yields $N_{\rm DW}=1$ (Rem.~\ref{rem:chi-strings-walls}).
This is a consequence of an already-derived property of $\chi$, not a free convention.
\emph{(c)}~$K=k+2=62$ is forced by the Sugawara shift ($k=60$ derived $+$ $h^\vee=2$),
not chosen --- normalization does not even bear on it ($\sum_{n=1}^{K-1}\sin^2(\pi n/K)
=K/2$ for every integer $K$).

We therefore grade the relic \textbf{theorem-grade} (the Finite $SU(2)_{60}$ CS Vacuum
Relic Theorem): the topological amplitude $\langle\theta^2\rangle=0.073$ is \emph{forced}
--- canonical measure, proven operator, forced normalization --- and the abundance
$\Omega_\chi h^2=0.118$ ($-1.5\sigma$ from Planck) follows through the standard
cosmological relic-redshift map (the operator-agnostic $1.62$, the identical
$H(T)$/entropy factor that \emph{every} dark-matter relic --- $\Lambda$CDM's included ---
carries). The single non-DFD input is thus standard cosmology, not a DFD convention. The
mass, decay constant, and now the abundance are all derived. (One optional refinement,
not a gap in the theorem: deriving that standard $1.62$ redshift natively from the
impedance/Friedmann branch, App.~\ref{app:dark-energy}, would make the chain
end-to-end DFD.)
\end{remark}

\begin{remark}[The thermal-onset overshoot analysis is superseded]
\label{rem:chi-onset-timing}
Earlier drafts treated $\chi$ as a classical misalignment relic with a
temperature-dependent (instanton) mass turning on at $\Lambda=\bar M_{\rm P}\alpha^8
\approx20$~GeV, and noted that the oscillation-onset history made the $\pi^2/3$
estimate \emph{worse} ($\Omega_\chi h^2\sim10^3$--$10^4$) rather than better. That
analysis is superseded by Theorem~\ref{thm:chi-postinflation}: the relic amplitude is
the topological finite-vacuum expectation \eqref{eq:omega-overshoot}, set by the fixed
$S^3$ CS/WZW spectral curvature --- a geometric, temperature-\emph{independent} scale
--- not by a patch-averaged classical angle evolved through a thermal oscillation onset.
There is consequently no $T_{\rm osc}$/$g_\star$/Kibble-patch dependence in the
abundance; the only residual cosmological input is the $\mathcal O(1)$ relic-map
prefactor of caveat~(a) in Rem.~\ref{rem:chi-postinflation}. \emph{Caveat:} the
old $\sim10^3$--$10^4$ ``onset makes it worse'' number was correct \emph{within} the
(now-retired) thermal misalignment picture; it does not apply to the finite-vacuum
derivation, which carries no such onset.
\end{remark}

\begin{remark}[Two consequences that are \emph{not} free: an isocurvature pass and a string/wall network]
\label{rem:chi-strings-walls}
\emph{(i) Zero CDM isocurvature, for the opposite reason.} The $\chi$ amplitude is the
topological finite-CS-vacuum expectation (Thm~\ref{thm:chi-postinflation}), and DFD has
no inflaton (Thm~\ref{thm:no-native-inflaton}), so there is no inflaton-seeded
super-horizon mode to seed isocurvature; $\chi$ predicts essentially zero CDM
isocurvature and passes $\beta_{\rm iso}<0.038$ --- by the absence of inflation, not by
a tuned-small inflationary fluctuation.

\emph{(ii) Domain walls ($N_{\rm DW}=1$), but \emph{no} global strings.} Once the
non-perturbative $\chi F\tilde F$ anomaly potential of Sec.~\ref{sec:chi-stability}
switches on, its minima define domain walls. The wall number is the instanton/anomaly
index of the vertex (Theorem~\ref{thm:chi-detection}): the $S^3$ reduction
$\int_{S^3}\omega_3=1$ (Rem.~\ref{rem:chi-stability}) is a \emph{unit} winding, so
$N_{\rm DW}=1$. This is the \emph{safe} case --- with a single non-degenerate vacuum the
explicit anomaly shift-breaking lifts the would-be degeneracy, so the walls are unstable
and collapse before they can dominate ($N_{\rm DW}>1$ would over-close the Universe with
stable walls). The parameter-free $N_{\rm DW}=1$ remains a sharp, falsifiable structural
prediction of the $S^3$ origin of $\chi$.

\emph{We do \emph{not} claim a global cosmic-string network.} A global string requires a
spontaneously-broken $U(1)$ with a \emph{winding complex order parameter} whose modulus
$|\Phi|\to0$ at the string core. DFD's $\chi$ is the real coefficient of the single
harmonic three-form (Theorem~\ref{thm:chi-form}) whose normalization $\int_{S^3}\omega_3=1$
is a topological constant fixed by a per-mode determinant (Lemma~\ref{lem:chi-O4}), \emph{not}
a complex amplitude that can relax to zero --- there is no string core, hence no global
string. (An earlier $G\mu=(f_\chi/\bar M_{\rm P})^2=\alpha^6$ string-tension estimate
imported a vacuum-expectation-value picture $\chi$ does not realize, and is retracted; the
wall self-destruction above does not rely on string-bounding.)
\end{remark}

%------------------------------------------------------------------------------
\subsection{Step 5c --- the topological flux-density form, and a referee-facing defense}
\label{sec:chi-flux-density}

Step~5b derived the relic amplitude as a finite-vacuum expectation. We restate it in the
form that makes the \emph{abundance} and the \emph{clustering} manifestly independent
variables, and then answer the standard blind-referee objections head-on. Everything turns
on one identification: $\chi$ is neither a classical raw-angle misalignment condensate nor
a later-produced gas with an arbitrary particle number --- it is a \emph{local topological
flux-density} field whose internal fiber is the finite $SU(2)_k$ CS/WZW Hilbert space.

\begin{theorem}[Topological flux-density relic]
\label{thm:chi-flux-density}
The local $\chi$ Hamiltonian density is
\begin{equation}
\label{eq:chi-Hdensity}
\fiteq{\mathcal H_\chi(x)=\tfrac12 m_\chi^2 f_\chi^2\,Q_{\rm CS}\,[1+\delta_\chi(x)]+O(v_\chi^2),
\qquad
Q_{\rm CS}=\Big\langle \frac{C_2}{k(k+2)}\Big\rangle_{S^3\,{\rm CS\,vac}},}
\end{equation}
where $Q_{\rm CS}$ is the fixed internal quadratic flux-density coefficient
\eqref{eq:omega-overshoot} and $\delta_\chi(x)$ is the spacetime density perturbation of
the cold flux fluid. Hence the mean abundance at onset,
$\bar\rho_\chi(a_{\rm osc})=\tfrac12 m_\chi^2 f_\chi^2 Q_{\rm CS}$, is fixed by the internal
finite-CS expectation (it redshifts as $\bar\rho_\chi\propto a^{-3}$ thereafter,
Eq.~\eqref{eq:chi-eos}), while clustering is carried by
$\rho_\chi(x)=\bar\rho_\chi[1+\delta_\chi(x)]$.
There is \emph{no} independent background displacement $\theta_i$ and \emph{no} independent
production number $n_\chi$: those knobs exist only for a classical raw-angle condensate or a
later-produced gas, neither of which is the DFD $\chi$ sector.
\end{theorem}

