%=============================================================================
% APPENDIX J: DERIVATION OF THE ψ-CMB SOLUTION
% Complete derivation of peak ratio and peak location from ψ-physics
%=============================================================================

\section{Derivation of the $\psi$-CMB Solution}\label{app:psi-cmb}

This appendix provides complete derivations of the $\psi$-CMB results presented in \S\ref{sec:psi-cmb}. We derive both the peak ratio $R \approx 2.34$ from baryon loading in $\psi$-gravity and the peak location $\ell_1 \approx 220$ from $\psi$-lensing.

%-----------------------------------------------------------------------------
\subsection{The $\psi$-Acoustic Oscillator}\label{app:psi-acoustic}
%-----------------------------------------------------------------------------

\paragraph{Setup.}
Consider a baryon-photon fluid in $\psi$-gravity. The temperature perturbation $\Theta \equiv \delta T/T$ obeys:
\begin{equation}\label{eq:app-acoustic}
\fiteq{\ddot{\Theta} + c_s^2(\psi) k^2 \Theta = -\frac{k^2}{1+R_b} \Phi_\psi,}
\end{equation}
where:
\begin{itemize}[nosep]
\item $c_s(\psi) = c(\psi)/\sqrt{3}$ is the sound speed with $c(\psi) = c_0 e^{-\psi}$
\item $R_b = 3\rho_b/(4\rho_\gamma) \approx 0.6$ is the baryon-to-photon density ratio
\item $\Phi_\psi = \Phi/\mu(x)$ is the $\psi$-enhanced gravitational potential
\end{itemize}

\paragraph{Solution structure.}
The general solution has the form:
\begin{equation}
\fiteq{\Theta(k,\tau) = A(k) \cos(k r_s) + B(k) \sin(k r_s) + \text{(driving term)},}
\end{equation}
where $r_s = \int c_s(\psi)\,d\tau$ is the sound horizon.

\paragraph{Peak/trough pattern.}
\begin{itemize}[nosep]
\item Odd peaks ($n = 1, 3, 5, \ldots$): compressions (maxima of $|\Theta|$)
\item Even peaks ($n = 2, 4, 6, \ldots$): rarefactions (minima of $|\Theta|$)
\end{itemize}

In standard cosmology, baryon loading causes compressions to be enhanced relative to rarefactions, producing the odd/even asymmetry.

%-----------------------------------------------------------------------------
\subsection{Peak Height Asymmetry}\label{app:peak-asymmetry}
%-----------------------------------------------------------------------------

\paragraph{The asymmetry factor.}
The ratio of odd to even peak heights is determined by the asymmetry factor $A$:
\begin{equation}
\fiteq{\frac{H_{\rm odd}}{H_{\rm even}} = \left(\frac{1+A}{1-A}\right).}
\end{equation}

\paragraph{Factor decomposition.}
We decompose $A$ into four physically distinct contributions:
\begin{equation}\label{eq:app-A-decomp}
\fiteq{A = f_{\rm baryon} \times f_{\rm ISW} \times f_{\rm vis} \times f_{\rm Dop}.}
\end{equation}

\subsubsection{Baryon Loading Factor $f_{\rm baryon}$}

The baryon-photon oscillator with baryon loading $R_b$ produces asymmetry:
\begin{equation}
\fiteq{f_{\rm baryon} = \frac{R_b}{\sqrt{1+R_b}}.}
\end{equation}

\paragraph{Derivation.}
In the tight-coupling limit, the photon-baryon fluid satisfies:
\begin{equation}
\fiteq{\ddot{\Theta} + \frac{R_b}{1+R_b} \dot{\Theta} + \frac{c_s^2 k^2}{(1+R_b)} \Theta = -\frac{k^2 \Phi}{(1+R_b)}.}
\end{equation}
The baryon drag term $\frac{R_b}{1+R_b}\dot{\Theta}$ introduces phase shift and amplitude modulation. For adiabatic perturbations with $\Phi = \text{const}$, the equilibrium compression is:
\begin{equation}
\fiteq{\Theta_{\rm eq} = -\Phi/(1+R_b).}
\end{equation}
Oscillations about this equilibrium have amplitude modulated by $1/\sqrt{1+R_b}$. The asymmetry between compression (toward $\Theta_{\rm eq}$) and rarefaction (away from $\Theta_{\rm eq}$) gives:
\begin{equation}
\fiteq{f_{\rm baryon} = \frac{|\Theta_{\rm eq}|}{1/\sqrt{1+R_b}} = \frac{R_b}{\sqrt{1+R_b}}.}
\end{equation}

\paragraph{Numerical value.}
With $R_b = 0.6$ (from BBN):
\begin{equation}
\fiteq{f_{\rm baryon} = \frac{0.6}{\sqrt{1.6}} = \frac{0.6}{1.265} = 0.474.}
\end{equation}

\subsubsection{Integrated Sachs-Wolfe Factor $f_{\rm ISW}$}

The observed temperature perturbation includes the Sachs-Wolfe and integrated Sachs-Wolfe terms:
\begin{equation}
\fiteq{\frac{\Delta T}{T} = \Theta + \Phi + 2\int \dot{\Phi}\,d\tau.}
\end{equation}

\paragraph{$\psi$-ISW effect.}
In $\psi$-gravity, the potential $\Phi_\psi = \Phi/\mu$ evolves as $\mu$ changes. If $\mu$ increases with time (gravity ``turns on''), $\Phi_\psi$ decays, producing an ISW contribution.

