%=============================================================================
% Appendix CL: Publication-Grade CMB Likelihood and Bayesian Model Comparison
%
% PURPOSE: Document the publication-grade primary-CMB likelihood (Planck
% plik-lite, full 613x613 Fortran covariance, cobaya-validated) and the
% Bayesian model comparison of DFD vs LCDM on real cosmological data. State the
% two-sided result: DFD is COMPETITIVE on CMB+BAO+SNe and WINS on parameter
% economy, but is DISFAVOURED once the Planck CMB-lensing bandpowers are added.
%
% SCOPE: this is a confrontation/adjudication appendix, not a new win.
% No DFD win is created. CORRECTION 2026-06-27 (CL.4): at the FORCED A_s=32pi alpha^5
% (sigma8=0.820) the CMB-lensing test is a FLAT deficit, A_L=1.0542+/-0.0263,
% +2.06 sigma PASS (chi2=12.66/9, dchi2=+3.9; 2026-07 correction of the stale
% "A_L=1.046, +1.8 sigma, chi2~11.7" line) -- the falsifier has NOT fired. The old +232/+217/>=6.5sigma "LCDM decisively
% favoured" was the sigma8=0.790 arm through a cobaya-path normalization bug (retracted
% inline in CL.3/CL.5, superseded by CL.4). The item-4 S8 prediction is the forced
% S8=0.784 (mild galaxy-lensing match, KiDS +1.1 sigma/DES +0.5 sigma), available
% SIMULTANEOUSLY with the lensing pass (one sigma8, Q=1). S8=0.755 retires to an
% alternative corpus normalization not simultaneously available with the pass.
%
% ENVIRONMENTS: uses only master-document environments (remark, falsifier,
% status, tcolorbox, booktabs). revtex4-2 twocolumn safe: wide table in table*,
% no p{} columns, \fittabFull shrink-to-fit.
%=============================================================================

\section{Publication-Grade CMB Likelihood and Bayesian Model Comparison}
\label{app:cl-cmb-likelihood}

\subsection{CL.1\quad Scope and the publication-grade primary CMB}

This appendix records the real-data confrontation of DFD with the standard
cosmological probes at \emph{likelihood} grade and the Bayesian model
comparison against $\Lambda$CDM. The primary-CMB step is publication-grade:
the Planck 2018 high-$\ell$ \textsf{plik\_lite} \textsf{TT/TE/EE} likelihood is
evaluated with its \emph{full $613\times613$ Fortran binned covariance} (not a
diagonal first pass), with the Planck calibration $A_{\rm planck}$ profiled, and
the pipeline is \emph{cobaya-validated}: feeding the screened DFD bandpowers
through cobaya's own \textsf{PlanckPlikLite.get\_chi\_squared} reproduces the
hand-rolled $\chi^2$ to the digit ($\Lambda$CDM $587.6$, DFD $610.9$ at
$A_{\rm planck}=1$), and the pipeline returns the published $\Lambda$CDM
\textsf{plik\_lite} value as a unit test.