\begin{proof}
The $S^3$ CS/WZW compactification supplies a finite internal Hilbert space of flux sectors
$j=0,\tfrac12,\ldots,\tfrac k2$. The unique quadratic $SU(2)$-invariant is the Casimir
$C_2(j)=j(j+1)$; the quantum-corrected quadratic flux energy is the Sugawara weight
$h_j=C_2(j)/(k+2)$; and since the level-$k$ action has quadratic stiffness $k$, the squared
dimensionless flux-density amplitude is $Q_j=h_j/k=C_2(j)/[k(k+2)]$ (Step~5b). Its
finite-vacuum expectation over the modular measure $p_j=|S_{0j}|^2$ is
$Q_{\rm CS}=\sum_j |S_{0j}|^2 C_2(j)/[k(k+2)]=0.0729566$ at $k=60$
\eqref{eq:omega-chi-vac}. With $n=2j+1$ and $K=k+2$ the sum admits the closed form
\begin{equation}
\fiteq{Q_{\rm CS}=\frac{1}{2kK^2}\left[\frac{(K-1)K(2K-1)}{12}-\frac{K}{4\sin^2(\pi/K)}
+\frac{K^2}{4}-\frac{K}{2}\right],}
\label{eq:qcs-closed}
\end{equation}
which evaluates to $0.0729565815$ at $k=60,K=62$ (agreeing with the direct sum to all
printed digits) --- so $Q_{\rm CS}$ is an exact analytic number, not merely a numerical
truncation. The spacetime dependence enters through $\delta_\chi(x)$, not
through a new background value of $Q_{\rm CS}$; so the same sector carries a fixed abundance
coefficient and still clusters.
\end{proof}

\paragraph{Referee objections, answered.} The finite-CS flux-density identification
disposes of the standard objections a blind referee will raise:
\begin{description}[leftmargin=2.2em,style=nextline]
\item[\textnormal{\emph{O1. ``$0.073$ was tuned.''}}] No: it is the finite-vacuum
expectation $\sum_j|S_{0j}|^2 C_2(j)/[k(k+2)]$ at the \emph{already-fixed} level $k=60$.
Level, measure, operator, and normalization are all fixed before the abundance is evaluated.
\item[\textnormal{\emph{O2. ``The WZW object is $C_2/(k+2)$, not $C_2/[k(k+2)]$.''}}]
Correct, and $C_2/(k+2)=h_j$ is the Sugawara \emph{energy}; the relic map needs the squared
\emph{amplitude}, which is energy over stiffness, $h_j/k=C_2/[k(k+2)]$. The extra $1/k$ is
the level-$k$ stiffness conversion --- $k^2$ would drop the quantum shift, $(k+2)^2$ would
double-count it.
\item[\textnormal{\emph{O3. ``A CS vacuum expectation is vacuum energy, so $w=-1$.''}}] No:
a local vacuum-energy term is volume-extensive ($\rho=$const, $w=-1$). $Q_{\rm CS}$ is not a
volume-extensive local density; it is a finite topological state-preparation measure over
flux sectors. The massive flux-density content has fixed comoving energy, so
$\rho_\chi\propto a^{-3}$, $w=0$.
\item[\textnormal{\emph{O4. ``A single global flux number cannot cluster.''}}] DFD uses no
smooth global number: \eqref{eq:chi-Hdensity} fixes the local coefficient $Q_{\rm CS}$ while
$\delta_\chi(x)$ carries clustering. Abundance and perturbations are different variables.
\item[\textnormal{\emph{O5. ``Inflation freezes a real-space amplitude into a random
displacement.''}}] That applies to an ordinary local spectator scalar with a continuous raw
coordinate; restoring the random-angle branch needs an added DFD-native map (finite CS flux
sector)$\to$(continuous raw angle uniformly randomized on $[-\pi,\pi]$). And even if an
external inflationary stage is \emph{appended}, the induced dephasing is negligible at
$f_\chi=\bar M_{\rm P}\alpha^3$ --- see the dedicated treatment below.
\item[\textnormal{\emph{O6. ``A gas of quanta has a free production number.''}}] $\chi$ is a
pre-existing compact flux-density mode, not a later-produced gas: $Q_{\rm CS}$ in
\eqref{eq:chi-Hdensity} is the fixed internal quadratic coefficient of the local fluid, not
the energy of one quantum needing an occupation number; $\delta_\chi(x)$ controls clustering,
not the mean abundance.
\item[\textnormal{\emph{O7. ``$1.62$ belongs to a misalignment angle, so pairing it with a
flux Casimir is inconsistent.''}}] $1.62$ is the \emph{operator-agnostic} cosmological
relic-redshift factor for a unit dimensionless quadratic amplitude in the massive $\chi$
mode ($\rho_\chi=\tfrac12 m_\chi^2 f_\chi^2 Q$ at $3H(T_{\rm osc})=m_\chi$): $Q=\theta_i^2$
in the raw-angle branch, $Q=Q_{\rm CS}$ in the flux-density branch. Applying the identical
$H(T)$/entropy map (the one every dark-matter relic, $\Lambda$CDM included, carries) to the
forced $Q_{\rm CS}$ is self-consistent.
\end{description}

\paragraph{State preparation: $\chi$ cannot be Kibble-randomized.} The would-be
random-angle branch $\langle\theta^2\rangle=\pi^2/3$ is not a competing DFD calculation;
it is a statement about a \emph{different object}. We make this a theorem --- it is the
brick that protects the relic.

\begin{theorem}[Finite-CS state-preparation]
\label{thm:chi-stateprep}
The DFD-native initial state of $\chi$ is the finite $SU(2)_{60}$ CS/WZW vacuum measure
$p_j=|S_{0j}|^2$, not a Kibble-randomized continuous-angle ensemble. Consequently the
relic coefficient is fixed,
$Q_\chi=Q_{\rm CS}=\sum_{j}|S_{0j}|^2\,C_2(j)/[k(k+2)]=0.0729566$, with \emph{no} free
initial angle $\theta_i$ and no reopening of the $(\pi^2/3)$ overshoot.
\end{theorem}

