%=============================================================================
% Appendix PK: The DFD-native production galaxy power-spectrum P(k) pipeline
%
% PURPOSE: close the *pipeline* half of open-problems item F. Appendix GR
% (growth regime) closed the regime QUESTION but explicitly left the pipeline
% obligation open: "Still program-grade: full production P(k)/Boltzmann
% pipeline." This appendix documents the end-to-end, window-convolved,
% covariance-weighted forward model that turns the frozen DFD cosmology into a
% predicted galaxy power spectrum and scores it against real BOSS DR12 data,
% head-to-head with LCDM through the IDENTICAL pipeline.
%
% SCOPE: this is an infrastructure + confrontation appendix, NOT a new
% win. MAXIMAL-FROZEN protocol result (2026-07-02, two independent
% reruns agree to all digits): at k<=0.15 DFD and Planck-LCDM are statistically
% indistinguishable (chi^2/dof 1.059 vs 0.967); at k<=0.20 DFD trails by
% Delta chi^2 = +21.5 over 124 points (1.311 vs 1.137) -- acceptable but
% disfavored, at the edge of the <=1.3 band (PK.11). The earlier ~1.27
% AP-marginalised figure held only under the retired sigma8->0.790
% normalization and is superseded. Not a full-shape loss, not a tie. The
% number is whatever the optimiser returned; nothing is hard-coded (guard
% against manufactured wins).
% 2026-07 re-audit (3/3): consistent-neutrino rerun gives +23.0 (m_nu=0
% protocol was ~1.5 units GENEROUS to DFD); headline +21.5 stands as-run.
%=============================================================================

\section{The DFD-Native Production Galaxy $P(k)$ Pipeline}
\label{app:pk-pipeline}

\subsection{PK.1\quad Purpose: closing the pipeline half of item F}

Appendix~\ref{app:growth-regime} (GR) closed the cosmological growth/velocity
\emph{regime question}: linear growth is carried by $\chi$-matter with
$Q\equiv G_{\rm eff}/G=1$, GR-like, giving the single primary prediction at the
\emph{forced} $A_s=32\pi\alpha^5$, $\sigma_8(0)=0.820$, $f(0)=0.487$,
$S_8=0.784$ (the alternative corpus normalization $\sigma_8=0.790$/$S_8=0.755$ is
asserted, not re-derived, and is contingent --- see GR.5/GR.7). It explicitly left open the
\emph{pipeline} obligation of open-problems item F: every prior DFD
power-spectrum confrontation
(\texttt{pk\_analysis\_pipeline.py}, \texttt{dfd\_Pk\_confrontation\_v2.py})
stopped at a linear Kaiser \emph{multipole-ratio} inversion ($\beta=f/b$ from
$P_2/P_0$) on mock-derived ratios. None forward-modelled a window-convolved,
covariance-weighted, full-spectrum $\chi^2$ against real data.

This appendix documents the script
(\texttt{dfd\_class/dfd\_pk\_pipeline.py}) that does. It takes the frozen DFD
cosmology to a predicted galaxy $P(k)$ multipole vector, convolves it through
the published survey window, and scores it against the real Beutler
\emph{et~al.} BOSS DR12 measurements under a real Patchy-mock covariance, with
a matched $\Lambda$CDM run through the identical code path.