The DFD cosmology is run on its \emph{derived/frozen} background parameters
($H_0=72.09$, $\omega_b=0.02237$, $\omega_\chi h^2=0.1199$, $n_s=0.9667$,
$A_s=2.08\times10^{-9}$, $\tau=0.054$) with the optical $\theta_*$ screen
applied as the $\theta_*$ multipole rescale at \emph{one fitted amplitude}
$A^*=0.0908$; \emph{no refit of the background}. The screen amplitude is
\textbf{fitted, not derived}: the fitted $A^*=0.0908$ sits $-26.6\%$ off the
$2\sqrt{\alpha}$-gate value $A=2\sqrt{\alpha}\,R_d/(1+R_d)=0.0667$, and the
frozen gate value does \emph{not} reproduce the fit --- evaluated at the
consistent (CAMB) convention it leaves $\theta_*$ misplaced at $+12.2\sigma$
and costs \textsf{plik\_lite} $\Delta\chi^2=+202/+172$ (per-bin variants). An
earlier ``frozen-amplitude validation'' claiming the gate value works as well
as the fit is \textbf{retracted}: it rested on a cross-pipeline calibration
transplant (a toy pipeline miscalibrated by $-0.32\%$ in absolute $\theta_*$
compared against a CAMB-convention measurement). What survives is the honest
headline: $\theta_*$ as a \emph{one-parameter fitted screen}.
A 2026-07 forward re-derivation (3/3 independent confirmations; conventions
pinned by first reproducing the anchors of this paragraph, including the
frozen-gate $+12.2\sigma$ and the fitted $A^*\to+0.0\sigma$) exhausted the
derivable-amplitude menu at the consistent CAMB convention: the drag-loaded
$2\sqrt{\alpha}$ gate ($+12.2\sigma$), the $\sqrt{f_b}$ loading
($+11.7\sigma$ drag / $-1.5\sigma$ saturated), the AQUAL
$f^{0.72}$/$f^{0.79}$ kernels ($+14.6\sigma$ to $+26.1\sigma$), the ungated
screens ($-41.7\sigma$/$-75.4\sigma$), and inserted gate multipliers
($\times\sqrt2\to-2.0\sigma$, $\times4/\pi\to+2.9\sigma$,
$\times2\to-22.1\sigma$, $\times\pi/4\to+19.6\sigma$,
$\times\tfrac12\to+29.3\sigma$) all fail to land on $A^*$; the near misses
(the saturated-loading gate at $-0.7\sigma$, $\times\sqrt2$ at $-2.0\sigma$)
are menu \emph{selections} --- a convention branch and an inserted multiplier
--- and promoting one to a derivation would incur exactly the trials charge
of App.~\ref{app:CE:combined-evidence}. The forced boost identity
$1/\mu-1=2\sqrt{\alpha}$ checks exactly (residual $0$) but is
$z$-independent, so it cannot generate the screen's $\Omega_\Lambda(z)$
loading shape, which enters as a separately inserted kernel. A
single-amplitude scan confirms the trade-off: the gate value $A=0.0667$
reproduces the BAO sector ($\chi^2=9.5/9.3$, matching the corpus $9.4$)
while misplacing $\theta_*$ at $+12.2\sigma$, whereas the fitted
$A^*=0.0904$ lands $\theta_*$ at $+0.0\sigma$ but degrades BAO to
$12.6/13.3$ (SNe essentially unchanged, $1390.2\to1388.6$): one amplitude
cannot serve both sectors from the gate. The fitted-screen status is
confirmed, not softened.
On the frozen
background with the single fitted amplitude DFD attains
$\chi^2/\mathrm{dof}=0.97$ ($593.5/613$, amplitude-profiled) versus $\Lambda$CDM
$0.96$ ($585.5/613$): $\theta_*$ lands at $+1.0\sigma$ and $\Lambda$CDM is
marginally preferred by $\Delta\chi^2=+23.3/+8.0$ (per-bin variants) over $613$
degrees of freedom (statistically minor). DFD is an
excellent \emph{absolute} fit to the full Planck spectrum.
Figure~\ref{fig:cl-cmb-tt} shows this forward-computed TT spectrum directly.

\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{fig_cmb_tt.png}
\caption{DFD forward-computed CMB TT power spectrum
$\mathcal{D}_\ell=\ell(\ell+1)C_\ell/2\pi$ at the \emph{frozen derived}
background parameters ($H_0=72.09$, $\omega_b=0.02237$, $\omega_\chi h^2=0.12$,
$n_s=0.9667$, $A_s=2.08\times10^{-9}$; background not refit, $\theta_*$ screen
at the single fitted amplitude $A^*=0.0908$) versus the Planck-2018
$\Lambda$CDM reference. This is the DFD-CLASS/CAMB \emph{forward} run of the
likelihood pipeline documented in this appendix (CL.1; script
\texttt{dfd\_class/figures/make\_cmb\_tt.py}): the first three acoustic peaks,
the third-to-first peak height ratio, and $100\,\theta_*$ are annotated for
each model.}
\label{fig:cl-cmb-tt}
\end{figure}

