%=============================================================================
% Appendix LS: Full 2-Halo / Realization Line-of-Sight Optical-Screen Amplitude
%
% PURPOSE: Close gap item 8 of the late-time optical-screen program -- the
% amplitude of the galaxy-weighted line-of-sight (LOS) refractive screen that
% the DFD optical metric imprints on BAO/CMB-distance observables -- by
% replacing the single-halo NFW+splashback TOY (excess ~0.034) with a full
% halo-model (1-halo + 2-halo) LOS computation AND an independent lognormal
% structure-realization. The appendix reports the derived amplitude central
% value and band, compares directly to the target 0.030 and data-preferred
% 0.039, and states plainly whether the 2-halo upgrade CONFIRMS, IMPROVES, or
% WEAKENS the in-range single-halo result.
%
% HEADLINE: the upgrade CONFIRMS the machinery and the structure
% convergence kappa_struct ~ 0.003-0.004, but it WEAKENS the clean "comfortably
% exceeds target" reading of the single-halo toy. The amplitude is NOT pinned
% by structure; it is governed by ONE undetermined quantity -- the MOND
% external-field (EFE) floor x_ext -- and only reaches/passes the target in a
% narrow weak-field corner. Nothing is tuned to 0.030.
%
% RELATIONSHIP TO APPENDICES AC / AE:
%   AC supplies the linear-growth response operator and adopts the Hubble-EFE
%   hypothesis x_EFE ~ 5.85 (named there as testable, not action-derived).
%   AE establishes that the FRW background gives |grad psi_bar| = 0, i.e. the
%   cosmological-perturbation regime is deep-MOND (x_ext -> 0). The LOS screen
%   sits between these two corpus-named regimes, and its amplitude swings by an
%   order of magnitude across them. This appendix maps that swing.
%
% SCRIPTS (reproducibility): dfd_class/los_screen_2halo.py (Strand 1, halo
% model), dfd_class/los_screen_mock.py (Strand 2, lognormal realization),
% dfd_class/halo_los_2halo_efe.py (Strand 3, EFE-floor sensitivity),
% dfd_class/make_fig_los_screen.py (figure). All deterministic.
%=============================================================================

\section{Line-of-Sight Optical-Screen Amplitude: Full 2-Halo and Realization Computation}
\label{app:los-screen}

\subsection{LS.1\quad Statement and Scope}

The DFD optical metric ($n=e^\psi$) makes the apparent distance to a background
source depend on the matter the line of sight (LOS) threads, through a refractive
screen
\begin{equation}
\fiteq{\begin{aligned}
&\Delta\psi_{\rm BAO}(z)\;=\;\int \! d\chi\; W_{\rm BAO}(\chi)\,
\kappa_{\rm struct}(\chi)\,\Big\langle \tfrac{1}{\mu\!\left(g/a_0\right)}\Big\rangle(\chi),\\
&\quad \mu(x)=\frac{x}{1+x},\quad a_0(z)=2\sqrt{\alpha}\,c\,H(z),
\end{aligned}}
\label{eq:LS-screen}
\end{equation}
where $\kappa_{\rm struct}$ is the projected matter fluctuation correlated with
the galaxy tracer and the factor $\langle 1/\mu\rangle$ is the DFD weak-field
(deep-MOND) \emph{boost} in the splashback/filament skin, with
$x=g_{\rm local}/a_0$. The late-time-screen program (Appendix~\ref{app:dark-energy}
and the four-case decomposition) requires this screen to carry an amplitude
$\Delta\psi_{\rm BAO}\simeq 0.030$ to move $\theta_\ast$ by $+1\sigma$ while
holding the BAO fit ($\chi^2/{\rm dof}=0.78$); the joint late-time data prefer a
slightly larger $\simeq 0.039$.

Earlier work fixed the amplitude only with a \emph{single-halo} NFW$+$splashback
toy that returned excess $\simeq 0.034$ with un-tuned ingredients, bracketing the
target but ``not nailed to three digits.'' This appendix replaces that toy with
two independent realistic-structure computations and reports the result with a
full, conservatively audited error budget.

\begin{tcolorbox}[colback=blue!5!white, colframe=blue!60!black, title=Scope and non-claims]
No parameter is tuned to $0.030$ or $0.039$. Every ingredient is a standard,
pre-registered literature object (Tinker08 mass function, Tinker10 bias, Duffy+08
concentration, standard HOD, CAMB $P(k)$, DFD cosmology, $a_0(z)=2\sqrt\alpha\,cH(z)$).
The target appears only in printed ratio lines. We report whatever the calculation
gives, central value \emph{and} band, including the regimes where it falls short.
\end{tcolorbox}

\subsection{LS.2\quad The two factors and what is firm}

The amplitude factorizes as $\Delta\psi_{\rm BAO}=\kappa_{\rm struct}\times Q$,
with $Q\equiv\langle 1/\mu(g/a_0)\rangle$. The upgrade settles the first factor and
exposes the second.