\noindent
\begin{tcolorbox}[colback=blue!5!white, colframe=blue!60!black,
  title=Summary of this appendix]
The production pipeline is built, runnable, and self-validating ($\Lambda$CDM
recovers $\chi^2/\nu\simeq1.0$ on the mock-covariance bins), with
self-consistent Patchy $P_0{+}P_2$ mock covariances built for \emph{all four}
BOSS DR12 bins (NGC/SGC $\times z_{1,3}$; each passes the single-mock-vs-mean
validation at $\chi^2/\mathrm{dof}\simeq0.55$--$0.93$).
Under a maximal-frozen protocol (no $\sigma_8$ renormalization,
$A_s{=}32\pi\alpha^5$ frozen, identical nuisance freedom for both models,
fixed model-predicted Alcock--Paczynski dilation; two independent
reruns agree to all digits): in the clean linear regime
$k\le0.15\,h\,\mathrm{Mpc}^{-1}$ DFD and Planck-$\Lambda$CDM are statistically
\emph{indistinguishable} (DFD $\chi^2/\mathrm{dof}=1.059$, $p=0.34$, vs
$0.967$); at $k\le0.20$ DFD trails by $\Delta\chi^2={+}21.5$ over 124 points
($1.311$ vs $1.137$) --- acceptable but disfavored, at the edge of the
$\le1.3$ competitive band (PK.11). The deficit is a \emph{broadband shape}
effect: the turnover sits at slightly too large a scale, the direct signature
of DFD's lower $\Omega_m=0.274$ at higher $H_0=72.1$ --- the same single
mechanism that resolves the $H_0$ and $S_8$ tensions. Identical 6-term
broadband marginalization for both models reduces the gap to $+9.7$ (DFD
$1.012$). An earlier AP-marginalised figure of $\chi^2/\mathrm{dof}\simeq1.27$
held only under the retired $\sigma_8{\to}0.790$ normalization and is
superseded by the frozen-protocol numbers above. A false-positive guard
confirms AP is a physical d.o.f.\ and not a $\chi^2$ eraser: a deliberately
broken $\omega_c=0.05$ cosmology stays at $\chi^2/\mathrm{dof}\sim28$ even
with free AP. \textbf{Verdict: indistinguishable at $k\le0.15$; acceptable but
disfavored at $k\le0.20$ --- not a full-shape ``loss'', not a tie.} The score
is reported as computed at every protocol level; the sign of $\Delta\chi^2$
never flips.
\end{tcolorbox}

\paragraph{Neutrino-consistency cross-check (2026-07 re-audit, 3/3).}
The maximal-frozen protocol above sets $m_\nu=0$ for \emph{both} models
(\texttt{dfd\_class/dfd\_pk\_pipeline.py}, CAMB call) --- a symmetric
simplification, not a hidden asymmetry. Re-running with each model's own
consistent neutrino sector (DFD: the derived normal-ordering chain of
App.~\ref{app:neutrino}, $(m_1,m_2,m_3)=(2.34,8.96,50.16)$~meV,
$\Sigma m_\nu=61.46$~meV, $\omega_\nu=0.00066$; $\Lambda$CDM: the
Planck-baseline $60$~meV, $\omega_\nu=0.00065$; $\omega_\nu$ carried in the
AP background) gives $\Delta\chi^2={+}23.0$ at $k\le0.20$ (DFD
$161.4/124=1.302$ vs $\Lambda$CDM $138.4/124=1.116$; AP-off base $+39.0$),
with one-sided brackets $+20.4$ (DFD-only massive) to $+24.1$
($\Lambda$CDM-only massive). The frozen $m_\nu=0$ headline of $+21.5$ was
therefore $\approx1.5$ $\chi^2$ units \emph{generous} to DFD, not biased
against it; the headline is retained as the registered protocol number, with
$+23.0$ recorded as the physically-consistent cross-check. Consistency
anchors: $\sigma_8(0)$ shifts $0.8235\to0.8114$ (DFD) and $0.8226\to0.8110$
($\Lambda$CDM, matching the published $0.8111$); $\Omega_m$: DFD
$0.2705\to0.2718$, $\Lambda$CDM $0.3138\to0.3152$ (matching Planck
$0.3153$); the recomputed AP factors match the logged run to
$|\Delta|\le6\times10^{-5}$. Artifacts:
\texttt{dfd\_class/downgrade\_reaudit\_2026-07/}.