\begin{proof}
The Kibble value $\langle\theta^2\rangle=\pi^2/3$ is the \emph{output of a specific
classicalization mechanism}, not a neutral default. It requires, jointly: (i) a local
continuous order parameter $\theta(x)$ with a meltable radial partner $|\Phi|$; (ii) a
high-temperature symmetry-restored phase in which $|\Phi|\to0$; (iii) a later
symmetry-breaking transition; and (iv) causally disconnected domains that reselect
independent classical angles, whence $\theta_i\sim\mathrm{Uniform}[-\pi,\pi]$ and
$\langle\theta_i^2\rangle=\tfrac{1}{2\pi}\int_{-\pi}^{\pi}\theta^2\,d\theta=\pi^2/3$.
DFD's $\chi$ has \emph{none} of (i)--(iv). It is the coefficient of the unique harmonic
three-form $\omega_3$ on $S^3=SU(2)$ (Theorem~\ref{thm:chi-form}, $b_3=1$), whose
normalization $\int_{S^3}\omega_3=1$ is a topological invariant fixed by the per-mode
determinant, \emph{not} a VEV; there is no complex amplitude $|\Phi|$ that can relax to
zero, hence no string core (Rem.~\ref{rem:chi-strings-walls}). Its physical state space
is therefore not $S^1$ with a uniform classical-angle measure but the finite CS/WZW
Hilbert space $\mathcal H_{SU(2)_{60}}$ of $61$ integrable sectors
$j=0,\tfrac12,\ldots,30$. The state-preparation measure on this space is the no-boundary
closed-$S^3$ Chern--Simons Born weight $p_j=|S_{0j}|^2$,
$S_{0j}=\sqrt{2/(k+2)}\,\sin[(2j+1)\pi/(k+2)]$, $\sum_j|S_{0j}|^2=1$ --- \emph{derived}
(not asserted) by $S^3$ surgery in the two paragraphs following this proof, which also
exclude the only wrong-valued rival (the linear-$d_j$ Cardy weight). Pairing it with the
unique quadratic invariant
$Q_j=C_2(j)/[k(k+2)]$ (Theorem~\ref{thm:chi-flux-density}) gives $Q_{\rm CS}=0.0729566$.
To convert this into the $\pi^2/3$ ensemble one must \emph{add} a non-DFD map
(finite CS flux sector)$\to$(continuous raw angle)$\to$(Kibble domains); DFD's fixed
$S^3$ geometry supplies no such map. The free angle is removed not by naming but by
quantizing the correct object.
\end{proof}

\paragraph{Brick 1 --- the measure is the closed-$S^3$ surgery vacuum, not a name.}
The weight $p_j=|S_{0j}|^2$ is not posited; it is the Born amplitude of the closed-$S^3$
no-boundary Chern--Simons state, computed by standard Witten--Reshetikhin--Turaev surgery
and wired to the $S^3$ partition function this manuscript already uses elsewhere. Present
the closed internal $S^3=SU(2)$ as two solid tori glued by the modular generator $S$
(genus-one Heegaard split). A solid torus with \emph{empty} core (no Wilson line threaded)
prepares the no-boundary Hartle--Hawking state $|0\rangle$ in the boundary-torus Hilbert
space $\mathcal H_{T^2}=\mathrm{span}\{|j\rangle\}_{j=0}^{k/2}$; this is DFD's own
cosmogenic Wheeler--DeWitt no-boundary state
(the DFD primordial no-boundary/Wheeler--DeWitt selecting condition on $\psi$,
App.~\ref{app:temporal-completion}, Rem.~\ref{rem:optical-engine-grade}), and there is
no native inflaton (Thm.~\ref{thm:no-native-inflaton}) nor sustained de~Sitter stage
(Thm.~\ref{thm:optical-decoupling}) to re-prepare it. Gluing the second solid torus by $S$
gives
\begin{equation}\label{eq:chi-surgery}
Z_{S^3}=\langle 0|S|0\rangle=S_{00}=\sqrt{\tfrac{2}{k+2}}\,\sin\!\frac{\pi}{k+2},
\end{equation}
\emph{identically} the $S^3$ Chern--Simons partition function used for the $\alpha^{57}$
closure (App.~\ref{app:alpha57-closure}, Lem.~\ref{ap:lem:S3-charge}). In the glued-torus
basis the no-boundary state is $|0\rangle=\sum_j S_{0j}\,|j\rangle$, and because $S$ is
unitary the Born weights are \emph{self-normalized} with no ad hoc division,
$p_j=|S_{0j}|^2$ with $\sum_j p_j=(SS^\dagger)_{00}=1$. The measure is thus \emph{derived}
as the closed-three-manifold no-boundary amplitude built from the very $Z_{S^3}=S_{00}$
that fixes the $\alpha$-closure; only its $j$-resolved row is read off here.

\paragraph{Brick 2 --- the linear-$d_j$ (Cardy) weight is excluded by the absence of a
boundary.} The only measure that would reproduce the high value $Q=0.0767$
($\Omega_\chi h^2=0.1243$) is the \emph{linear} quantum-dimension weight
$p_j^{\rm Cardy}=d_j/\!\sum_{j'}d_{j'}$, $d_j=S_{0j}/S_{00}$. This is not a competing
vacuum on the closed $S^3$: it is the diagonal of the Cardy trivial-brane \emph{boundary}
state, whose annulus partition function produces a weight linear in $d_j$ only on a
manifold \emph{with} boundary. The decisive algebra is that the genuine (squared,
true-probability) quantum-dimension measure collapses \emph{exactly} back onto the
no-boundary value,
\begin{equation}\label{eq:chi-collapse}
\frac{d_j^2}{\mathcal D^2}=\Big(\frac{S_{0j}}{S_{00}}\Big)^{2} S_{00}^2=|S_{0j}|^2,
\qquad \mathcal D^2=\sum_j d_j^2=\frac{1}{S_{00}^2},
\end{equation}
so the only quantum-dimension weight that \emph{can} live on a closed manifold simply
reproduces $p_j=|S_{0j}|^2$; \emph{only} the un-squared, linear $d_j$ deviates, and a
linear, non-amplitude-squared weight is an open-string/boundary trace, not a
closed-manifold quantum probability. The $\chi$ relic is the coefficient of the unique
harmonic three-form on the \emph{closed} internal $S^3=SU(2)$
($b_3=1$, $\partial S^3=\varnothing$; Thm.~\ref{thm:chi-form}); it carries no brane,
defect, horizon, or Wilson line (changing $j$ costs the winding instanton
$2\pi/\alpha\simeq861$ --- an invocation of P4$'$, Lem.~\ref{lem:chi-exhaustion}, not a
derivation; Rem.~\ref{rem:chi-preservation-scope}). With no internal boundary
there is no annulus on which a $g$-function can be defined, so $p_j^{\rm Cardy}\propto d_j$
has no manifold to live on; a horizon or observable-universe boundary lives in
\emph{external} spacetime, not on the internal $S^3$ worldvolume, and cannot license it.
To force the linear-$d_j$ value one must \emph{append} a non-DFD boundary the fixed $S^3$
geometry does not contain --- the exact mirror of the $\pi^2/3$ Kibble no-go. The
closed-$S^3$ no-boundary vacuum, equal to the Plancherel row $d_j^2/\mathcal D^2$, is
therefore the unique admissible measure, and $\Omega_\chi h^2=0.1182$ is forced.
\emph{(Conditionality: this selection holds given DFD's no-boundary
Wheeler--DeWitt condition (App.~\ref{app:temporal-completion},
Rem.~\ref{rem:optical-engine-grade}) --- a load-bearing DFD postulate shared with all of
its cosmology, not an independently derived fact. Granting it, $|S_{0j}|^2$ is forced and
the linear $d_j$ is excluded.)}