\paragraph{$\kappa_{\rm struct}$ (firm).}
In Strand~1 (\texttt{los\_screen\_2halo.py}) the projected matter convergence is a
genuine Limber RMS of the halo-model power spectrum $P_{\rm mm}=P_{1h}+P_{2h}$,
with $P_{1h}$ the NFW-$u(k)$ one-halo term integrated over the Tinker08 mass
function and $P_{2h}$ the Tinker10-bias two-halo term times the linear $P(k)$. This
\emph{derives} $\sigma_\kappa=0.0027$ (one-halo share $81\%$) and the effective
galaxy bias $b_g^{\rm eff}=2.00$ (HOD-weighted Tinker10, not set), giving
\begin{equation}
\kappa_{\rm struct}=b_g^{\rm eff}\,r_{\rm corr}\,\sigma_\kappa
=2.00\times 0.70\times 0.0027 = 0.0038
\end{equation}
(one-halo $0.0031$ plus two-halo $0.0007$).
The lognormal realization (Strand~2, \texttt{los\_screen\_mock.py}) independently
measures $\sigma_\kappa=0.0027\pm0.0004$ on the resolution-independent
$4\,{\rm Mpc}/h$-smoothed convergence map at a mock seed $\sigma_8=0.79$ (the screen
output $\sigma_\kappa\sim0.003$ is normalization-insensitive, so this seed value
does not propagate into the prediction; DFD's forced normalization is
$\sigma_8=0.820$), and a BOSS-anchored
estimate (Strand~3) gives the per-LOS RMS $0.0029$. All three methods agree:
\[
\fiteq{\boxed{\;\kappa_{\rm struct}\simeq 0.003\text{--}0.004\;\text{is robust and measured, not assumed.}\;}}
\]

\paragraph{$Q$ (governed by an undetermined floor).}
The local self-acceleration in the filament/splashback skin is genuinely
deep-MOND ($g_{\rm self}/a_0\sim 0.01$--$0.2$), so $Q$ is dominated by whatever
\emph{floors} the argument of $\mu$ -- the MOND external-field-effect (EFE) value
$x_{\rm ext}=g_{\rm ext}/a_0$. The action does not fix $x_{\rm ext}$ at LOS-skin
scales: Appendix~\ref{app:sigma8-optical} shows the FRW background gives
$|\nabla\bar\psi|=0$ (cosmological deep-MOND, $x_{\rm ext}\!\to\!0$), while
Appendix~\ref{app:linear-growth-closure} adopts the Hubble-EFE hypothesis
$x_{\rm ext}=1/(2\sqrt\alpha)\simeq5.85$ for galactic/cluster scales (named there
as a testable hypothesis the action does not produce). The LOS screen lives between
these two corpus-named regimes.

\subsection{LS.3\quad Derived amplitude: central value and band}

\paragraph{Fiducial (deep-MOND, $x_{\rm ext}\!\to\!0$).}
At the cosmological deep-MOND floor that Strand~1/2 use by default
($x=\max(g/a_0,10^{-4})$), the full $1h{+}2h$ halo model gives a $\rho$(trace)-weighted
boost $Q=13.5$ and
\begin{equation}
\fiteq{\begin{aligned}
&\Delta\psi_{\rm BAO}^{\rm fid}=0.052,\\
&\quad\text{band over pre-registered variations }=[0.035,\,0.071].
\end{aligned}}
\end{equation}
The lognormal realization independently returns $0.040\pm0.004$. The band is built by
varying one standard axis at a time over its literature range, summarized in
Table~\ref{tab:LS-band}.

\begin{table}[H]
\centering
\caption{Strand~1 halo-model amplitude at the cosmological deep-MOND floor
($x_{\rm ext}\!\to\!0$), varying one pre-registered axis at a time. Nothing tuned;
the target appears only in the final ratio column.}
\label{tab:LS-band}
\begin{tabular}{lccccc}
\toprule
axis & value & $b_g^{\rm eff}$ & $\kappa_{\rm str}$ & $Q$ & $\Delta\psi$ \\
\midrule
mass function   & Tinker08      & 2.00 & 0.0038 & 13.5 & 0.052 \\
mass function   & Sheth--Tormen & 2.03 & 0.0039 & 13.5 & 0.052 \\
concentration   & $\times0.7$   & 2.00 & 0.0038 & 14.5 & 0.055 \\
concentration   & $\times1.4$   & 2.00 & 0.0039 & 12.6 & 0.049 \\
splashback $R_{\rm sp}$ & $1.0\,R_{200m}$ & 2.00 & 0.0038 & 9.2  & 0.035 \\
splashback $R_{\rm sp}$ & $2.0\,R_{200m}$ & 2.00 & 0.0038 & 18.5 & 0.071 \\
HOD $\log M_{\min}$ & 12.0 & 1.73 & 0.0033 & 17.2 & 0.057 \\
HOD $\log M_{\min}$ & 13.0 & 2.29 & 0.0044 & 11.3 & 0.050 \\
infall slope    & $r^{-1.0}$ & 2.00 & 0.0038 & 13.8 & 0.053 \\
infall slope    & $r^{-2.0}$ & 2.00 & 0.0038 & 13.2 & 0.051 \\
\bottomrule
\end{tabular}
\end{table}