\subsection{PK.2\quad Stage 1 --- linear matter power at the frozen DFD cosmology}

The linear matter transfer function and power spectrum are produced by CAMB
(version 1.6.6) at the frozen DFD parameters: $H_0=72.09$, $\omega_b=0.02237$,
$\omega_\chi=0.1182$ (the $\chi$-matter field of Appendix~\ref{app:chi-matter}
supplies the cold component $\omega_{cdm}$), $n_s=0.9667$, with the primordial
amplitude $A_s$ rescaled so that the realised present-day amplitude matches the
GR appendix normalisation,
\begin{equation}
\fiteq{A_s\ \longrightarrow\ A_s\,\Big(\tfrac{\sigma_8^{\rm target}}{\sigma_8^{\rm CAMB}(0)}\Big)^{2},\qquad
\sigma_8^{\rm target}=0.820\ \ (\text{forced } A_s=32\pi\alpha^5).}
\label{eq:pk-As-rescale}
\end{equation}
This yields $P_{\rm lin}(k,z{=}0)$ on $10^{-3}\le k\le 2\,h/\mathrm{Mpc}$.
The primary normalization is the forced $\sigma_8(0)=0.820$ ($S_8=0.784$); the
alternative corpus value $\sigma_8=0.790$ ($S_8=0.755$) is an explicitly
contingent low-arm (asserted, not re-derived; GR.5). The BOSS full-shape verdict
below is \emph{insensitive} to this choice: the deficit is a broadband
\emph{shape} effect (set by $\Omega_m=0.274$, $H_0=72.1$), not amplitude ---
re-normalising $\sigma_8$ over $0.79$--$0.82$ moves the combined
$\chi^2/\mathrm{dof}$ by $<0.03$ (PK.8), so the head-to-head numbers stand at
either normalization.
The matched $\Lambda$CDM twin uses Planck-2018 parameters
($H_0=67.36$, $\omega_c=0.12$, $n_s=0.9649$, $\sigma_8=0.8111$).
Figure~\ref{fig:pk-linear-matter} shows the resulting linear spectrum and the
$Q{=}1$ growth history; Fig.~\ref{fig:pk-bao-hubble} shows the BAO distance
ladder of the \emph{same} frozen background (the geometry whose predicted
$D_V/r_d$ shift reappears as the AP dilation of PK.9/PK.11; BAO likelihood
scored in App.~\ref{app:cl-cmb-likelihood}).

\begin{figure*}[t]
\centering
\includegraphics[width=0.96\textwidth]{fig_matter_pk.png}
\caption{Linear matter power and growth at the frozen DFD parameters
(DFD-CLASS/CAMB forward run, Stage~1--2 of this pipeline; script
\texttt{dfd\_class/figures/make\_matter\_pk.py}). \emph{Left:} linear matter
power spectrum $P(k)$ at $z=0$ for DFD versus the Planck-2018 $\Lambda$CDM
twin, with $S_8$ annotated. \emph{Right:} the growth history
$f\sigma_8(z)$ carried by $\chi$-matter at $Q=1$ (Eq.~\eqref{eq:pk-growth})
overlaid on the BOSS/eBOSS redshift-space-distortion measurements, with the
$\Lambda$CDM curve for contrast.}
\label{fig:pk-linear-matter}
\end{figure*}

\begin{figure}[t]
\centering
\includegraphics[width=\columnwidth]{fig_bao_hubble.png}
\caption{BAO/$H(z)$ distance ladder at the frozen DFD background
(DFD-CLASS forward run; script
\texttt{dfd\_class/figures/make\_bao\_hubble.py}): the predicted
$D_V/r_d$, $D_M/r_d$ and $D_H/r_d$ curves, using the \emph{derived}
$r_{\rm drag}$, overlaid on the $12$ consensus BAO data points (error bars),
with the Planck-$\Lambda$CDM curves for contrast and the BAO $\chi^2$
annotated (likelihood-level scoring in App.~\ref{app:cl-cmb-likelihood}).}
\label{fig:pk-bao-hubble}
\end{figure}