\subsection{CL.2\quad Parameter count}

$\Lambda$CDM carries six free parameters: $\omega_b$, $\omega_c$,
$\theta_*$ (or $H_0$), $A_s$, $n_s$, $\tau$. DFD \emph{derives} most of these
from its tower and freezes them as sharp targets, not fits:
\begin{itemize}\itemsep1pt
\item $n_s=1-2/60=0.9667$ (mode-count tilt),
\item $\omega_\chi h^2=0.12$ (the $16/3\to$ misalignment chain),
\item $H_0=72.09$ (the $\alpha^{28.5}/t_P$ relation),
\item $\omega_b$ (BBN/baryon trace) and the screen amplitude
      $A_{\rm screen}=2\sqrt{\alpha}\,f_b$.
\end{itemize}
The genuinely \emph{fitted} DFD knobs are the misalignment angle $\theta_i$
(setting $\Omega_\chi$) and the $A_s$ overall coefficient (the $32\pi$
normalization); the reionization optical depth $\tau$ is an astrophysical
nuisance shared with $\Lambda$CDM. We therefore quote two counts:
a conservative $k_{\rm DFD}=3$ ($\theta_i$, $A_s$ coeff, $\tau$) used for the
table below, and $k_{\rm DFD}=4$, the value used by the recorded joint pipeline
(and cross-checked by its own $\Delta\mathrm{AIC}=+228$). A maximal skeptic who
rejects \emph{every} derivation drives the count to $k_{\rm DFD}=8$. In all
counts DFD frees \emph{fewer} parameters than $\Lambda$CDM, which is the
source of its parameter-economy (BIC) credit.

\subsection{CL.3\quad Bayesian model comparison}

Table~\ref{tab:cl-modelcomp} collects the recorded per-sector best-fit $\chi^2$
(all from real data: \textsf{plik\_lite} full covariance, $12$ BAO consensus
points, the $1580$-SN Pantheon$+$ cosmology sample with full stat$+$sys
covariance, and the full Planck 2018 nine-bin lensing bandpower likelihood) and
the model-comparison statistics
\begin{equation}
\fiteq{
\mathrm{BIC}=\chi^2+k\ln N,\qquad
\mathrm{AIC}=\chi^2+2k,}
\end{equation}
for the two data combinations:
\textbf{(A)} CMB$+$BAO$+$SNe (no lensing) and
\textbf{(B)} CMB$+$BAO$+$SNe$+$lensing. Deltas are $\Delta\equiv(\mathrm{DFD})-(\Lambda\mathrm{CDM})$,
so a \emph{negative} $\Delta\mathrm{BIC}$ favours DFD.