\paragraph{The floor sensitivity (the decisive finding).}
Strand~3 (\texttt{halo\_los\_2halo\_efe.py}) holds $\kappa_{\rm struct}$ fixed and
maps the amplitude as a function of $x_{\rm ext}$ with two independent boost
estimators (a halo-profile integral and a lognormal-PDF path integral), which agree
across the grid. The result (Table~\ref{tab:LS-efe}) is a $\sim$10$\times$ swing:
\begin{table}[H]
\centering
\caption{Screen amplitude vs the undetermined EFE floor $x_{\rm ext}$
($\kappa_{\rm gal}=b_g r\,\sigma_\kappa=0.0052$ from the BOSS-anchored RMS). The
amplitude reaches $0.030$ only for $x_{\rm ext}\!\lesssim\!0.11$ and $0.039$ only at
$x_{\rm ext}\!\simeq\!0.03$; at the corpus Hubble floor it is $5\times$ short.}
\label{tab:LS-efe}
\fittab{\begin{tabular}{lccc}
\toprule
$x_{\rm ext}$ & regime & $Q$ & $\Delta\psi$ \\
\midrule
$\to 0$  & cosmological deep-MOND (App.~AE)      & $13$--$25$ & $0.040$--$0.052$ \\
$0.031$  & in-range edge                         & $7.5$ & $0.039$ \\
$0.107$  & in-range edge                         & $5.8$ & $0.030$ \\
$0.24$   & screen-module floor                   & $4.1$ & $0.021$ \\
$5.85$   & Hubble-EFE (App.~AC, galactic)        & $1.17$ & $0.006$ \\
\bottomrule
\end{tabular}}
\end{table}

At the corpus screen-module floor $x_{\rm ext}=0.24$, the band over all un-tuned
\emph{structure} ingredients is $[0.017,0.033]$ -- the target $0.030$ sits at the
\emph{upper edge}, and $0.039$ is not reached. The dominant uncertainty is therefore
not the halo ingredients (they move the answer by only $\pm0.006$) but the
order-of-magnitude swing from the undetermined $x_{\rm ext}$.

\subsection{LS.4\quad Verdict: does the 2-halo upgrade confirm, improve, or weaken?}

\begin{tcolorbox}[colback=orange!6!white, colframe=orange!75!black,
                  title=Plain verdict]
The 2-halo upgrade \textbf{CONFIRMS the machinery and the structure factor}
($\kappa_{\rm struct}\simeq0.003$--$0.004$, three independent methods), and it
\textbf{IMPROVES} on the toy by replacing a hand-set $\kappa_{\rm gal}=0.006$ and a
single profile with a derived $P_{\rm mm}=P_{1h}+P_{2h}$ and a population-averaged
boost. But on the \emph{headline question} -- ``is the amplitude comfortably
$\geq0.030$?'' -- it \textbf{WEAKENS} the clean single-halo reading. The single-halo
toy's in-range result was an artifact of an implicit $x_{\rm ext}=0$. The proper
calculation exposes that the amplitude is a steep function of the undetermined EFE
floor: it brackets $0.030$ only for $x_{\rm ext}\lesssim0.11$ (a narrow weak-field
corner), does not robustly reach $0.039$, and falls $5\times$ short at the
corpus-named Hubble floor.
\end{tcolorbox}

\noindent The bottom line, with the floor stated explicitly:
\begin{equation}
\fiteq{\boxed{\;
\Delta\psi_{\rm BAO}=\kappa_{\rm struct}\times Q(x_{\rm ext})
\in[\,0.006,\,0.052\,]\;\;\text{as}\;\;x_{\rm ext}:5.85\!\to\!0
\;}}
\end{equation}
(with $\kappa_{\rm struct}\simeq0.0038$ fixed),
in range ($\simeq0.020$--$0.030$) only for the weak void-skin floor
$x_{\rm ext}\simeq0.1$--$0.24$. The amplitude is \emph{not} pinned by structure;
it is pinned by the EFE floor, which the corpus does not derive at LOS-skin scale.
Figure~\ref{fig:LS} shows both the halo-model build-up and the floor sensitivity.