\subsection{PK.3\quad Stage 2 --- DFD growth carrier ($\chi$-matter, $Q=1$)}

Per Proposition~\ref{prop:geff-not-carrier}, the linear-growth carrier is
$\chi$-matter at $Q=1$ (GR-like, scale-independent on linear scales), so the
linear growth obeys the standard ODE
\begin{equation}
\fiteq{\frac{d^2D}{da^2}+\Big(\frac{3}{a}+\frac{d\ln H}{da}\Big)\frac{dD}{da}
-\frac{3}{2}\frac{\Omega_m(a)}{a^2}\,Q\,D=0,\qquad Q=1,}
\label{eq:pk-growth}
\end{equation}
on the DFD background $H(z)$. Because $Q=1$, CAMB's own $D(z)$ at these
parameters \emph{is} this solution; the pipeline reads CAMB's redshift outputs
directly and extracts $f(z)=d\ln D/d\ln a$ and $\sigma_8(z)$. A CAMB
array-ordering convention (the $\sigma_8(z)$/$f\sigma_8(z)$ arrays are returned
in decreasing-$z$ order, reversed relative to the ascending power-spectrum
redshifts) was corrected; the realised values
$f(0.38)=0.681$, $f(0.61)=0.763$ are consistent with $\Omega_m(z)^{0.55}$ and
with the GR appendix.

\subsection{PK.4\quad Stage 3 --- nonlinear correction}

The mildly nonlinear regime is supplied by CAMB's HMcode (\texttt{mead2020})
halo-model prescription, $P_{\rm nl}(k,z)$. On the fitted range
$k\le 0.20\,h/\mathrm{Mpc}$ the boost is modest --- rising from
$\sim\!2\%$ at $k=0.1$ to a factor $\sim\!4.5$ only by $k=1$ --- but it is
included in full rather than truncated to the linear spectrum. This is a
standard (not DFD-native) nonlinear closure; a DFD-native nonlinear correction
remains open (PK.9).

\subsection{PK.5\quad Stage 4 --- galaxy redshift-space multipoles}

The galaxy spectrum in redshift space uses linear bias $b_1$, Kaiser
redshift-space distortions, and a Lorentzian Fingers-of-God damping:
\begin{equation}
\fiteq{P_g(k,\mu)=\frac{\big(b_1+f\mu^2\big)^2}{\big[1+\tfrac12(k\mu\sigma_v)^2\big]^2}\,
P_{\rm nl}(k)\;+\;P_{\rm shot},}
\label{eq:pk-galaxy}
\end{equation}
with $f=f(z)$ \emph{fixed} by the model growth (PK.3) and the nuisances
$\{b_1,\sigma_v,P_{\rm shot}\}$ profiled per bin. The Legendre multipoles are
\begin{equation}
\fiteq{P_\ell(k)=(2\ell+1)\int_0^1 P_g(k,\mu)\,\mathcal{L}_\ell(\mu)\,d\mu,\qquad \ell=0,2,}
\label{eq:pk-multipoles}
\end{equation}
evaluated by 16-node Gauss--Legendre quadrature, giving the monopole $P_0$ and
quadrupole $P_2$.

\subsection{PK.6\quad Stage 5 --- survey-window convolution}

The finite survey footprint mixes the true multipoles. The fine theory vector
$[\,P_0(k_{\rm fine}),P_2(k_{\rm fine})\,]$ ($1000$ points each, $dk=0.001$) is
convolved through the published Beutler $200\times2000$ window matrix $W$,
\begin{equation}
\fiteq{\mathbf{d}^{\rm model}_{200}=W_{200\times2000}\;\mathbf{t}_{2000},\qquad
\mathbf{t}=\big[\,P_0^{\rm fine},\,P_2^{\rm fine}\,\big],}
\label{eq:pk-window}
\end{equation}
returning the measured $200$-vector $[P_0(100),P_2(100)]$ on the $dk=0.01$ grid.
The matrix's undocumented column layout (input ordered as two multipoles over
$1000$ fine-$k$, not five multipoles over $400$) was reverse-engineered and
\emph{validated} against the Patchy mock mean: a windowed CAMB spectrum
reproduces both the mock $P_0$ and $P_2$. Setting $W=0$ makes $\chi^2$ explode
($\sim\!2\times10^4$), confirming the window is genuinely wired into the fit.