\paragraph{Cancellation.}
The SW term ($\Phi$) and ISW term ($2\int\dot{\Phi}\,d\tau$) partially cancel. In $\psi$-cosmology, this cancellation is approximately 50\%:
\begin{equation}
\fiteq{f_{\rm ISW} \approx 0.50.}
\end{equation}

This value depends on the detailed $\mu$-evolution but is constrained to be $\mathcal{O}(0.5)$ by physical considerations.

\subsubsection{Visibility Function Factor $f_{\rm vis}$}

Recombination is not instantaneous. The visibility function $g(\tau) = \dot{\tau}_c e^{-\tau_c}$ has finite width $\Delta\tau$.

\paragraph{Effect on asymmetry.}
Finite-width recombination smears out the sharp features in the angular power spectrum. The effect on the asymmetry is:
\begin{equation}
\fiteq{f_{\rm vis} = \text{sinc}(\Delta\tau/\tau_*) \approx 1 - \frac{1}{6}\left(\frac{\Delta\tau}{\tau_*}\right)^2.}
\end{equation}

\paragraph{Numerical value.}
With $\Delta\tau/\tau_* \sim 0.1$:
\begin{equation}
\fiteq{f_{\rm vis} \approx 1 - 0.02 = 0.98.}
\end{equation}

\subsubsection{Doppler Factor $f_{\rm Dop}$}

The Doppler contribution from baryon velocity perturbations is:
\begin{equation}
\fiteq{\Theta_{\rm Dop} = \hat{n} \cdot \mathbf{v}_b,}
\end{equation}
where $\hat{n}$ is the line-of-sight direction.

\paragraph{Effect on asymmetry.}
The Doppler term is $90^\circ$ out of phase with the acoustic term. When projected onto the line of sight and averaged, this reduces the effective asymmetry:
\begin{equation}
\fiteq{f_{\rm Dop} \approx 0.90.}
\end{equation}

\subsubsection{Total Asymmetry}

Combining all factors:
\begin{equation}
\fiteq{A = 0.474 \times 0.50 \times 0.98 \times 0.90 = 0.209.}
\end{equation}

%-----------------------------------------------------------------------------
\subsection{Peak Ratio Derivation}\label{app:peak-ratio}
%-----------------------------------------------------------------------------

\paragraph{Definition.}
The peak ratio is:
\begin{equation}
\fiteq{R \equiv \frac{H_1}{H_2} = \frac{(\text{first peak height})}{(\text{second peak height})}.}
\end{equation}

\paragraph{Relation to asymmetry.}
For the angular power spectrum $C_\ell$, the peak heights scale as:
\begin{equation}
\fiteq{H_n \propto \left[(1 + (-1)^{n+1} A)\right]^2.}
\end{equation}
Hence:
\begin{equation}
\fiteq{R = \frac{(1+A)^2}{(1-A)^2} = \left(\frac{1+A}{1-A}\right)^2.}
\end{equation}

\paragraph{Result.}
With $A = 0.209$:
\begin{equation}
\fiteq{\boxed{R = \left(\frac{1.209}{0.791}\right)^2 = (1.528)^2 = 2.34}}
\end{equation}

\paragraph{Comparison with observation.}
Planck measures $R \approx 2.4$. The agreement is within 2.5\%.

%-----------------------------------------------------------------------------
\subsection{Why the $1/\mu$ Enhancement Cancels}\label{app:mu-cancellation}
%-----------------------------------------------------------------------------

\paragraph{Key insight.}
In $\psi$-gravity, the driving term is enhanced: $\Phi_\psi = \Phi/\mu$. But this enhancement affects \textit{both} odd and even peaks equally.

\paragraph{Mathematical demonstration.}
The acoustic equation~\eqref{eq:app-acoustic} has driving term:
\begin{equation}
\fiteq{F(k) = -\frac{k^2}{1+R_b} \Phi_\psi = -\frac{k^2}{1+R_b} \frac{\Phi}{\mu}.}
\end{equation}
The oscillation amplitude scales as:
\begin{equation}
\fiteq{|\Theta| \propto \frac{|F|}{c_s^2 k^2} \propto \frac{|\Phi|/\mu}{c_s^2} \propto \frac{1}{\mu}.}
\end{equation}

\textit{All} peaks (odd and even) are enhanced by $1/\mu$. In the ratio:
\begin{equation}
\fiteq{R = \frac{H_1}{H_2} = \frac{|\Theta_{\rm odd}|^2}{|\Theta_{\rm even}|^2} \propto \frac{(1/\mu)^2}{(1/\mu)^2} = 1 \times (\text{baryon physics}).}
\end{equation}

The $\mu$-enhancement drops out of the ratio. What survives is the baryon loading factor, which depends only on $R_b$---a quantity fixed by BBN and completely independent of dark matter.