\begin{table*}[t]
\centering
\caption{Bayesian model comparison, DFD vs $\Lambda$CDM, on real Planck$+$BAO$+$SNe$+$lensing data.
Per-sector $\chi^2$ are recorded best-fit values; the $\theta_*$ screen carries the single
fitted amplitude $A^*=0.0908$ ($-26.6\%$ off the $2\sqrt{\alpha}$-gate value $0.0667$; the
frozen gate value fails at $+12.2\sigma$, $\Delta\chi^2=+202/+172$, and its earlier
``validation'' is retracted --- background not refit). $\Delta\equiv$DFD$-\Lambda$CDM. Combination (A) excludes the Planck CMB-lensing
bandpowers; (B) includes them. \textbf{NOTE (SUPERSEDED by CL.4):} the Planck-lensing row and the
Combination-(B) totals below are the \emph{$\sigma_8=0.790$ arm run through a cobaya-path
normalisation bug}. The from-scratch CAMB recompute at the \emph{forced} $\sigma_8=0.820$ gives a
flat $C_L^{\phi\phi}$ ratio, $A_L=1.0542\pm0.0263$, $\Delta\chi^2_{\rm lens}=+3.9$
($\chi^2=12.66/9$, $+2.06\sigma$ PASS), so
the corrected Combination (B) reverts to the Combination-(A) near-tie. The $+212/+232/+217$ figures
are retained here only to document the retraction.}
\label{tab:cl-modelcomp}
\fittabFull{%
\begin{tabular}{lrrrr}
\toprule
Sector (real data) & $N$ & $\chi^2_{\rm DFD}$ & $\chi^2_{\Lambda{\rm CDM}}$ & $\Delta\chi^2$ \\
\midrule
CMB \textsf{plik\_lite} (full $613{\times}613$ cov) & $613$ & $593.5$ & $585.5$ & $+8.0$ \\
BAO ($12$ consensus pts)                            & $12$  & $9.4$   & $13.2$  & $-3.8$ \\
Pantheon$+$ SNe ($1580$, full cov)                  & $1580$& $1390.4$& $1387.2$& $+3.2$ \\
Planck lensing ($9$-bin, \emph{$\sigma_8{=}0.790$ arm, cobaya-bug; SUPERSEDED}) & $9$ & $221.0$ & $8.7$ & $+212.3$ \\
Planck lensing ($9$-bin, \textbf{forced $\sigma_8{=}0.820$; CL.4})      & $9$   & $12.66$ & $8.7$   & $+3.9$ \\
\midrule
\multicolumn{5}{l}{\textbf{Combination (A): CMB$+$BAO$+$SNe} \quad ($N=2205$, $k_{\rm DFD}=3$, $k_{\Lambda{\rm CDM}}=6$)}\\
\quad joint $\chi^2$  & $2205$ & $1993.3$ & $1986.0$ & $+7.3$\ \ ($\sim+20$ profiled)\\
\quad joint BIC       &        & $2016.4$ & $2032.2$ & $\mathbf{-15.8}$\\
\midrule
\multicolumn{5}{l}{\textbf{Combination (B): CMB$+$BAO$+$SNe$+$lensing} \quad ($N=2214$, $k_{\rm DFD}=3$, $k_{\Lambda{\rm CDM}}=6$)}\\
\multicolumn{5}{l}{\quad\emph{(B, SUPERSEDED --- $\sigma_8{=}0.790$ arm, cobaya-bug lensing row):}}\\
\quad joint $\chi^2$  & $2214$ & $2214.3$ & $1994.6$ & $+219.6$\ \ ($+232$ profiled)\\
\quad joint BIC       &        & $2237.4$ & $2040.9$ & $\mathbf{+196.5}$\ \ ($+217$ profiled)\\
\multicolumn{5}{l}{\quad\textbf{(B, CORRECTED --- forced $\sigma_8{=}0.820$, flat $C_L^{\phi\phi}$, CL.4):}}\\
\quad joint $\chi^2$  & $2214$ & $\sim1996$ & $1994.6$ & $\sim+1$ (near-tie)\\
\quad joint BIC       &        & $\sim2018$ & $2040.9$ & $\mathbf{\sim-15}$ (mildly DFD)\\
\bottomrule
\end{tabular}}
\end{table*}

\paragraph{Combination (A), no lensing.}
The joint $\chi^2$ gap is small ($\Delta\chi^2=+7.3$ per-sector, $\approx+20$ in
the amplitude-profiled pipeline, on $N\sim2.2\times10^3$). DFD's parameter
economy ($3$ fewer free parameters, a BIC credit of
$3\ln N\simeq+23$) outweighs that gap:
$\Delta\mathrm{BIC}_{\rm(A)}=-15.8$ (conservative $k_{\rm DFD}=3$) to
$-8.1$ ($k_{\rm DFD}=4$). Both are negative: \textbf{DFD is competitive with,
and on parameter economy mildly favoured over, $\Lambda$CDM on CMB$+$BAO$+$SNe.}
We flag that the no-lensing $\Delta\mathrm{BIC}$ sign is borderline and
prior/count dependent; the proper Laplace evidence
$\ln B(\mathrm{DFD}/\Lambda\mathrm{CDM})\approx-1.2$ to $+0.8$ makes this an
\emph{evidence-level tie}, not a clean DFD win. The bare ``strong DFD'' BIC
headline is a $1/\sqrt N$ shortcut and is not claimed.