\subsection{PK.7\quad $\chi^2$, covariance, and the validation data}

For each cap--redshift bin the score is the profiled, Hartlap-corrected
covariance $\chi^2$ over $0.03\le k\le0.20\,h/\mathrm{Mpc}$,
\begin{equation}
\fiteq{\chi^2=\big(\mathbf{d}-\mathbf{m}\big)^{\!\top}\,
\frac{N_{\rm mock}-p-2}{N_{\rm mock}-1}\,C^{-1}\,
\big(\mathbf{d}-\mathbf{m}\big),}
\label{eq:pk-chi2}
\end{equation}
minimised over $\{b_1,\sigma_v,P_{\rm shot}\}$ ($p$ = number of fitted data
points; the Hartlap factor de-biases a mock-estimated inverse covariance). The
$\Lambda$CDM twin is fit through the \emph{identical} code (same window, same
covariance per bin, same $k$-mask, same three nuisances); only the input
cosmology differs.

\paragraph{Validation data.} Real, published Beutler \emph{et~al.} deconvolved
BOSS DR12 galaxy $P(k)$ multipoles
(\texttt{ps1D\_BOSS\_DR12\_\{NGC,SGC\}\_\{z1,z3\}}, $z_{\rm eff}=0.38,0.61$),
with the published $200\times2000$ survey-window matrices.

\paragraph{Covariance status (load-bearing).} The published
$200\times200$ Patchy covariance has a \emph{correct} monopole block (its $P_0$
errors match the mock scatter to $\sim\!1\%$) but a \emph{mis-normalised}
quadrupole block (its $P_2$ errors are $\sim\!5$--$45\times$ smaller than the
true mock scatter; a single mock against the mock mean returns
$\chi^2/\nu\simeq32$--$47$ with it). To use the quadrupole we rebuild
the covariance directly from the Patchy mock suite. The size of the mock-suite
data volume permitted rebuilding only the NGC $z_1$ mock covariance for this release
($231$ mocks; data-vs-mock-mean $\chi^2/\nu=1.87$, $\Lambda$CDM best-fit
$\chi^2/\nu\simeq1.0$, validating the whole pipeline). The remaining three bins
fall back to a \emph{$P_0$-only} fit with the verified-correct published $P_0$
covariance --- a standard, conservative choice that drops the (unusable)
quadrupole rather than trusting a mis-normalised error.

\subsection{PK.8\quad Legacy-input run (stale; retained for the record): DFD vs $\Lambda$CDM}

\textbf{Provenance note (2026-07).} The run recorded in this subsection is a
\emph{stale legacy-input run} and is retained only for the record. The canonical
headline is the maximal-frozen protocol result of PK.1/PK.11 at the derived
inputs: $\Delta\chi^2={+}21.5$ over 124 points ($1.311$ vs $1.137$) at
$k\le0.20$ --- unchanged by this relabelling.
Table~\ref{tab:pk-result} reports the per-bin and combined
$\chi^2/\mathrm{dof}$ from the legacy run; Fig.~\ref{fig:pk-pipeline} overlays the
window-convolved best-fit spectra on the data.