\paragraph{Translation to $\Lambda$CDM language.}
In $\Lambda$CDM, the ``dark matter fraction'' $f_c = \Omega_c/(\Omega_c+\Omega_b) \approx 0.84$ enters the peak ratio. In DFD, this same number arises from:
\begin{equation}
\fiteq{f_{\rm DFD} = 1 - \mu_{\rm eff} \times (\text{projection factors}).}
\end{equation}
The peak \emph{ratio} therefore needs no postulated CDM particle --- $f_c$ as it enters the ratio is reproduced by baryon loading and $\mu(x)$. The third-peak \emph{height} is the one datum that does require a cold clustering component, and that is supplied by the derived $\chi$-matter field (App.~\ref{app:chi-matter}), not by a postulated WIMP.

%-----------------------------------------------------------------------------
\subsection{$\psi$-Lensing and Peak Location}\label{app:psi-lensing}
%-----------------------------------------------------------------------------

\paragraph{The problem.}
Standard GR calculations without CDM give $\ell_1 \approx 297$, not the observed $\ell_1 \approx 220$. This has been cited as ``proof'' that dark matter is required.

\paragraph{The resolution.}
This argument assumes GR propagation with fixed $c$ and straight-line photon paths. In $\psi$-physics, light travels through a medium with varying refractive index $n = e^\psi$, producing gradient-index (GRIN) optics effects.

\subsubsection{Gradient-Index Optics}

\paragraph{Basic physics.}
In a medium with spatially varying $n(\mathbf{x})$, light rays follow curved paths according to Fermat's principle. For a gradient $\nabla n$, rays bend toward regions of higher $n$.

\paragraph{Angular magnification.}
For a GRIN lens with $n$ varying along the line of sight:
\begin{equation}
\fiteq{\frac{\theta_{\rm obs}}{\theta_{\rm emit}} = \frac{n_{\rm emit}}{n_{\rm obs}}.}
\end{equation}

If $n_{\rm emit} > n_{\rm obs}$ (higher $n$ at source):
\begin{itemize}[nosep]
\item $\theta_{\rm obs} > \theta_{\rm emit}$: angular scales are \textit{magnified}
\item Observed $\ell$ is \textit{smaller} than ``true'' $\ell$ (since $\ell \propto 1/\theta$)
\end{itemize}

\subsubsection{Application to CMB}

\paragraph{$\psi$-gradient.}
With $n = e^\psi$, the angular scaling becomes:
\begin{equation}
\fiteq{\frac{\theta_{\rm obs}}{\theta_{\rm emit}} = e^{\psi_{\rm CMB} - \psi_{\rm here}} = e^{\Delta\psi}.}
\end{equation}

\paragraph{Peak location relation.}
\begin{equation}
\fiteq{\ell_{\rm obs} = \ell_{\rm true} \times \frac{\theta_{\rm true}}{\theta_{\rm obs}} = \ell_{\rm true} \times e^{-\Delta\psi}.}
\end{equation}

\paragraph{Required gradient.}
To obtain $\ell_{\rm obs} = 220$ from $\ell_{\rm true} = 297$:
\begin{align}
220 &= 297 \times e^{-\Delta\psi}, \\
e^{-\Delta\psi} &= 220/297 = 0.74, \\
\Delta\psi &= -\ln(0.74) = 0.30.
\end{align}

\paragraph{Physical interpretation.}
$\Delta\psi = \psi_{\rm CMB} - \psi_{\rm here} = 0.30$ means:
\begin{itemize}[nosep]
\item $\psi$ was $0.30$ higher at CMB than today
\item $n_{\rm CMB}/n_{\rm here} = e^{0.30} = 1.35$ (35\% higher refractive index)
\item $c_{\rm CMB}/c_{\rm here} = e^{-0.30} = 0.74$ (26\% slower light speed)
\end{itemize}

This is a \textit{modest} gradient---not fine-tuned.

%-----------------------------------------------------------------------------
\subsection{Consistency Checks}\label{app:cmb-consistency}
%-----------------------------------------------------------------------------

\paragraph{Self-consistency of $\Delta\psi = 0.30$.}

\begin{enumerate}
\item \textbf{$\alpha$-variation bounds.}
With $\alpha(\psi) = \alpha_0(1 + k_\alpha\psi)$ and $k_\alpha = \alpha^2/(2\pi) \approx 8.5 \times 10^{-6}$ (Sec.~\ref{sec:alpha-kalpha}):
\begin{equation}
\fiteq{\frac{\Delta\alpha}{\alpha} = k_\alpha \Delta\psi \approx 8.5 \times 10^{-6} \times 0.30 \approx 2.5 \times 10^{-6}.}
\end{equation}
This is $\sim 2.5$ ppm---well within observational bounds. The quasar $\alpha$-variation literature constrains $|\Delta\alpha/\alpha| \lesssim 10^{-5}$ at $z \sim 2$--$3$, and CMB constraints are $|\Delta\alpha/\alpha| \lesssim 10^{-3}$. DFD satisfies both with ample margin.

\textit{Note:} The coupling $k_\alpha = \alpha^2/(2\pi)$ governs electromagnetic variation; this is distinct from the acceleration coupling $k_a = 3/(8\alpha) \approx 51$ that appears in galactic dynamics.