\paragraph{Combination (B), with CMB lensing. [SUPERSEDED by CL.4 --- retained to document the retraction.]}
The figures in this paragraph are the \emph{$\sigma_8=0.790$ arm computed through a
cobaya-path normalisation bug} and are \textbf{withdrawn}. As recorded, the Planck
CMB-lensing sector \emph{appeared} to contribute $\Delta\chi^2=+212.3$
($221$ vs $8.7$ over $9$ bins), giving
\begin{equation}
\fiteq{
\Delta\mathrm{BIC}_{\rm(B)}\ (\text{buggy},\ \sigma_8{=}0.790)
=+232-2\ln 2214=+217.}
\end{equation}
\textbf{Corrected (CL.4, from-scratch CAMB at the forced $\sigma_8=0.820$):} the
$C_L^{\phi\phi}$ ratio is \emph{flat} (below $\Lambda$CDM, shape scatter
$0.3\%$), $A_L=1.0542\pm0.0263$, $\Delta\chi^2_{\rm lens}=+3.9$
($\chi^2=12.66/9$, $+2.06\sigma$ PASS), so the
lensing sector does \emph{not} swamp the parameter-economy credit; Combination (B)
reverts to the Combination-(A) near-tie ($\Delta\mathrm{BIC}_{\rm(B)}\approx
\Delta\mathrm{BIC}_{\rm(A)}$, mildly DFD-favoured on parameter economy, an
evidence-level tie). The retracted ``$\Lambda$CDM decisively favoured /
$\ge6.5\sigma$'' headline was the buggy $\sigma_8=0.790$ arm; $\sigma_8=0.790$ is
not the forced amplitude and is not simultaneously available with the lensing pass.

\paragraph{The $\sigma_8$ amplitude: derived chain, shared data anomaly (2026-07-02).}
The DFD amplitude chain --- $A_s=32\pi\alpha^5$ (the $\alpha^5$ exponent derived; the
$32\pi$ coefficient is the single fitted normalization), $n_s=1-2/\Omega=0.9667$, derived
$H_0=72.09$ and $\Omega_m=0.274$, gravity-universal growth --- yields $\sigma_8=0.820$ and
$S_8=0.784$ with no further freedom (forward CAMB at the canonical inputs: $\sigma_8=0.8193$,
$S_8=0.7843$; with the strictly derived-native $\omega_\chi=0.118$ in place of the
$\theta_i$-fitted $0.1199$, $\sigma_8\simeq0.811$ --- still mid-range, still no new knob).
This resolves the cosmic-shear tension ($\chi^2=0.73$ vs $\Lambda$CDM's $10.15$;
$+0.4$--$1.3\sigma$ vs $+3$--$4.9\sigma$) at the cost of a mild Planck $\phi\phi$
clean-window pull ($+3.9\,\chi^2$; $\phi\phi$ alone prefers $\sigma_8\approx0.842$).
Crucially, this lensing-high/shear-low split is \emph{internal to the data}: on identical
clean-window data no single-amplitude model fits both --- the joint fit carries an
irreducible residual of $\approx3.9\,\chi^2$ ($\sim2\sigma$) at DFD's $\Omega_m$ and
$\approx13.6\,\chi^2$ ($\sim3.7\sigma$) at Planck's $\Omega_m$ --- and DFD's mid value sits
within $1.2\,\chi^2$ of the joint optimum while $\Lambda$CDM pays $\approx4.9$ over its own
floor ($27.3$ vs $13.9$ in total). \emph{Falsifier:} if future CMB lensing (SO/CMB-S4) and
shear converge on $\sigma_8\approx0.84$ at DFD's background, $\sigma_8=0.820$ is disfavoured
and the $32\pi$ normalization cannot absorb it without breaking the shear win.
\emph{(Scope, hardened by computation: the $\sim50/9$ figure from the fiducial-referenced
likelihood is an artifact of applying the CMB-linear correction --- anchored to the Planck
fiducial sky --- to DFD's unscreened theory TT outside the correction's regime (per-bin
shifts up to $2.5\sigma_b$, $1.0$--$2.6\times$ beyond the $\pm2\sigma$ $\Lambda$CDM-posterior
envelope). Evaluated with Planck's own CMB-marginalized consext8 likelihood --- the
prescribed variant for non-fiducial backgrounds --- DFD gives $\chi^2=14.0/9$
($A_L=1.067\pm0.029$, $+2.3\sigma$) versus $\Lambda$CDM $9.3/9$: the same shared-anomaly
regime as the clean-window channel, closing the lensing item without appeal to internal
judgment. DFD's residual $A_L$ pull ($+2.3\sigma$ vs $\Lambda$CDM's $+0.8\sigma$) remains
within the shared-anomaly interpretation; the SO/CMB-S4 falsifier above stands.)}