\begin{table}[h]
\centering
\caption{[Stale legacy-input run; retained for the record --- the canonical
maximal-frozen headline is $\Delta\chi^2={+}21.5$ ($1.311$ vs $1.137$), PK.1/PK.11.]
Full-shape BOSS DR12 $P(k)$ scores from
\texttt{dfd\_pk\_pipeline.py} (window-convolved, Patchy covariance,
Hartlap-corrected, nuisances $\{b_1,\sigma_v,P_{\rm shot}\}$ profiled,
$0.03\le k\le0.20\,h/\mathrm{Mpc}$). Numbers are exactly what the optimiser
returns; nothing is hard-coded. NGC $z_1$ uses the self-consistent
$P_0{+}P_2$ Patchy mock covariance; the other three use the published
$P_0$-only block.}
\label{tab:pk-result}
\begin{ruledtabular}
\fittab{%
\begin{tabular}{lccccc}
bin & $z$ & mult. & $N_{\rm bin}$ & DFD $\chi^2/\nu$ & $\Lambda$CDM $\chi^2/\nu$ \\
\midrule
NGC $z_1$ & $0.38$ & $P_0{+}P_2$ & $34$ & $1.65$ & $1.01$ \\
NGC $z_3$ & $0.61$ & $P_0$       & $17$ & $2.14$ & $1.56$ \\
SGC $z_1$ & $0.38$ & $P_0$       & $17$ & $0.93$ & $0.79$ \\
SGC $z_3$ & $0.61$ & $P_0$       & $17$ & $1.54$ & $1.07$ \\
\midrule
\textbf{combined} & & & $73$ & $\mathbf{1.59}$ & $\mathbf{1.09}$ \\
\end{tabular}%
}
\end{ruledtabular}
\end{table}

\noindent
The combined scores of the legacy-input run are
\begin{equation}
\fiteq{\chi^2_{\rm DFD}/\mathrm{dof}=115.8/73=1.59,\quad
\chi^2_{\Lambda\rm CDM}/\mathrm{dof}=79.2/73=1.09,\quad
\Delta\chi^2=+36.5\ \ \text{(stale legacy input).}}
\label{eq:pk-verdict}
\end{equation}
The best-fit nuisances are physical for both models
($b_1\simeq1.95$--$2.1$, $\sigma_v\simeq1$--$4\,\mathrm{Mpc}/h$). That
$\Lambda$CDM scores $\simeq1.0$ on this real data validates the pipeline; that
DFD scores $\simeq1.6$ is the adverse result. A diagnostic confirms the
deficit is \emph{broadband shape}, not amplitude: re-normalising DFD's
$\sigma_8$ up to the $\Lambda$CDM value $0.8111$ does not improve the fit
($1.586\!\to\!1.608$), so no bias/amplitude knob can absorb it --- it is driven
by DFD's lower $\Omega_m=0.274$ and higher $H_0=72.1$. This is fully consistent
with the GR appendix's own verdict ($f\sigma_8$ fit
``acceptable, not superior,'' $\chi^2/N\simeq1.2$--$1.6$ vs $0.8$--$1.1$).

\begin{figure*}[t]
\centering
\includegraphics[width=0.96\textwidth]{fig_pk_pipeline.png}
\caption{DFD-native production galaxy $P(k)$ pipeline against real BOSS DR12
data. Window-convolved best-fit monopole $P_0$ (and, left panel, quadrupole
$P_2$) for DFD (red dashed) and $\Lambda$CDM (blue solid) overlaid on the
Beutler measurements (black/grey points, Patchy-mock error bars), with
residual sub-panels $\Delta P_0/\sigma$ and the $\pm1\sigma$ band shaded. Left:
NGC $z=0.38$ with the self-consistent $P_0{+}P_2$ Patchy mock covariance
(the bin that validates the pipeline, $\Lambda$CDM $\chi^2/\nu\simeq1.0$).
Right: NGC $z=0.61$, $P_0$-only. DFD tracks the data acceptably but sits
systematically low relative to $\Lambda$CDM, the broadband-shape signature of
its lower $\Omega_m$ and higher $H_0$.}
\label{fig:pk-pipeline}
\end{figure*}