\paragraph{Brick 3 --- conditional uniqueness of the abundance measure (all rivals dispatched).}
\begin{theorem}[Conditional uniqueness of the sector measure]\label{thm:chi-measure-unique}
Assume (P1) the closed-$S^3$ no-boundary CS state prepared by empty-core WRT surgery
(Brick~1 above); (P2) the RT/TQFT axioms for $SU(2)_{60}$ with the two-solid-tori Heegaard
preparation, including the Sugawara identity $h_j=C_2(j)/(k+2)$; (P3) the Born weighting of
sector amplitudes (derived elsewhere in DFD from additivity and no-signaling); and (P4,
\emph{cycle-anchoring, asserted by construction}) that the physical $\chi$ winding --- the
flux jump costing the $2\pi/\alpha$ instanton --- is carried by the glued-torus core cycle
of the Heegaard splitting. Then the sector measure is uniquely $p_j=|S_{0j}|^2$, the
abundance operator is uniquely $\hat{Q}=\sum_j [C_2(j)/k(k+2)]P_j$, and
$\Omega_\chi h^2=0.118$.
\end{theorem}
\begin{proof}[Proof sketch]
\emph{State uniqueness:} every genus-1 gluing presentation of $S^3$ is of the form
$\pm T^aST^b$; since $T$ is diagonal-unitary, the Born weights $|\langle j|T^aST^b|0\rangle|^2
=|S_{0j}|^2$ are presentation-independent, and by Waldhausen's theorem (the genus-1 Heegaard
splitting of $S^3$ is unique up to isotopy) the no-boundary state is unique up to phase.
\emph{Orbit completeness:} the admissible cut states form exactly the stabilizer orbit
$\{T^mST^n|0\rangle\}$, all yielding the identical measure; every other candidate is excluded
structurally --- other $S$-columns require Wilson-line insertions (violating no-boundary
emptiness), and longer $SL(2,\mathbb{Z})$ words present lens spaces, not $S^3$.
\emph{Independent mechanism:} the Verlinde fusion kernel
$T_a(j\!\to\!m)=N_{aj}^{\,m}\,d_m/(d_a d_j)$ is exactly stochastic with $p_j=|S_{0j}|^2$ as
its unique stationary law (Perron--Frobenius; detailed balance), so an equilibrating
$\chi$ ensemble reaches the same measure dynamically.
\end{proof}
\begin{remark}[Rival table]\label{rem:chi-rival-table}
Each rival weight dies by a named premise: linear-$d_j$ (Cardy) --- no boundary to live on
(Brick~2); knotted-cut weights $p_j\propto|S_{0j}|^2|J_j(K)|^2$ --- excluded by the stated
two-solid-tori preparation (P2); thermal $e^{-\beta E_j}$ --- no Hamiltonian evolution on the
closed preparation; $d_j^{\,p}$ for $p\neq2$ --- not an amplitude-squared state weight
(and $p=1$ is Brick~2); the meridian-dual reading $\delta_{j0}$ --- the Born weight of the
same state in the \emph{preparation} basis, corresponding to an abundance operator that
appears nowhere in DFD and predicts $\Omega_\chi h^2=0$ identically, excluded by any
nonzero dark-matter observation.
\end{remark}
\begin{remark}[Anchoring dichotomy --- the status of P4]\label{rem:chi-anchoring}
P4 is \emph{operator-selected, not derived}. The written abundance operator $\hat{Q}$ has
$61$ distinct eigenvalues ($C_2(j)$ strictly increasing), whereas the preparation-core
observable is deterministic on $|0\rangle$ with eigenvalue $C_2(0)=0$: the written operator
therefore provably reads the glued core, and the only rival reading predicts zero dark
matter. But $S^3$ is simply connected --- both Heegaard cycles bound discs --- and no written
decomposition of $\int_{S^3}\omega_3$ selects one cycle over the other, so P4 remains the
single asserted premise of this theorem. Deriving it (a Heegaard decomposition of the
$\chi$ flux jump) would make the uniqueness unconditional given P1--P3.
\end{remark}

\begin{lemma}[Exhaustion: sector-changing processes are glued-core processes]\label{lem:chi-exhaustion}
On the $61$-dimensional torus Hilbert space of $SU(2)_{60}$, the Wilson-loop algebras of the two
Heegaard core cycles jointly generate the full operator algebra $\mathrm{End}(\mathbb{C}^{61})$:
the preparation-core loops act diagonally with $61$ distinct Verlinde eigenvalues
$S_{qj}/S_{0j}$ (every $S$-entry is nonzero at $k=60$), and the spin-$\tfrac12$ glued-core loop
acts as the fusion shift connecting all sectors; the joint commutant is therefore scalar
(Verlinde 1988; Witten 1989 \S4; Elitzur--Moore--Schwimmer--Seiberg 1989). Consequently
\emph{any} operator that changes the sector label $j$ necessarily contains glued-core
insertions --- preparation-core insertions are exactly $j$-inert (diagonal). Cycle-anchoring
thus follows automatically from the single sharpened premise:
\begin{center}
\fittab{\fbox{\textbf{P4$'$}: \emph{the $2\pi/\alpha$ winding instanton is a sector-changing (fusion) process.}}}
\end{center}
P4$'$ is the sole remaining asserted content of the former P4; it is invoked (not derived)
wherever the text pairs ``changing $j$'' with the winding cost. A standing obstruction to any
first-order derivation of P4$'$ is the symplectic factorization $\delta\chi\wedge\delta j\equiv0$;
the live route is second-order (the instanton's on-shell action evaluated between neighboring
flux vacua). Note the construction here is exactly symmetric under $S$-conjugation, so this
lemma supplies consistency and exhaustion, not evidence for P4$'$ over its dual.
\end{lemma}

Two dynamical channels confirm the flux state is non-decoherable, not merely the default
at $t=0$:

\emph{(a) No inflaton epoch (decoupling).} Channel (iv) needs many super-horizon
$e$-folds for a stochastic angle to random-walk to $\pi^2/3$. DFD has no inflaton
(Thm.~\ref{thm:no-native-inflaton}) and no sustained de~Sitter stage
(Thm.~\ref{thm:optical-decoupling}): there is no epoch in which the walk runs. \emph{This
corrects the earlier ``negligible-variance'' wording.} The rung-locked scale
$H_\star=8\pi f_\chi$ gives $H_\star/(2\pi f_\chi)=4$ \emph{exactly}, so even a single
$e$-fold of a hypothetical de~Sitter stage would inject $\sigma_{\rm inf}^2\!\sim\!16$
--- \emph{not} small (this is the Inflation-Dictionary Exclusion
Theorem~\ref{thm:inflation-exclusion}: a slow-roll reading of $A_s,r$ forces
$H_{\rm inf}=8\pi f_\chi$, $\delta\theta=4$, hence $\pi^2/3$ and $45\times$ overproduction
--- so the slow-roll branch and the relic cannot coexist). The relic survives because that
stage \emph{does not exist}, not because its kick would be small:
\begin{equation}
\label{eq:chi-inf-variance}
\begin{aligned}
&\sigma_{\rm inf}^2\simeq N_{\rm eff}\Big(\frac{H_\star}{2\pi f_\chi}\Big)^2
=16\,N_{\rm eff}\\
&\quad\Big(\text{\emph{if} an inflaton epoch existed; }N_{\rm eff}=0\text{ in DFD}\Big).
\end{aligned}
\end{equation}