\subsection{CL.4\quad Summary assessment}

\noindent
\begin{tcolorbox}[colback=blue!5!white, colframe=blue!60!black,
  title=Summary assessment (two-sided), breakable]
DFD is \textbf{competitive with $\Lambda$CDM on CMB$+$BAO$+$SNe} (joint
$\Delta\chi^2\sim+20/634$ in the recorded pipeline; a near-tie) and \textbf{wins
on parameter economy} (it derives, rather than fits, several $\Lambda$CDM
parameters).
\textbf{CORRECTION (2026-06-27, from-scratch rebuild; this CL.4 box SUPERSEDES the
CL.3 table headline and the CL.5 remark below).} The earlier recorded CMB-lensing
penalty ($\Delta\chi^2=+232$, $A_L=1.40$, ``$\Lambda$CDM decisively favoured'') is
\emph{withdrawn}. The correct reconciliation is \emph{two separate facts}, not a single
inflation factor: \textbf{(a)} the $+232$ lived at the input $\sigma_8=0.790$ (an
alternative corpus normalization, \emph{not} the forced amplitude) and was further
distorted by a \emph{normalisation bug in the inherited cobaya-path pipeline}; and
\textbf{(b)} the PASS belongs to a \emph{different input} --- the genuinely forced
$A_s=32\pi\alpha^5$, whose CAMB-native $\sigma_8=0.820$ gives a \emph{flat}
$C_L^{\phi\phi}$ ratio. (We do not quote a single ``$\times N$'' deflation of the
$+232$: dividing it by any factor still leaves a fail; the resolution is that the PASS
is a different $\sigma_8$ arm, not a rescaled $0.790$ arm.) An independent
reconstruction of the Planck 2018 9-bin lensing likelihood from the official
bandpowers/covariance/windows (no cobaya), validated to reproduce the published
$\Lambda$CDM $\chi^2=8.7/9$ and $A_L=1.0$, and a direct CAMB recompute, both give: at
the forced $\sigma_8=0.820$ the lensing power is a flat deficit below $\Lambda$CDM
across $L=8$--$400$ (shape scatter $0.3\%$: pure amplitude, no shape penalty); the
nine-bin likelihood returns $A_L=1.0542\pm0.0263$, \textbf{a mild $+2.06\sigma$ lean
that PASSES} ($\chi^2=12.66/9$, $\Delta\chi^2=+3.9$). The joint accordingly reverts from ``decisively
$\Lambda$CDM'' to the \textbf{near-tie}. The cause of the (mild) deficit is structural
--- DFD's high derived $H_0=72.09$ forces low $\Omega_m=0.274$ and its $\chi$-matter
growth is GR-like ($Q\simeq1$), so $C_L^{\phi\phi}\propto\Omega_m\sigma_8^2$ runs
$\sim4\%$ low --- but it is \emph{mild, not a catastrophe}. The cost of using the
forced $\sigma_8=0.820$: $S_8=0.784$, so the ``$S_8$-low'' galaxy-lensing win of
App.~\ref{app:growth-regime} softens to a \emph{mild match} (KiDS-1000 $+1.0\sigma$,
DES-Y3 $+0.5\sigma$). The sharper $\sigma_8=0.790$/$S_8=0.755$ is \emph{not
simultaneously available with this pass}: at $\sigma_8=0.790$ the deficit grows to
$\sim11\%$ and the lensing likelihood disfavours DFD at $>3.6\sigma$. Because DFD's
growth is $Q=1$, one $\sigma_8$ sets both arms, so the lensing pass and the
$S_8=0.755$ win cannot be claimed together.
\end{tcolorbox}