\subsection{PK.9\quad Status: production-grade now vs.\ what remains}

\noindent
\begin{tcolorbox}[breakable, enhanced jigsaw, colback=green!5!white, colframe=green!60!black,
  title=Production $P(k)$ pipeline --- Status]
\textbf{Production-grade now:}
\begin{itemize}[nosep]
\item End-to-end forward model: CAMB linear$+$HMcode $\to$ $\chi$-matter $Q{=}1$
growth $\to$ Kaiser$+$FoG multipoles $\to$ published $200\times2000$ window
convolution $\to$ Hartlap-corrected covariance $\chi^2$, run identically for
DFD and $\Lambda$CDM.
\item Validated against real published Beutler BOSS DR12 multipoles; window
layout reverse-engineered and verified against the Patchy mock mean; a
self-consistent $P_0{+}P_2$ Patchy mock covariance built for NGC $z_1$, on
which $\Lambda$CDM recovers $\chi^2/\nu\simeq1.0$.
\item Reproducible head-to-head score
(Eq.~\eqref{eq:pk-verdict}, Table~\ref{tab:pk-result}), now stress-tested
under a maximal-frozen protocol (no $\sigma_8$ renormalization,
$A_s{=}32\pi\alpha^5$ frozen, identical nuisance freedom, fixed
model-predicted Alcock--Paczynski dilation; two independent
reruns agree to all digits): at
$k\le0.20\,h\,\mathrm{Mpc}^{-1}$ DFD trails Planck-$\Lambda$CDM by
$\Delta\chi^2{=}{+}21.5$ over 124 points ($\chi^2/\mathrm{dof}=1.311$,
$p=0.012$, vs $1.137$, $p=0.14$) --- acceptable but disfavored, at the edge
of the $\le1.3$ band; at $k\le0.15$ the two are statistically
indistinguishable ($1.059$ vs $0.967$, $\Delta\chi^2{=}{+}7.7$,
$p_{\rm DFD}=0.34$). The earlier ``competitive $\simeq1.27$'' figure holds
only under the corpus $\sigma_8{\to}0.790$ normalization, which the frozen
protocol forbids; with frozen $A_s$ the deficit is real, concentrated in the
NGC bins as a low-$k$ ($0.03$--$0.08$) broadband overshoot (turnover at too
large a scale, the direct signature of DFD's lower $\Omega_m$ at higher
$H_0$), with a secondary NGC\,$z_1$ BAO-band phase mismatch; high-$k$ slopes
are comparable. Identical 6-term broadband marginalization for both models
reduces the gap to $+9.7$ (DFD $\chi^2/\mathrm{dof}=1.012$). The claim is
valid only for $k\le0.20$: at $k_{\max}{=}0.25$ both models fail
($\chi^2/\mathrm{dof}\sim5$), i.e.\ the minimal Kaiser+FoG model, not either
cosmology, breaks first. Not a full-shape loss; not a tie either.
\end{itemize}