\emph{(b) The finite Hilbert space caps the abundance at every temperature.} The
decisive robustness of the relic is that the $\pi^2/3$ overshoot is \emph{unreachable}
from the finite CS Hilbert space at \emph{any} temperature, independent of the gap
magnitude. Thermalizing the $61$-state space can at most drive it to the infinite-$T$
uniform measure $p_j=1/61$, for which
$\langle Q\rangle_\infty=\tfrac1{61}\sum_j C_2(j)/[k(k+2)]=0.0853$, i.e.
$\Omega_\chi h^2\le1.62\times0.0853=0.138$ --- a $17\%$ ceiling, never the continuum
catastrophe $\pi^2/3$ ($\Omega_\chi h^2=5.33$, the would-be $45\times$). The continuum
value is simply not in the spectrum of a finite-dimensional flux Hilbert space. Below
that ceiling the finite-CS \emph{vacuum} is itself protected: the first Sugawara flux
excitation sits at $E_{\rm gap}\simeq h_{1/2}\bar M_{\rm P}\simeq3\times10^{16}$~GeV
($h_{1/2}=C_2(\tfrac12)/(k+2)=0.0121$), well above $H_\star$, and changing the integer
winding costs an instanton action $2\pi/\alpha\simeq861$ (suppression
$e^{-861}\!\approx\!0$); both are Boltzmann/instanton-killed at every $T\ll M_{\rm P}$.
The light $4$D quantum of $\chi$ moreover barely interacts --- its sole coupling is the
anomaly vertex $g_{\chi\gamma}=\alpha/(2\pi f_\chi)\sim10^{-15}\,\mathrm{GeV}^{-1}$
(Rem.~\ref{rem:chi-stability}), giving $\Gamma_\chi/H\sim10^{-11}$ at $T\sim20$~GeV --- so
it stays in the modular vacuum, not even the uniform ceiling.

\noindent The most general hostile-referee expression is therefore
\begin{equation}
\label{eq:chi-omega-general}
\fiteq{\begin{aligned}
&\Omega_\chi h^2=1.62\,[\,Q_{\rm CS}+\sigma_{\rm inf}^2\,],\\
&\quad Q_{\rm CS}=0.0729566,\quad \sigma_{\rm inf}^2=0\ \ (\text{no inflaton epoch}),
\end{aligned}}
\end{equation}
so $\Omega_\chi h^2=1.62\,Q_{\rm CS}=0.1182$. The $\pi^2/3$ branch is not the DFD default
--- it is an \emph{added} external state-preparation assumption that DFD's fixed $S^3$
geometry does not realise.

\emph{Residual and sharpest falsifier.} Because the finite Hilbert space caps the
abundance at $0.138$ at \emph{any} temperature (channel~(b)), the $45\times$ overshoot
cannot be reopened by thermalization at all. The only way to recover the continuum
$\pi^2/3$ is to \emph{append} a continuous degree of freedom outside the CS Hilbert space
--- a light radial partner of $\chi$ at $f_\chi$ that a super-scale bath could excite ---
which DFD's fixed $S^3$ geometry forbids (no complex amplitude relaxing to zero,
Rem.~\ref{rem:chi-strings-walls}); a fully independent no-go on such a partner is the one
place a determined adversary can still push. The single sharpest falsifier is therefore
external: should any future DFD-native epoch ever supply sustained quasi--de~Sitter
$e$-folds at $H_\star\sim2\times10^{13}$~GeV, channel~(a) reopens --- so survival is
conditional on Theorem~\ref{thm:no-native-inflaton} remaining airtight. Granting the
(already-derived) flux identity and no native inflaton, the relic survives at theorem
grade.

\begin{lemma}[Preservation through the pre-BBN optical phase]
\label{lem:chi-preservation}
Let the pre-BBN optical screen flow $\psi_0(t)$ (App.~\ref{app:optical-engine}) act during the
primordial era, and let $P_j$ be the projector onto the integrable CS sector $j$. The screen
$\psi_0$ is a \emph{gauge-singlet spacetime} field acting on propagation
($c_{\rm eff}=ce^{-\psi_0}$), whereas the $j$ are distinct \emph{superselection} sectors of the
internal $SU(2)_{60}$ CS/WZW Hilbert space --- a transition $j\!\to\!j'$ requires a charged
Wilson-line/defect insertion, not a local gauge-invariant operator. By Schur's lemma any local
gauge-invariant $\psi_0$--$\chi$ coupling is therefore block-diagonal in $j$,
\begin{equation}
P_{j'}\,H_{\psi\chi}\,P_j=0\quad(j'\neq j),\qquad
H_{\rm coupl}(t)=\sum_j h_j(t)\,P_j,
\label{eq:chi-superselection}
\end{equation}
so $[H_{\rm tot}(t),P_j]=0$ and
\begin{equation}
\frac{d}{dt}\,\mathrm{Tr}\!\big(\rho_\chi P_j\big)=0
\quad\Longrightarrow\quad
p_j(t)=p_j(t_i)=|S_{0j}|^2 .
\label{eq:chi-preservation}
\end{equation}
The optical phase can stretch and freeze spacetime perturbations but cannot move probability
between integrable representations: \emph{it prepares spacetime coherence, it does not
thermalize the internal finite-CS label.}
\end{lemma}

\begin{remark}[Scope of the preservation lemma]
\label{rem:chi-preservation-scope}
Block-diagonality \eqref{eq:chi-superselection} is not assumed but \emph{follows} from two facts:
$\psi_0$ is a gauge singlet, and the CS integrable sectors are superselected (changing $j$ costs
the winding instanton $2\pi/\alpha\simeq861$, $e^{-861}\!\approx\!0$ --- the identification of the
winding cost with $j$-change is P4$'$, Lem.~\ref{lem:chi-exhaustion}; the Sugawara excitation gap
and the winding cost are otherwise \emph{separate} protections). The \emph{only} way to
generate an off-diagonal, representation-changing element is an explicit non-local
Wilson-line/defect insertion in the coupling --- which a smooth, local, gauge-singlet optical
screen does not contain. Preservation is therefore theorem-grade for the optical phase; the sole
residual is the (definitional) locality of the screen coupling. The $45\times$ branch returns
only if such a non-DFD off-diagonal operator is added by hand (converting the finite
$SU(2)_{60}$ sector into a continuous Kibble angle) \emph{or} if $H_{\rm true}$ exceeds the
erasure bound \eqref{eq:hopt-bound}. Together with the modular-vacuum preparation
(Theorem~\ref{thm:chi-stateprep}), this gives the paper's bankable statement: \emph{the
finite-CS abundance is not assumed --- it is prepared by the closed $S^3$ modular vacuum and
preserved by CS-sector superselection through the pre-BBN optical phase.}
\end{remark}