The CAMB-only figure \texttt{fig\_cmb\_lensing.png} (script
\texttt{cmb\_lensing\_figure.py}; one quick spectrum evaluation per model, no
sampler) makes the cause explicit: across the entire Planck lensing band
$L=8$--$400$ the DFD $C_L^{\phi\phi}$ curve sits \emph{slightly below}
$\Lambda$CDM, by a \emph{flat} amount (shape scatter $0.3\%$). The
signal-weighted lensing-amplitude ratio is
\begin{equation}
\fiteq{
\frac{A_L^{\rm DFD}}{A_L^{\Lambda{\rm CDM}}}=
\begin{cases}
0.956 & (\sigma_8=0.820,\ \textbf{forced } A_s=32\pi\alpha^5),\\[2pt]
0.890 & (\sigma_8=0.790,\ S_8=0.755,\ \text{alt.\ corpus norm.}),
\end{cases}}
\end{equation}
a $\sim4\%$ deficit at the forced normalization (growing to $\sim11\%$ only at the
alternative $\sigma_8=0.790$). Because DFD sits flatly below, the data prefer
$A_L=1.0542\pm0.0263$ to match it $\Rightarrow$ a small \emph{positive} $\chi^2$ penalty,
\textbf{$+2.06\sigma$, a mild PASS} at the forced $\sigma_8=0.820$.
\textbf{[SUPERSEDED:} the earlier statement that this was ``the correct sign and
magnitude bracket / per-mode shadow of the full 9-bin $\gtrsim6.5\sigma$
$\Delta\chi^2=+232$ result'' is \emph{withdrawn}: those numbers are the
$\sigma_8=0.790$ arm run through a cobaya-path normalisation bug, not the forced
$\sigma_8=0.820$ result, which is a flat-ratio $+2.06\sigma$ pass --- see the CL.4
box.\textbf{]} The separate ``DFD wins lensing'' \emph{proxy} line used the
$\sigma_8\,\Omega_m^{0.25}$ proxy, whose weak matter weighting let low
$\Omega_m$ over-compensate low $\sigma_8$; the full bandpower kernel weights matter
correctly. The proxy is retired in favour of the direct $C_L^{\phi\phi}$ ratio
($A_L=1.0542$, $+2.06\sigma$ pass).

\subsection{CL.5\quad Reconciliation of the item-4 $S_8$ prediction (corrected)}

\begin{remark}[The forced $S_8=0.784$ is a mild match; the $S_8=0.755$ ``win'' retires]
\label{rem:cl-s8-twosided}
\textbf{[CORRECTED, consistent with CL.4.]} DFD's \emph{forced} structure-growth
prediction ($A_s=32\pi\alpha^5$, $\sigma_8=0.820$, $\Omega_m=0.274$) is
\begin{equation}
\fiteq{
S_8=\sigma_8\sqrt{\Omega_m/0.3}=0.8193\sqrt{0.2749/0.3}=0.784.}
\end{equation}
This is a \emph{mild match to galaxy weak lensing on the low side}: KiDS-1000
($0.759\pm0.024$) at $+1.0\sigma$ and DES-Y3 $3\times2$pt ($0.776\pm0.017$) at
$+0.5\sigma$ --- both within $\sim1\sigma$, a mild match rather than a dramatic
win. Because DFD's growth is $Q\equiv G_{\rm eff}/G=1$ (LCDM-like), this
\emph{same} $\sigma_8=0.820$ also sets the CMB-lensing amplitude, where it gives a
flat $A_L=1.0542$ ($+2.06\sigma$ PASS, CL.4). The galaxy-lensing match and the
CMB-lensing pass are therefore \textbf{simultaneously available} at one
$\sigma_8=0.820$ --- there is no two-sided contradiction at the forced
normalization.

The earlier ``$S_8=0.755$ win'' is \emph{retired}: it is just $\sigma_8=0.790$
restated ($S_8=\sigma_8\sqrt{\Omega_m/0.3}$), and $\sigma_8=0.790$ is an
alternative corpus CMB-normalization (asserted, not re-derived; it requires
shrinking the forced $A_s$ by $7.2\%$). At $\sigma_8=0.790$ the CMB-lensing
deficit grows to $\sim11\%$ and the nine-bin likelihood disfavours DFD at
$>3.6\sigma$. Since one $\sigma_8$ ($Q=1$) sets \emph{both} arms, the sharper
$S_8=0.755$ galaxy-lensing number and the CMB-lensing pass \textbf{cannot} be
claimed together. The single prediction is the forced $S_8=0.784$ mild
match with a $+2.06\sigma$ CMB-lensing pass. The previously recorded
``$A_L=1.40$, $+14.7\sigma$, $\chi^2=221/9$, $\Delta\mathrm{BIC}=+217$,
$\Lambda$CDM favoured'' is \emph{withdrawn} (CL.4): that was the $\sigma_8=0.790$
arm through a cobaya-path normalisation bug. The old ``DFD has no $S_8$ tension''
phrasing is correctly scoped: DFD is a mild low-side match to galaxy lensing and
a mild pass of CMB lensing, jointly.
\end{remark}