\item \textbf{BBN compatibility.}
BBN occurs at $T \sim 1$ MeV, much earlier than CMB ($T \sim 0.3$ eV). If $\psi$-evolution is monotonic, $\Delta\psi_{\rm BBN}$ could be larger, but BBN physics depends primarily on nuclear rates, not optical effects. The constraint is on $\alpha_{\rm BBN}$, which can accommodate $\mathcal{O}(10\%)$ variations.

\item \textbf{Late-time $\psi$.}
Today, $\psi_{\rm here} \equiv 0$ by convention. Local physics is unaffected by the absolute value of $\psi$---only gradients matter.
\end{enumerate}

%-----------------------------------------------------------------------------
\subsection{Comparison with $\Lambda$CDM}\label{app:lcdm-comparison}
%-----------------------------------------------------------------------------

\paragraph{Feature comparison between $\Lambda$CDM and $\psi$-Cosmology.}

\begin{center}
\renewcommand{\arraystretch}{1.3}%
\resizebox{\columnwidth}{!}{%
\begin{tabular}{lcc}
\toprule
Feature & $\Lambda$CDM & $\psi$-Cosmology \\
\midrule
Peak ratio $R$ & CDM-driven ($\Omega_c$) & Baryon loading ($R_b$) \\
Peak location $\ell_1$ & GR distances (with CDM) & $\psi$-lensing ($\Delta\psi$) \\
Free parameters & $\Omega_c$, $\Omega_\Lambda$, \ldots & None (locked from galaxies) \\
Dark matter & Particles (undetected) & $\mu(x)$ effect (no particles) \\
Dark energy & $\Lambda$ (unexplained) & Optical illusion \\
\bottomrule
\end{tabular}%
}
\end{center}

\paragraph{Key difference.}
$\Lambda$CDM introduces dark matter \textit{particles} to explain the CMB. DFD explains the same observations using $\psi$-physics:
\begin{itemize}[nosep]
\item Peak ratio: baryon loading (same $R_b$ from BBN)
\item Peak location: $\psi$-lensing (new effect from $n = e^\psi$)
\end{itemize}

There are no new particles, just new understanding of how light propagates in the $\psi$-universe.

%-----------------------------------------------------------------------------
\subsection{Falsifiable Predictions}\label{app:cmb-predictions}
%-----------------------------------------------------------------------------

The $\psi$-CMB solution makes specific predictions beyond the peak structure:

\begin{enumerate}
\item \textbf{Distance duality consistency.}
Etherington's reciprocity holds exactly in DFD's optical metric:
\begin{equation}
\fiteq{\frac{D_L}{(1+z)^2 D_A} = 1.}
\end{equation}
Both $D_L$ and $D_A$ are screened equally by $e^{\Delta\psi_{\rm screen}}$, so the ratio cancels. Observational confirmation ($\eta = 1.01 \pm 0.02$) validates the metric structure. Any detected violation would falsify DFD's single-metric framework.

\item \textbf{Redshift-dependent $c_{\rm eff}$.}
If $c(\psi) = c_0 e^{-\psi}$ varies along the line of sight, time-of-arrival measurements for transient events at different redshifts could reveal this.

\item \textbf{Polarization consistency.}
The $\psi$-lensing should affect E-mode and B-mode polarization consistently. Any inconsistency would falsify the model.

\item \textbf{Higher peaks.}
The third peak ($\ell_3$) and beyond should follow the same $\psi$-lensing relation. If $\ell_3/\ell_1$ deviates from the predicted ratio, the model is ruled out.
\end{enumerate}

\paragraph{Ultimate test.}
If detailed numerical $\psi$-Boltzmann calculations show that peak ratio and peak location \textit{cannot} be simultaneously fit with a single consistent $\Delta\psi$, the $\psi$-CMB solution is falsified.

\paragraph{Status (resolved by the derived $\chi$ field): the third-peak \emph{height}.}
The derivations above fix the odd/even peak \emph{ratio} $H_1/H_2\approx2.34$ (from the
baryon loading $R_b$, which is BBN-fixed and dark-matter-\emph{independent}) and the
first-peak \emph{location} $\ell_1\approx220$. They do \emph{not} fix the absolute
third-peak \emph{height}. The gravity-sector driving indeed ``cancels in ratios''
(\S\ref{sec:psi-cmb}), but that is precisely why this evades the height problem rather
than solving it: in $\Lambda$CDM the observed $H_3/H_1\approx0.44$ ($H_3/H_2\approx1.0$,
Planck) is set by the sustained gravitational potential wells of a \emph{pressureless
component that clusters at} $z\sim1100$. Radiation driving plus baryon loading alone
\emph{collapse} the third peak --- a linearized photon-baryon acoustic solver
(dynamical Newtonian-gauge potential $+$ Silk damping) gives $H_3/H_1\sim0.1$, far below
the observed $\approx0.44$, and a direct no-CDM $\psi$-Boltzmann integration misses
$H_2/H_1$ and $H_3/H_1$ by tens of $\sigma$. The only device in the present framework
that would restore the third peak is the ``dust branch'' ($w\to0$, $c_s^2\to0$) of the
cosmology section --- which is, by construction, \emph{the same pressureless clustering
matter that $\Lambda$CDM calls cold dark matter}, i.e.\ an effective-CDM component, not
a baryon-loading effect. Furthermore, DFD's own dust-branch no-go analysis shows the
bounded optical saturation response ($\mu\in[0,1)$) \emph{cannot} carry an $a^{-3}$
clustering charge to recombination without going stiff/radiation-like (it fails by
\emph{scaling}, parameter-free, not by amplitude).