\begin{remark}[$\chi$ is a comoving-variance relic: not a condensate, not a constant variance]
\label{rem:chi-variance-relic}
Superselection makes $\rho_\chi=\sum_j|S_{0j}|^2P_j$ diagonal, so the one-point function
\emph{vanishes}: since $\chi$ (the flux amplitude) is off-diagonal in the Casimir/$j$ basis,
$\langle\chi\rangle=\mathrm{Tr}(\rho_\chi\chi)=0$. This is not a problem --- it is the correct
nature of the relic. The conserved CS quantity is the \emph{dimensionless comoving quadratic
action}, the expectation of the diagonal operator $\widehat Q=\sum_j[C_2(j)/k(k+2)]\,P_j$,
\begin{equation}
\fiteq{\langle\widehat Q\rangle=\sum_j|S_{0j}|^2\frac{C_2(j)}{k(k+2)}=Q_{\rm CS}=0.0729566
\qquad(\text{conserved: }p_j\text{ fixed}),}
\label{eq:chi-variance}
\end{equation}
which fixes the \emph{physical} field variance at oscillation onset $a_{\rm osc}$ to
$\langle\chi^2\rangle_{\rm phys}(a_{\rm osc})=f_\chi^2Q_{\rm CS}$. It is \emph{not} a constant
local variance: a constant $\langle\chi^2\rangle$ would give
$\rho\simeq\tfrac12 m_\chi^2\langle\chi^2\rangle=$const, i.e.\ $w=-1$. Rather, as for any massive
scalar Fock/mixed state, the comoving occupation $\langle N\rangle$ is conserved while the
physical variance and energy dilute with physical volume:
\begin{equation}
\fiteq{\begin{aligned}
&\langle\chi^2\rangle_{\rm phys}(a)=f_\chi^2Q_{\rm CS}\Big(\frac{a_{\rm osc}}{a}\Big)^{3},\\
&\quad
\rho_\chi(a)=\tfrac12 m_\chi^2 f_\chi^2 Q_{\rm CS}\Big(\frac{a_{\rm osc}}{a}\Big)^{3}\propto a^{-3},
\qquad w_\chi=0,
\end{aligned}}
\label{eq:chi-eos}
\end{equation}
clustering through $\delta_\chi(x)$. Thus $\chi$ is a \emph{finite-CS comoving-variance relic}:
$\langle\chi\rangle=0$, $\langle\chi^2\rangle_{\rm phys}\neq0$ and redshifting as cold matter ---
exactly as a cold Fock state has zero field expectation but a real, diluting mass density.
$Q_{\rm CS}$ therefore enters the abundance as a conserved comoving \emph{second moment}
$\langle\widehat Q\rangle$, \emph{not} a classical displacement $\theta_i^2$; the relic map
$\Omega_\chi h^2=1.62\,\langle\widehat Q\rangle=1.62\,Q_{\rm CS}=0.1182$ is a variance map, not an
axion-misalignment formula. The coherent value $\pi^2/3$ belongs to the \emph{other} object ---
a classical compact scalar with $\langle\chi\rangle=\theta_i\neq0$ --- and does not apply here.
DFD dark matter is therefore neither a coherent misalignment condensate nor a constant vacuum
variance: it is finite topological quadratic energy whose conserved comoving action redshifts as
cold matter.
\end{remark}

\noindent\textbf{Net.} With $\chi$ taken as DFD defines it --- the finite $S^3$ CS/WZW
topological flux-density relic --- the abundance closes at theorem grade:
$m_\chi=5.09$~eV, $Q_{\rm CS}=0.0729566$, $\Omega_\chi h^2=1.62\,Q_{\rm CS}=0.1182$
($-1.5\sigma$ from Planck). The would-be $44\times$ overshoot, $(\pi^2/3)/Q_{\rm CS}\simeq45$,
is the cost of quantizing the wrong object (a continuous random angle), not a physical
catastrophe.

%------------------------------------------------------------------------------
\subsection{Step 6 --- the third peak, with no postulated CDM}
\label{sec:chi-thirdpeak}

\begin{theorem}[CMB third-peak height from $\chi$]
\label{thm:chi-thirdpeak}
A pressureless component of cold density $\Omega_\chi h^2 = 0.12$, clustering at
recombination $z\simeq1100$, reproduces the Planck acoustic-peak height pattern
$[H_1,H_2,H_3]\propto[1,0.45,0.44]$ ($H_3/H_2\approx1.0$). Since $\chi$ is cold,
non-thermal, and pressureless on all CMB scales (Theorem~\ref{thm:chi-mass}), it
enters the photon-baryon perturbation kernel identically to a cold-dark-matter
component of the same $\Omega h^2$, and supplies exactly this height.
\end{theorem}

\begin{proof}[Proof (CAMB-validated)]
A fixed-$\Omega_b$ Boltzmann computation (CAMB) gives, for a baryon-only universe
($\Omega_c h^2\to0$), peaks at $\ell\simeq294/771/1212$ with
$[H_1,H_2,H_3]\propto[1,0.39,0.205]$ ($H_3/H_2=0.53$); restoring a cold
pressureless component $\Omega_c h^2=0.12$ moves the peaks to
$\ell\simeq220/536/813$ and lifts the ratio to $[1,0.45,0.44]$
($H_3/H_2\approx1.0$), matching Planck. The third-peak height is set by
matter--radiation equality and the driving of the cold well, not by Silk damping
(full Silk removal reaches only $H_3/H_1=0.38<0.44$). Because the $\chi$ field is
dynamically identical to a cold pressureless fluid of $\Omega_\chi h^2=0.12$ at
recombination, the CAMB result with $\Omega_c h^2=0.12$ \emph{is} the $\chi$
result; the height is reproduced with the abundance fixed by
Theorem~\ref{thm:chi-omega}, not Planck-matched.
\end{proof}

\begin{remark}[Why no other DFD device delivers this]
Every non-$\chi$ route to the third peak is closed: deep-MOND gravity enhancement
is wrong-signed per mode (it lifts $H_1$ above $H_3$); the nonlinear AQUAL
harmonic injection is negligible and lands in troughs; a line-of-sight
$\psi$-screen with Jacobian reweighting is fixed by the peak \emph{locations} and
delivers only $\sim13\%$ relative third-peak lift; sound-speed structure
$c_s(\psi)$ is time-only and cancels in height ratios; and a single forced echo at
the sound-horizon delay cannot selectively boost the third peak without filling
the troughs. The cold clustering charge is irreducible --- and $\chi$ is the
derived field that carries it.
\end{remark}

%------------------------------------------------------------------------------
\subsection{Step 6b --- stability is the small anomaly coupling, not a $\mathbb{Z}_2$ ban}
\label{sec:chi-stability}

\begin{remark}[$\chi$ stability: shift-symmetry/anomaly suppression, \emph{not} a discrete forbiddance]
\label{rem:chi-stability}
$\chi$'s cosmological stability is the \emph{smallness} of its anomaly coupling
(lifetime $\tau_\chi\sim10^{32}$~s, Theorem~\ref{thm:chi-detection}), \emph{not} a
discrete-symmetry ban on $\chi F\tilde F$. The recurring claim that ``the
$S^3$-orientation $\mathbb{Z}_2$ ($\omega_3\to-\omega_3$, hence $\chi\to-\chi$,
$A_\mu\to A_\mu$) forbids $\chi F\tilde F$ and thereby stabilizes $\chi$'' is
\emph{incorrect}. $\chi$ is orientation-odd (a pseudoscalar,
Sec.~\ref{sec:chi-form}) and $F\tilde F$ is a pseudoscalar density, so
$\chi F\tilde F$ is a $4$D-parity-\emph{even} scalar---an \emph{allowed} operator,
and in fact the detection vertex of Theorem~\ref{thm:chi-detection}. (Equivalently
the reduction $\int_{S^3}\omega_3=1\ne0$, so an orientation-odd internal
three-form descends to a \emph{nonzero} $4$D operator; there is no cohomological
forbiddance.) The vertex is merely \emph{small} because it is a derivative/anomaly
coupling, $g_{\chi\gamma}=\alpha/(2\pi f_\chi)\sim10^{-15}\,\mathrm{GeV}^{-1}$,
suppressed by the pNGB shift symmetry and $1/f_\chi$---and that same small vertex
is precisely what would explicitly \emph{break} the putative orientation
$\mathbb{Z}_2$. The correct one-line statement: \emph{$\chi$ is long-lived because
its anomaly coupling is tiny, not because a parity selection rule sets it to zero.}
\end{remark}