\begin{figure*}[t]
\centering
\includegraphics[width=0.92\textwidth]{fig_los_screen.png}
\caption{\textbf{Left (a):} cumulative LOS screen $\Delta\psi_{\rm BAO}(<\chi)$ from
the full halo model at $z\simeq0.5$, decomposed into 1-halo (dark) and 2-halo
(light), reaching the fiducial $0.052$ in the deep-MOND ($x_{\rm ext}\!\to\!0$)
limit; target $0.030$ (dashed) and data-preferred $0.039$ (dotted) marked.
\textbf{Right (b):} the sensitivity -- amplitude vs the undetermined MOND
external-field floor $x_{\rm ext}$. The amplitude collapses by an order of magnitude
between the cosmological deep-MOND regime ($x_{\rm ext}\!\to\!0$) and the corpus
Hubble-EFE floor ($x_{\rm ext}=5.85$); the shaded band is the narrow corner where
$0.030$--$0.039$ are met.}
\label{fig:LS}
\end{figure*}

\begin{tcolorbox}[colback=green!4!white, colframe=green!55!black,
                  title=Reproducibility]
\small\sloppy
All numbers from \texttt{dfd\_class/}, deterministic (\texttt{camb} $\geq$1.6,
\texttt{dfd\_params.py}: $H_0=72.09$, $\alpha=0.00730$, $\Omega_m=0.274$,
$\Omega_\chi h^2=0.120$):
\begin{itemize}\itemsep1pt
\item \texttt{los\_screen\_2halo.py} -- Strand~1, halo model ($\sim$6\,s):
fiducial $0.052$, band $[0.035,0.071]$, $b_g^{\rm eff}=2.00$, $\sigma_\kappa=0.0027$.
\item \texttt{los\_screen\_mock.py} -- Strand~2, lognormal realization:
$0.040\pm0.004$, $\sigma_\kappa=0.0027\pm0.0004$, $\langle1/\mu\rangle_{\rm HOD}=11.7$.
\item \texttt{halo\_los\_2halo\_efe.py} -- Strand~3, EFE-floor scan: amplitude
$0.052\!\to\!0.006$ as $x_{\rm ext}:0\!\to\!5.85$; band at $x_{\rm ext}=0.24$ is
$[0.017,0.033]$.
\item \texttt{make\_fig\_los\_screen.py} -- regenerates \texttt{fig\_los\_screen.png}.
\end{itemize}
\end{tcolorbox}

\begin{status}[LS: LOS optical-screen amplitude]
\textbf{Genuine 2-halo:} yes -- $P_{\rm mm}=P_{1h}+P_{2h}$ from Tinker08 MF $+$
Tinker10 bias $+$ Duffy+08 $c(M)$ $+$ analytic NFW $u(k)$, with a cross-checking
lognormal realization; not a relabeled single-halo toy. \textbf{Tuned:} no.
\textbf{Structure factor $\kappa_{\rm struct}$:} firm, $\simeq0.003$--$0.004$.
\textbf{Amplitude:} \emph{not pinned by structure} -- governed by the undetermined
EFE floor $x_{\rm ext}$; central spans $\simeq0.006$--$0.052$. \textbf{Brackets
0.030:} yes, but only marginally and only for $x_{\rm ext}\lesssim0.11$.
\textbf{Reaches 0.039:} no, except at the near-divergent deep-MOND edge.
\textbf{Net effect of the upgrade:} CONFIRMS machinery, IMPROVES rigor, WEAKENS the
clean ``comfortably exceeds target'' claim. The amplitude is settled only up to the
EFE floor; the final arbiter is the sky-map cross-correlation, not more halo modeling.
\end{status}

\begin{falsifier}[LS]
The screen mechanism is falsified if either (i) the structure convergence is shown to
be $\sigma_\kappa\ll0.003$ at $z\simeq0.5$ (it is currently $0.0027$--$0.0029$ across
three methods), removing the first factor; or (ii) the EFE floor at LOS-skin scale is
pinned $\geq0.5$ (e.g. the corpus Hubble-EFE value $5.85$ is shown to apply to
cosmological filaments), collapsing $Q\leq1.5$ and $\Delta\psi_{\rm BAO}\leq0.008$,
$\sim4\times$ below the target. The decisive direct test is the predicted
galaxy$\times$LOS-convergence sky-map cross-correlation sign and amplitude
(Appendix~\ref{app:sn-lensing-discriminator}); a null or wrong-sign measurement at the forecast
significance falsifies the late-time screen as the third-peak / $\theta_\ast$ lever.
\end{falsifier}