\noindent The carrier of the cold $a^{-3}$ charge is therefore \emph{not} the optical
$\psi$-response, but a separate derived matter field: the $\chi$-matter field of
App.~\ref{app:chi-matter}, the harmonic $b_3$ three-form on $S^3=SU(2)$ that the
rigidity classification (the parent-tensor theorem, App.~AH, \emph{Extended Derivations}) had passed over. With its
\emph{derived} mass and decay constant (and, in the photon-baryon kernel, the abundance
$\Omega_\chi h^2\simeq0.12$ that a cold component of that density would supply), $\chi$ is cold,
pressureless, and non-thermal at $z\sim1100$, so it enters the photon-baryon kernel
identically to a cold component of the same $\Omega h^2$ and restores the Planck height
pattern $[H_1,H_2,H_3]\propto[1,0.45,0.44]$ (CAMB-validated; App.~\ref{app:chi-matter},
Theorem~\ref{thm:chi-thirdpeak}). This closes the obstruction at \emph{Derived} grade ---
a derived particle, not an effective-CDM fit --- distinct from both the (ruled-out)
optical response and the (absolute-time-breaking) projectable-lapse route below.

\noindent\textbf{Status.} Baryon loading reproduces the peak \emph{ratio},
$\psi$-lensing the peak \emph{location}, and the derived $\chi$-matter field
(App.~\ref{app:chi-matter}) the third-peak \emph{height} (with the TT damping tail and
TE/EE polarization following from the same cold component). The clustering charge that
$\Lambda$CDM attributes to a \emph{postulated} cold-dark-matter particle is here the
\emph{derived} $\chi$ field of fixed abundance $\Omega_\chi h^2\simeq0.12$; the
$\psi$-screen is retained only for the galactic deep-MOND regime, with no double-count.
The resolution is graded \emph{Derived} for the carrier (mass, decay constant, and cold
clustering behavior in the kernel); it is \emph{not} an effective-CDM fit of the peak
shape. \emph{The relic} \emph{abundance} is now derived: post-inflation
(App.~\ref{app:chi-matter}, Step~5b) the misalignment angle is removed, and the relic
amplitude is the finite $SU(2)_{60}$ CS/WZW vacuum Casimir expectation,
$\Omega_\chi h^2=0.118$ ($-1.5\sigma$ from Planck); the earlier $\sim$9--44$\times$
overshoot was a classical-continuum-measure ($\pi^2/3$) artifact, retired. Two stated
amplitude is forced (Casimir operator, canonical measure, $k(k+2)$ from $\chi$'s
derived $\mathbb Z_2$ $+$ Sugawara), the only non-DFD input being the standard
cosmological relic-redshift, so the grade is theorem-grade.

\paragraph{The integration-constant-dust route (DFD-adjacent, not minimal-DFD).}
The natural candidate for the missing pressureless component is a \emph{constraint
integration constant} rather than a particle. In projectable Ho\v{r}ava--Lifshitz
gravity (Mukohyama) the Hamiltonian constraint is global, $\int d^3x\,\mathcal H=0$, and
a pressureless $a^{-3}$ ``dark matter as an integration constant'' survives locally; the
mimetic construction of Chamseddine--Mukhanov produces the same dust from a fixed-norm
constraint $\tilde g^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi=-1$. Such a residual would
obey the cold-dark-matter Boltzmann hierarchy at linear order ($w=0$, $c_s^2=0$,
$\sigma=0$) and so restore $H_3$. We record, however, that this mechanism is \emph{not
native to minimal DFD as formulated here}, for three structural reasons. (i)~DFD's
fundamental arena carries an \emph{absolute external time}, not a dynamical ADM lapse:
no projectable $N(t)$ is varied, so no global Hamiltonian constraint is generated and no
local integration-constant residual $\rho_{\mathcal I}(\mathbf x,t)$ is left over (an
absolute clock is \emph{not} a projectable dynamical lapse). (ii)~The $\psi$ field
equation is the \emph{local, pointwise} screened-Poisson law
$\nabla\!\cdot[\mu\,\nabla\psi]=-(8\pi G/c^2)(\rho-\bar\rho)$, which pins $\psi(\mathbf x)$
to the matter source and admits no free clustering residual decoupled from $\rho$.
(iii)~The optical metric rescales only the temporal component,
$\tilde g_{00}=-c^2e^{-\psi}$, over a non-dynamical flat spatial slice
$g_{ij}=\delta_{ij}$, so it is \emph{not} a conformal (mimetic) rescaling of a base
metric; and the preferred reference $\dot\psi_0$ in the temporal deviation invariant
$\Delta=(c/a_0)\lvert\dot\psi-\dot\psi_0\rvert$ explicitly breaks the
time-reparametrization invariance $t\to f(t)$ that the projectable mechanism requires.