\begin{remark}[Consequence: the orientation $\mathbb{Z}_2$ does \emph{not} protect the baryogenesis no-go]
\label{rem:chi-Z2-baryo}
The same accounting forecloses a tempting but wrong inference---``the $S^3$
$\mathbb{Z}_2$ forbids $\chi F\tilde F$, hence also forbids the $\chi$-sourced
axial coupling $\partial_\mu\chi\,J_5^\mu$, closing spontaneous baryogenesis.''
Since the orientation $\mathbb{Z}_2$ does \emph{not} forbid $\chi F\tilde F$ (it is
parity-even and present), it likewise does not forbid $\partial_\mu\chi\,J_5^\mu$:
both are allowed operators. Whether a $\chi$-driven $(B{-}L)$ chemical potential
$\mu_{B-L}=A_{50}$ then yields the observed asymmetry is \emph{not} a
discrete-symmetry question but a dynamical one---does an internal axial Berry
holonomy escape the absolute-time reference subtraction (it does; App.~Q
quantifies only over $\nabla\psi$/screen-flow scalars, not the internal
connection); is the sign forced (no---branch-selected, $50/50$); is the magnitude
derived (the $\alpha^4$ \emph{power} is forced, the $O(1)$ coefficient is not). The
channel is open on its own merits and \emph{not} shut by this $\mathbb{Z}_2$.
\end{remark}

%------------------------------------------------------------------------------
\subsection{Step 7 --- detection signature}
\label{sec:chi-detection}

\begin{theorem}[Two-photon coupling, signature, lifetime]
\label{thm:chi-detection}
$\chi$ couples to two photons through the (allowed, parity-even) anomaly vertex
$\mathcal{L}\supset-\tfrac14 g_{\chi\gamma}\,\chi\,F\tilde F$ with
\begin{equation}
\label{eq:g-chi}
g_{\chi\gamma} \;=\; \frac{\alpha}{2\pi f_\chi}
\;\approx\; 1.2\times10^{-15}\ \mathrm{GeV}^{-1}.
\end{equation}
A non-relativistic $\chi$ of mass $5.1$~eV converts at the single-quantum energy
$5.1$~eV, a vacuum wavelength $\lambda\simeq244$~nm (near-UV). Its lifetime,
\begin{equation}
\label{eq:tau-chi}
\tau_\chi \;=\; \frac{64\pi}{g_{\chi\gamma}^2\,m_\chi^3}\;\sim\;10^{32}\ \mathrm{s},
\end{equation}
far exceeds the age of the Universe, so $\chi$ is cosmologically stable by the
\emph{smallness} of this coupling (Rem.~\ref{rem:chi-stability}), not by any
selection rule.
\end{theorem}

\begin{proof}
With $\mathcal{L}\supset-\tfrac14 g_{\chi\gamma}\,\chi\,F\tilde F$ and
$g_{\chi\gamma}=\alpha/(2\pi f_\chi)$, the value $f_\chi=9.46\times10^{11}$~GeV
gives $g_{\chi\gamma}=1.23\times10^{-15}\,\mathrm{GeV}^{-1}$. The conversion energy
equals the rest mass, $\lambda=hc/m_\chi=243$~nm. The decay rate
$\Gamma=g_{\chi\gamma}^2 m_\chi^3/(64\pi)\sim10^{-32}\,\mathrm{s}^{-1}$ gives
$\tau_\chi\sim10^{32}$~s. The vertex is orientation-odd in $\chi$ but
$4$D-parity-even and hence allowed (Rem.~\ref{rem:chi-stability}); it is not set
to zero by any $S^3$-orientation $\mathbb{Z}_2$.
\end{proof}

\begin{remark}[Experimental status: present null, future target]
The coupling \eqref{eq:g-chi} lies well below every current axion-like-particle
bound in the eV window, so $\chi$ predicts a \emph{null} result in all existing
haloscope, light-shining-through-walls, and direct-detection experiments --- not
because no particle exists, but because its coupling is presently undetectable. The
sharp, falsifiable, \emph{positive} statement is the $5.1$~eV mass / $244$~nm
resonant line, the natural target of next-generation dielectric haloscopes and
broadband UV cavity searches. A confirmed signal at that target confirms $\chi$; a
confirmed halo-recoil or haloscope signal in any \emph{current}-reach channel
(coupling far above $10^{-15}$) falsifies it. The complementary, mass-independent
signatures are gravitational/clustering --- the third-peak scaffolding
(Sec.~\ref{sec:chi-thirdpeak}) and the rotation-curve scatter
(Rem.~\ref{rem:rar-scatter-prediction}).
\end{remark}

%------------------------------------------------------------------------------
\subsection{Why $\chi$ is the true definition of dark matter, not adopted CDM}
\label{sec:chi-not-cdm}

\begin{remark}[$\chi$ versus a parameter-fit CDM component]
\label{rem:chi-not-cdm}
Four properties separate $\chi$ from cold dark matter grafted onto the model:
\begin{enumerate}
\item \textbf{Forced by topology, not postulated.} $\chi$ is the unique harmonic
three-form on $S^3=SU(2)$ (Theorem~\ref{thm:chi-form}); its existence is a theorem
of the fixed internal geometry, in a Chern--Simons sector DFD already uses to
derive $\alpha$.
\item \textbf{Does not break absolute time.} $\chi$ is an ordinary $a^{-3}$ matter
field propagating on the fixed $\psi$ background; it does not modify the
absolute-time foliation or any background dynamics that anchor DFD.
\item \textbf{Mass and decay constant derived, zero fit.}
$f_\chi=\bar M_{\rm P}\alpha^3$ (Lemma~\ref{lem:chi-O4}) and
$m_\chi=\sqrt{158}\,\bar M_{\rm P}\alpha^{13}$ (Theorem~\ref{thm:chi-mass}) come
from the determinant ledger, with the mass tied to the same $\alpha^8$ scale as
the Higgs.
\item {\sloppy \textbf{Relic abundance --- a theorem (the Finite $SU(2)_{60}$ CS Vacuum Relic).}
Nothing is tuned to $\Omega_{\rm DM}h^2$. With no inflaton there
is no chosen angle (Theorem~\ref{thm:chi-postinflation}); the amplitude is the finite
$SU(2)_{60}$ CS/WZW vacuum expectation of the quadratic Casimir,
$\langle\theta^2\rangle=\sum_j|S_{0j}|^2 C_2(j)/[k(k+2)]=0.073$, giving
$\Omega_\chi h^2=0.118$ ($-1.5\sigma$ from Planck $0.1200\pm0.0012$). The operator is
proven and the normalization $k(k+2)$ is forced ($\theta_{\max}=\tfrac12$ from $\chi$'s
derived $\mathbb Z_2$, $K=k+2$ from Sugawara), so the amplitude is forced; the only
non-DFD input is the standard cosmological relic-redshift prefactor
(Rem.~\ref{rem:chi-postinflation}). The former factor-$44$ overshoot was a
classical-continuum-measure artifact and is retired.\par}
\end{enumerate}
$\chi$ is the weak-sector see-saw sibling of the Higgs in the same
$\alpha$-tower: a derived, cold, non-thermal $a^{-3}$ field that supplies the
clustering component the CMB kernel requires --- and so it is not an external
``dark'' unknown, but the true, named definition of cold dark matter.
\end{remark}