\textbf{Still open (not yet production-grade):}
\begin{itemize}[nosep]
\item \emph{Quadrupole on all bins --- now CLOSED.} All four bins now carry
self-consistent Patchy $P_0{+}P_2$ mock covariances ($300$ mocks each for
SGC\,$z_1$, NGC\,$z_3$, SGC\,$z_3$; $231$ for NGC\,$z_1$), each validated at
single-mock $\chi^2/\mathrm{dof}\simeq0.55$--$0.93$ (builder
\texttt{build\_mock\_cov.py}). The fixed-template $P_0{+}P_2$ combined score
(DFD $1.30$ vs $\Lambda$CDM $1.11$) narrowed the earlier $P_0$-only fallback
($1.59$ vs $1.09$); both are superseded by the maximal-frozen protocol score
above.
\item \emph{RSD model.} The nuisance model is minimal (linear bias $+$ Kaiser
$+$ Lorentzian FoG); a full TNS/EFT-of-LSS one-loop counterterm treatment, as
in published BOSS full-shape analyses, is not yet implemented.
\item \emph{Geometry.} The Alcock--Paczynski dilation is now folded in as the
fixed model-predicted shift (executed 2026-07-02 under the maximal-frozen
protocol above); the remaining geometry gap is the explicit wide-angle
$M$-matrix only (the $v1$ averaged window partially absorbs it).
\item \emph{Nonlinear closure} is standard HMcode, not DFD-native; and
eBOSS QSO ($z=1.52$) and the raw DESI DR1 catalogs (which would need a $P(k)$
estimator from scratch) are not yet included.
\end{itemize}
The qualitative verdict --- DFD acceptable, $\Lambda$CDM preferred on
pre-recon full shape, gap closing as $k_{\max}$ tightens toward the linear
regime --- is robust across covariance choices, AP treatment, and broadband
marginalization; the sign of $\Delta\chi^2$ never flips.
\end{tcolorbox}

\subsection{PK.10\quad Reproducibility}

\noindent
\begin{tcolorbox}[colback=gray!5!white, colframe=gray!60!black,
  title=Reproducibility]
The pipeline is a single self-contained script
(\texttt{dfd\_class/dfd\_pk\_pipeline.py}, CAMB~1.6.6 $+$ numpy/scipy). Run
\begin{quote}\ttfamily
python3 dfd\_class/dfd\_pk\_pipeline.py
\end{quote}
It loads the four BOSS DR12 bins, runs CAMB for DFD and $\Lambda$CDM, fits each
bin, and prints the per-bin and combined $\chi^2/\mathrm{dof}$ of
Table~\ref{tab:pk-result} ($\sim$few seconds). The companion
\texttt{make\_fig\_pk\_pipeline.py} regenerates Fig.~\ref{fig:pk-pipeline}
(\texttt{fig\_pk\_pipeline.png}) by reusing the same fit. The Patchy mock
covariance for NGC $z_1$ is cached in \texttt{mock\_cov\_cache.npz}.
\emph{Caveat:} no $\chi^2$ literal appears in the source; every score comes
from a live \texttt{scipy.minimize} over a live CAMB $P(k)$. Perturbation
checks confirm a real fit (scaling the model, the data, or the covariance moves
$\chi^2$ as expected; $W{=}0\Rightarrow\chi^2\sim2\times10^4$; a deliberately
broken cosmology $\omega_c=0.05$ explodes to $\chi^2/\nu\simeq21$).
\end{tcolorbox}

\subsection{PK.11\quad Falsifier}

\noindent
\begin{tcolorbox}[colback=red!5!white, colframe=red!60!black,
  title=Production $P(k)$ pipeline --- Falsifier]
The DFD prediction tested here is falsified if, on the full-shape galaxy
$P(k)$:
\begin{enumerate}
\item \textbf{Decisive exclusion.} A complete full-shape analysis (all four
bins with self-consistent $P_0{+}P_2$ covariances, plus eBOSS QSO) returns a
DFD $\chi^2/\mathrm{dof}$ that is unacceptable in absolute terms
($\gg\!2$ with physical nuisances) while $\Lambda$CDM remains $\simeq1$ --- i.e.\
DFD is not merely worse but \emph{ruled out}.
\item \textbf{Shape, not nuisance.} Since the deficit is broadband shape, a DFD
spectrum that, with a TNS/EFT model and AP marginalisation, \emph{cannot} be
brought to $\chi^2/\mathrm{dof}\lesssim1.3$ would confirm the lower
$\Omega_m$/higher $H_0$ background is excluded by clustering --- a genuine
tension with the frozen DFD background.
\end{enumerate}
Conversely, recovering $\chi^2/\mathrm{dof}\lesssim1.3$ under the fuller model
would upgrade DFD from ``acceptable, worse'' to ``competitive'' on LSS.
\end{tcolorbox}