\subsection{CL.6\quad Status and falsifier}

\begin{status}[CMB-likelihood package]
\label{stat:cl-package}
\emph{Publication-grade:} primary CMB (\textsf{plik\_lite} \textsf{TT/TE/EE},
full $613\times613$ covariance, $A_{\rm planck}$-profiled, cobaya-validated);
Planck 2018 nine-bin lensing bandpower (cobaya native CMBlikes,
$\Lambda$CDM-validated at $\chi^2=8.7/9$, $A_L=1.004\pm0.024$); BAO
($\chi^2/\mathrm{dof}=0.78$); Pantheon$+$ SNe (full stat$+$sys covariance).
\emph{Remaining for full publication maximality:} (i) the full \textsf{plik}
\textsf{TT/TE/EE} with explicit foregrounds (we use the cobaya-validated
\textsf{plik\_lite}); (ii) the BAO full covariance (we use consensus points with
quoted errors); (iii) nested-sampling Bayesian evidence (PolyChord/MultiNest not
available here) --- the proper Laplace evidence is the substitute and gives an
evidence-level tie; with the corrected CMB lensing ($+2.06\sigma$ flat-ratio PASS
at the forced $\sigma_8=0.820$, CL.4) folded in, the joint stays a near-tie
(the earlier ``$\Delta\mathrm{BIC}=+217$ with lensing'' was the $\sigma_8=0.790$
cobaya-bug arm and is withdrawn).
\end{status}

\begin{falsifier}[CMB lensing --- mild PASS at the forced normalization]
\label{fals:cl-lensing}
The frozen DFD prediction at the \emph{forced} amplitude ($A_s=32\pi\alpha^5$,
$\sigma_8=0.820$, $\Omega_m=0.274$, $S_8=0.784$, GR-like $Q=1$ growth) gives a
\emph{flat} $C_L^{\phi\phi}$ ratio below $\Lambda$CDM across
$L=8$--$400$, i.e.\ $A_L=1.0542\pm0.0263$, a mild \textbf{$+2.06\sigma$ PASS} of the Planck
2018 nine-bin lensing likelihood ($\chi^2=12.66/9$, $\Delta\chi^2=+3.9$).
\textbf{The falsifier has NOT fired at the forced normalization.} The earlier
``$A_L=1.40$, $+14.7\sigma$, $\Delta\mathrm{BIC}=+217$, decisively
$\Lambda$CDM'' statement is \emph{withdrawn} (CL.4): it was the $\sigma_8=0.790$
arm run through a cobaya-path normalisation bug. The genuine \emph{open} item is
not a fired falsifier but a scoping question (App.~\ref{app:growth-regime}, GR.7):
the ``one $\sigma_8$ sets both arms'' lock rests on $\chi$ clustering Newtonianly
($Q=1$) on linear scales --- asserted, not yet derived. If instead the AQUAL
screen leaked into linear growth, $Q$ would run with $z$, $C_L^{\phi\phi}$ would
shift, and the sharper $S_8=0.755$ galaxy-lensing win could revive (while the
flat-ratio CMB-lensing pass would change). This connects to the long-standing
``needs-clustering / effective-CDM'' wall of the third-peak hunt; it is the
deciding open dark-sector growth item, not a present falsification.
\end{falsifier}

\noindent\textbf{One-line standing.} DFD is globally competitive with
$\Lambda$CDM on CMB$+$BAO$+$SNe$+$lensing with fewer derived inputs: at the
forced $\sigma_8=0.820$ it gives a mild $+2.06\sigma$ CMB-lensing pass and a mild
$S_8=0.784$ galaxy-lensing match \emph{jointly}, so the joint fit is a near-tie.
The active frontier is the (asserted, not yet derived) $Q=1$-vs-deep-MOND scoping
of $\chi$-matter linear growth (GR.7), which decides whether the sharper
$S_8=0.755$ win can revive.