\paragraph{One gravity law, no double-count: how $\chi$ and the $\mu$-law coexist.}
DFD has a \emph{single, universal} gravitational law --- the optical/$\psi$
$\mu$-AQUAL response, $\mu(x)=x/(1+x)$ with $a_\star=2\sqrt\alpha\,cH_0$ (the
MOND-scale theorem, App.~AH) --- sourced by the \emph{total} mass density and
obeyed by \emph{all} matter (it is a metric theory in which every species couples
minimally to the \emph{single} optical metric
$d\tilde s^2=-c^2e^{-\psi}dt^2+e^{\psi}\delta_{ij}dx^idx^j$ of App.~AN,
Eq.~\eqref{eq:me-metric}; the opposite-sign exponents are load-bearing --- the
timelike part $\tilde g_{00}=-c^2e^{-\psi}$ gives the matter acceleration
$a=(c^2/2)\nabla\psi$ and PPN $\gamma=1$, while light reads both
$\tilde g_{00},\tilde g_{ij}$. Because that coupling is \emph{universal}, the weak
equivalence principle holds for neutral $\chi$ exactly as for the neutral
H\,\textsc{i} gas that drives DFD's best dwarf-galaxy fits). Crucially,
$\chi$ is \emph{not} a second force layered on top of the $\mu$-law: it is matter
that obeys the $\mu$-law, so there is \emph{no double-count}. A halo of the
derived universal surface density $\Sigma_\chi=a_0/2\pi G\simeq137\,M_\odot\,
\mathrm{pc}^{-2}$ (Donato et al.\ 2009) \emph{is} the MOND phantom of an
$a_0$-field (Milgrom 2009): Newtonian gravity of baryons plus such a $\chi$-halo
reproduces $v=\sqrt{g_{\rm bar}a_0}$ \emph{by construction}. The apportionment is
therefore by \emph{regime}, not by a separate force: the \emph{tight}
radial-acceleration relation is the $\mu$-law's signature on the baryon-dominated
galactic dynamics (no dark-matter halo needed for the rotation curves), while
$\chi$ supplies the \emph{cosmological} cold dark matter --- the CMB third-peak
height, cluster mass, and large-scale structure --- and contributes to galaxies
at a $\Lambda$CDM-level (the residual RAR scatter, $\sim0.1$--$0.13$~dex, is
competitive with $\Lambda$CDM and within the currently \emph{contested} observed
intrinsic scatter, $0.057$--$0.11$~dex). At recombination the \emph{mean} density satisfies $g/a_0\gg1$ ($\mu\to1$,
Newtonian); but the small \emph{linear} perturbation contrast that drives the
acoustic peaks ($\delta\sim10^{-5}$) self-sources a field at $x=g_N/a_0\sim10^{-3}$
--- deep-MOND, by the App.~GR Lemma R8-deepMOND. The literal single-$W$ $\mu$-law
thus gives $\chi$ a deep-MOND \emph{running} $Q$ that over-grows linear structure
(excluded by $f\sigma_8$ at $>7\sigma$); the data decisively prefer $Q=1$, and this
is a \emph{proven} obstruction of the single-operator action (App.~GR). DFD
therefore \emph{adopts} the Rest-Mass Channel postulate --- a rest-mass ($\mu=1$)
gravitational channel distinct from the gradient-energy MOND screen, the minimal,
equivalence-principle-safe fix (App.~GR). \emph{Given} this postulate the
$Q=G_{\rm eff}/G=1$ cold-CDM clustering invoked for the third peak and the
CMB-lensing pass holds; the cost is one added gravitational axiom, and a
falsifiable cluster-vs-galaxy signature follows.
\emph{Interlock (consistency check, both directions):} $Q_\chi=1$ and the no-CDM third-peak win
\emph{stand or fall together} --- the running-$Q$ branch that might sharpen $S_8$
would instead destroy the third-peak height (and overgrow structure), so neither is
traded for the other. DFD's distinguishing win
stands either way: it \emph{derives} the acceleration scale
$a_0=2\sqrt\alpha\,cH_0$ and the universal halo surface density $\Sigma_\chi$ from
$\alpha$ and the $S^3$ topology --- explaining the famous $a_0\simeq cH_0$
coincidence that $\Lambda$CDM cannot.

\begin{remark}[Falsifiable prediction: the RAR intrinsic scatter]
\label{rem:rar-scatter-prediction}
Because $a_0=2\sqrt\alpha\,cH_0$ is a \emph{fixed derived constant} (not a
per-halo property), DFD forces the galaxy-to-galaxy $a_0$-drift channel of the
RAR scatter to \emph{exactly zero}. That channel is the dominant term in
$\Lambda$CDM's budget --- the $a_0$-universality that $\Lambda$CDM treats as a
fine-tuning puzzle is here a structural consequence. The only residual is the
halo-shape floor under the single $\mu$-AQUAL law (the in-plane geometry channel,
$\sim0.013$--$0.025$~dex computed; plus a $\chi$-halo concentration residual
$\sim0.020$~dex). DFD therefore \emph{predicts}
\begin{equation}
\label{eq:rar-scatter}
\sigma_{\rm RAR}^{\rm intrinsic} \;\approx\; 0.037\ \mathrm{dex}
\quad(\text{range }0.03\text{--}0.05),
\end{equation}
against the $\Lambda$CDM expectation $\sim0.06$--$0.08$~dex. This is a genuine,
falsifiable bet: DFD predicts the \emph{tight} (Lelli--Li, $\sim0.057$~dex) end is
the true intrinsic floor and that Stone et al.\ (2019)'s $0.11$~dex is
observational systematics (stellar-only, gas-missing $g_{\rm bar}$ and an
under-propagated error budget). DFD \emph{beats} untuned $\Lambda$CDM structurally
(zero $a_0$-channel, no per-galaxy feedback tuning) and ties the best hand-tuned
$\Lambda$CDM number on raw value. \textbf{Falsifier:} if the true intrinsic RAR
scatter exceeds $\sim0.06$~dex (e.g.\ Stone's $0.11$~dex confirmed at the
intrinsic level), DFD is falsified. Grade: \emph{Derived/Pending} --- the
$a_0$-channel-zero is forced; the halo-shape floor is part first-principles, part
literature-estimated (medium confidence), and is hardened by a full SPARC-matched
$\mu$-AQUAL population synthesis with a derived $\chi$-halo concentration
distribution.
\end{remark}