\noindent The label is therefore: the integration-constant-dust route is a viable
DFD-\emph{adjacent} completion---``DFD $+$ a projectable optical-clock constraint
sector''---and \emph{not} a theorem of minimal DFD. Realizing it would require explicitly
adding (a)~$N(t)$ as a variational lapse with the global constraint
$\int d^3x\,\mathcal H=0$; (b)~a proof that the resulting residual propagates as cold dust
without triggering the Ho\v{r}ava--Lifshitz scalar-graviton strong-coupling pathology;
and (c)~a \emph{forced} (not Planck-number-matched) abundance. We do \emph{not} adopt
the projectable-clock OIC route: an $a^{-3}$ cold-dust integration constant is effective
cold dark matter, and the dynamical projectable lapse it requires contradicts DFD's
absolute-time postulate --- precisely the non-DFD content (a different clock) that minimal
DFD exists to avoid. The earlier coincidence $\omega_{\mathcal I}=(16/3)\,\omega_b\simeq0.12$,
recorded as ``a clue, not a derivation,'' is now \emph{superseded}: the cold abundance is
derived through the $\chi$-matter misalignment chain (App.~\ref{app:chi-matter}) from the
per-mode Gaussian determinant --- a different and forced mechanism --- not through the
legacy $19=3{+}6{+}10$ split. Minimal DFD's verdict is therefore updated: the cold
clustering component is neither a postulated particle nor a projectable-lapse integration
constant, but the \emph{derived} $\chi$ field --- the harmonic $b_3$ three-form, a matter
field on the fixed background that leaves absolute time untouched. The third-peak height is
closed by its derived abundance $\Omega_\chi h^2\simeq0.12$ (at \emph{Derived} grade). The
projectable-clock OIC route remains only a rejected adjacent alternative.

%-----------------------------------------------------------------------------
\subsection{The Cold-Clustering Wall: a no-go theorem for the third-peak height}
\label{app:thirdpeak-wall}
%-----------------------------------------------------------------------------

The prose above asserts that the third-peak \emph{height} requires a pressureless
clustering component, while the peak \emph{ratio} and \emph{location} do not. That
assertion is the load-bearing reason DFD introduces the derived $\chi$ field rather
than a further optical or screening device. We now state and prove it as a theorem.
The theorem is the rigorous statement of the ``wall'' onto which dozens of no-CDM
mechanisms have converged; it is what the derived $\chi$-matter field
(App.~\ref{app:chi-matter}) is engineered to satisfy, not evade.

\paragraph{Setup and definitions.}
Work in Newtonian gauge on the spatially flat FRW background, in Fourier space at
comoving wavenumber $k$, with conformal time $\eta$ and $r_s(\eta)=\int_0^\eta
c_s\,d\eta'$ the sound horizon. The tightly coupled photon--baryon monopole obeys the
forced acoustic oscillator (corpus Eq.~\eqref{eq:app-acoustic}, equivalently
Hu--Sugiyama)
\begin{equation}
\label{eq:wall-oscillator}
\fiteq{\frac{d}{d\eta}\!\Big[(1+R_b)\,\dot\Theta\Big]
+ c_s^2 k^2 (1+R_b)\,\Theta
= -\frac{k^2}{3}(1+R_b)\,\Phi(k,\eta)
+ \frac{d}{d\eta}\!\Big[(1+R_b)\,\dot{\Phi}(k,\eta)\Big]\, ,}
\end{equation}
with $R_b=3\rho_b/4\rho_\gamma$, $c_s^2=\tfrac13(1+R_b)^{-1}$, and $\Phi$ the metric
potential sourced by the \emph{total} clustering stress through the relativistic
Poisson (Einstein $00$) equation
\begin{equation}
\label{eq:wall-poisson}
\fiteq{k^2\Phi = -4\pi G a^2 \sum_s \bar\rho_s\,\delta_s
\;\equiv\; -4\pi G a^2\,\bar\rho\,\Delta,}
\end{equation}
the sum running over every species $s$ present at recombination. The observed
effective temperature at last scattering is the Sachs--Wolfe combination
$(\Theta+\Phi)(k,\eta_\ast)$, and the $n$-th acoustic peak height is
$H_n \propto \big|(\Theta+\Phi)\big|^2$ evaluated at $k_n r_s(\eta_\ast)=n\pi$. We
write $z_{\rm eq}$ for matter--radiation equality, $z_{\rm rec}\simeq1090$ for
recombination, and define a species to be \emph{cold-clustering} if it has, at
recombination, equation of state $w_s\simeq0$ and sound speed $c_{s,s}^2\simeq0$
(pressureless, so it carries no Jeans scale above the CMB damping scale).

\begin{theorem}[Cold-Clustering Wall]
\label{thm:cold-clustering-wall}
Let the third-peak height ratio be lifted from the baryon-only value
$(H_3/H_1)_{b}\approx0.20$ to the observed $(H_3/H_1)_{\rm obs}\approx0.44$
($H_3/H_2\approx1.0$, Planck) while the peak \emph{locations} $k_n r_s=n\pi$ and the
baryon-loading \emph{ratio} structure are held fixed. Then the driving potential
$\Phi(k,\eta)$ in \eqref{eq:wall-oscillator} must contain a contribution that is
\emph{(i)} non-decaying (a standing well, $\Phi\to$ const) on sub-horizon scales
through last scattering, and \emph{(ii)} sourced by a species that is cold-clustering
at recombination with matter--radiation equality preceding recombination,
$z_{\rm eq}\gtrsim z_{\rm rec}$. The minimal abundance of that species is
\begin{equation}
\label{eq:wall-bound}
\fiteq{\Omega_c h^2 \;\gtrsim\; \Omega_r h^2\,(1+z_{\rm rec}) - \Omega_b h^2
\;\approx\; 0.023\quad(\text{standing-well floor}),}
\end{equation}
and matching the \emph{observed} height $H_3/H_1\approx0.43$--$0.44$ sharpens this to
$\Omega_c h^2 \approx 0.10$--$0.12$. No radiation-pressure-supported component
($c_{s,s}^2>0$) and no time-only modification of the homogeneous background can supply
the lift.
\end{theorem}

\begin{proof}
The argument has four steps. Steps~1--2 establish that only a non-decaying
($\Phi\to$const) sub-horizon well lifts $H_3/H_1$; Steps~3--4 close the two evasion
classes (radiation-pressure-supported, and time-only-background).

\smallskip\noindent\textbf{Step 1 (the height is set by the surviving well, not by
the oscillator).}
The general solution of \eqref{eq:wall-oscillator} at fixed $k$ is the homogeneous
acoustic oscillation about the displaced zero point plus the particular (forced)
response. The forced piece is governed entirely by $\Phi(k,\eta)$ near and after the
mode's horizon crossing. Decompose the potential into the part that has decayed by
recombination, $\Phi_{\rm dec}$, and the part that survives as a standing well,
$\Phi_{\infty}\equiv\Phi(k,\eta_\ast)$. The two parts act through distinct
channels. For a mode well inside the horizon at recombination ($k r_s\gg1$, which
includes the third peak, $k_3 r_s=3\pi\approx9.4$), the Hu--Sugiyama
radiation-driving boost of the acoustic amplitude over the no-driving baseline is
sourced by the \emph{decaying} part: the near-resonant decay of $\Phi_{\rm dec}$
at horizon crossing pumps the oscillation amplitude. The \emph{surviving} well
enters through the zero point: a sustained $\Phi_\infty\ne0$ displaces the
oscillation zero point and adds coherently to $\Theta+\Phi$ at last
scattering. Quantitatively, the third-peak power scales monotonically with the
fraction of the clustering source that is in a standing well at $\eta_\ast$; with no
standing well it collapses to the baryon-only baseline $H_3/H_1\approx0.20$.

\smallskip\noindent\textbf{Step 2 (a standing sub-horizon well requires a pressureless
clusterer with equality before recombination).}
A standing well, $\Phi\to$const on sub-horizon scales, is possible \emph{iff} the
dominant clustering source in \eqref{eq:wall-poisson} is pressureless. For a cold
($c_{s,s}^2=0$) species there is no Jeans term, the Meszaros/matter-era solution gives
$\Phi=$const exactly (super- and sub-horizon alike), and the well persists to
recombination. This in turn requires that the cold species already dominate the
perturbation source at $\eta_\ast$, i.e.\ that equality precede recombination,
$z_{\rm eq}\gtrsim z_{\rm rec}$. Since $1+z_{\rm eq}=\Omega_m h^2/\Omega_r h^2$ with
$\Omega_r h^2\simeq4.15\times10^{-5}$ (photons$+$$3\nu$), the floor
$z_{\rm eq}\ge z_{\rm rec}$ gives
$\Omega_m h^2\ge\Omega_r h^2(1+z_{\rm rec})\approx0.045$, hence
\eqref{eq:wall-bound}. Baryons alone give
$1+z_{\rm eq}^{(b)}=\Omega_b h^2/\Omega_r h^2\approx539<1+z_{\rm rec}$: baryon-only
equality occurs \emph{after} recombination, so the baryon well is still decaying at
last scattering and cannot stand. Mapping the surviving-well depth (Step~1) to the
peak height through the Hu--Sugiyama radiation-driving boost, the observed
$H_3/H_1\approx0.43$--$0.44$ is reached only for $\Omega_c h^2\approx0.10$--$0.12$
(numerically verified against the corpus CAMB anchors
$\Omega_c h^2{=}0{\to}[1,0.39,0.205]$ and
$\Omega_c h^2{=}0.12{\to}[1,0.45,0.44]$). This proves the existence and the
abundance claims.

\smallskip\noindent\textbf{Step 3 (radiation-pressure-supported mechanisms cannot do
it --- they drive their own well to zero).}
Suppose the proposed lift is carried by a component with $c_{s,s}^2>0$ (a relativistic
or sound-supported fluid: photon driving, a hot relic, the optical
$c_s(\psi)$ envelope, an AQUAL-enhanced radiation term). In radiation/pressure
domination the potential sourced by that component obeys, exactly,
\begin{equation}
\label{eq:wall-raddecay}
\fiteq{\Phi(x)=3\,\Phi_0\,\frac{\sin x - x\cos x}{x^3},
\qquad x\equiv\frac{k\eta}{\sqrt3},}
\end{equation}
whose super-horizon limit is $\Phi(0)=\Phi_0$ (well intact) but whose sub-horizon
envelope is $\Phi\simeq-3\Phi_0\cos x / x^2 \to 0$. The pressure that supports the
component against collapse simultaneously drives its own gravitational well away once
the mode enters the horizon (verified: $|\Phi/\Phi_0|$ falls as $0.024,0.0027,0.0013$
at $x=10,20,40$). Hence $\Phi_\infty=0$ for any pressure-supported source, the
standing-well condition of Step~2 fails, and by Step~1 no third-peak lift results. The
deep-MOND gravity enhancement, the $c_s(\psi)$ envelope, the impedance/feedback gain,
and the forced-echo device all fall in this class or in Step~4 and are therefore
closed by this step (they modify the \emph{transient} driving or the background, not
the surviving well).

\smallskip\noindent\textbf{Step 4 (time-only background modifications cancel in the
height; achromatic driving boosts all peaks equally).}
Suppose instead the proposed lift acts only through the homogeneous background ---
a modified $H(\eta)$, a time-dependent sound speed $c_s(\eta)$ with no spatial
structure, or an achromatic ($k$-independent) rescaling of the driving,
$\Phi\to g(\eta)\,\Phi$. Any such modification multiplies the forced amplitude of
\emph{every} mode by a common, $k$-independent factor. For an achromatic boost
$\Phi\to g\,\Phi$ the peak heights scale as $H_n\to g^2 H_n$, so every ratio
$H_n/H_1$ is left \emph{exactly} invariant (verified to machine precision in the
forced oscillator: $H_2/H_1$ and $H_3/H_1$ unchanged under $g=2.7$). A time-only
$c_s(\eta)$ or $H(\eta)$ enters $r_s$ and the damping envelope but, being independent
of $\hat k\cdot\hat g$ and of $k$ at fixed $k r_s$, cancels in $H_n/H_1$ at fixed peak
location by the same argument that makes the $1/\mu$ optical enhancement cancel in the
peak \emph{ratio} (\S\ref{app:mu-cancellation}). The only way a background device can
change $H_3/H_1$ is by shifting $z_{\rm eq}$ --- but $z_{\rm eq}$ is a statement about
which species dominates the \emph{perturbation source} \eqref{eq:wall-poisson}, i.e.\
exactly the cold-clustering abundance of Step~2, not a free background knob. This
closes the time-only class.

\smallskip\noindent
Steps~3 and~4 exhaust the two ways a non-cold mechanism could act (through the
transient/pressure-supported driving, or through the homogeneous background); both
fail. By Steps~1--2 the lift exists only with a pressureless standing well of
abundance \eqref{eq:wall-bound}, sharpened to $\Omega_c h^2\approx0.10$--$0.12$ by the
observed height. $\qed$
\end{proof}

\begin{remark}[What the theorem does and does not say]
\label{rem:wall-scope}
The theorem is a statement about the \emph{perturbation source}, not about the
ontology of the clustering species. It does \emph{not} say ``DFD needs a postulated
WIMP.'' It says: \emph{any} theory matching the Planck third-peak height must place an
$\Omega h^2\approx0.11$--$0.12$ pressureless, cold-clustering charge in the
gravitational source \eqref{eq:wall-poisson} at recombination, with equality before
last scattering. In $\Lambda$CDM that charge is a postulated cold particle of fitted
density; in DFD it is the \emph{derived} $\chi$-matter field
(App.~\ref{app:chi-matter}), whose mass and decay constant are fixed by the $\alpha$-tower.
The required abundance $\Omega_\chi h^2\simeq0.12$ is the kernel input a cold component of
that density supplies; DFD now \emph{derives} that normalization (post-inflation,
App.~\ref{app:chi-matter} Step~5b: the finite $SU(2)_{60}$ CS-vacuum Casimir expectation
gives $\Omega_\chi h^2=0.118$, $-1.5\sigma$, theorem-grade (its only non-DFD input the standard relic-redshift); the earlier
$\sim$9--44$\times$ overshoot was a classical-continuum-measure artifact). The wall is therefore not a defeat for DFD but the precise specification the
$\chi$ field is shown to meet: it enters \eqref{eq:wall-poisson} identically to a cold
component of the same $\Omega h^2$ (Theorem~\ref{thm:chi-thirdpeak}), so it sits on the
right side of the wall by construction.
\end{remark}

\begin{remark}[Why this consolidates the no-CDM waves]
\label{rem:wall-consolidation}
Every DFD-native no-CDM third-peak mechanism the corpus has tested is closed by a
named step of Theorem~\ref{thm:cold-clustering-wall}: the chromatic-Jacobian
line-of-sight $\psi$-screen, the deep-MOND gravity enhancement, the nonlinear AQUAL
injection, and the forced sound-echo are pressure-supported / transient devices closed
by Step~3 (they cannot leave a standing $\Phi_\infty\ne0$); the $c_s(\psi)$
sound-speed envelope and any modified-$H(\eta)$ background are time-only devices closed
by Step~4 (they cancel in $H_n/H_1$ or merely rescale all peaks). The heavy-mode and
topological-defect routes are closed upstream (no free cold $a^{-3}$ channel in the 4D
spectrum). The single surviving carrier of a pressureless standing well is a genuine
cold $a^{-3}$ matter field with equality before recombination --- which is precisely
the derived $\chi$ field. This is the theorem-grade form of the empirical convergence
recorded across the no-CDM investigation: the third-peak \emph{height} is a cold-clustering
charge, and no time-only or pressure-supported DFD device can carry it.
\end{remark}
