%=============================================================================
% APPENDIX SL: THE SUPERNOVA-LENSING SIGN DISCRIMINATOR
% A pre-registered, parameter-free sign test that separates DFD from LambdaCDM.
% DFD forces d(delta_mu)/d(kappa_fg) = +s_DFD > 0 (a SN seen THROUGH foreground
% mass looks FARTHER, never closer); GR/LambdaCDM lensing forces the OPPOSITE
% sign s_LCDM <= 0. The slope MAGNITUDE carries a mu-law boost band; the SIGN does
% not. Status is DATA-GATED: existing data is ~1-1.5 sigma (map-noise
% limited, NOT theory limited); the appendix establishes the discriminator as a
% rigorous PRE-REGISTERED prediction awaiting LSST/Roman convergence maps.
% Grades:
%   (T) THEOREM        -- closed-form consequence of a printed DFD object.
%   (P) PREDICTION     -- parameter-free, pre-registerable data confrontation.
% Supporting code: dfd_class/sn_lensing_discriminator_forecast.py
%                  dfd_class/anisotropic_screen.py, scripts/preflight_forecast.py.
%=============================================================================

\section{The Supernova--Lensing Sign Discriminator}
\label{app:sn-lensing-discriminator}

\noindent\emph{This appendix isolates the single cleanest falsifiable separation between
DFD and the $\Lambda$CDM concordance model that is testable with standard-candle data: the
\emph{sign} of the correlation between standardized Type~Ia supernova (SN~Ia) Hubble residuals
$\delta\mu$ and the foreground weak-lensing convergence $\kappa_{\rm fg}$ along each
line of sight. In DFD the vacuum optical index $n=e^{\psi}$ makes a SN seen \emph{through} a
foreground overdensity appear \emph{farther} (positive residual); in $\Lambda$CDM
gravitational magnification makes the same SN appear \emph{brighter / closer} (negative
residual). The two predictions are \emph{opposite-signed}, and the DFD sign is a theorem given
the founding optical postulate. The slope magnitude is $s_{\rm DFD}\simeq+11$ mag per unit
$\kappa$ (band $[+7,+28]$ from the $\mu$-law boost), versus $s_{\Lambda{\rm CDM}}=-5/\ln10
=-2.17$ mag per unit $\kappa$, a definite sign flip with a $\sim$13~mag$/\kappa$ physical
separation. We give the proofs, a pre-registered power forecast
($\sigma_s=\sigma_{\rm int}/[\sqrt{N}\,\sigma_{\kappa,\rm eff}]$), and an explicit status:
existing data is $\sim1$--$1.5\sigma$ (under-powered, limited by convergence-map noise, not by
theory); the decisive $\geq3\sigma$ test is a pre-registered prediction awaiting LSST/Roman.}

%-----------------------------------------------------------------------------
\subsection{The line-of-sight optical screen and its sign}\label{app:sl-screen}
%-----------------------------------------------------------------------------

The DFD optical postulate is that the refractive index of the vacuum is $n=e^{\psi}$, with
$\psi$ the optical-density field that satisfies the screened Poisson equation
\begin{equation}\label{eq:sl-fieldeq}
\fiteq{\begin{aligned}
&\nabla\!\cdot\!\bigl[\mu(g/a_0)\,\nabla\psi\bigr]=\frac{8\pi G}{c^2}\,\rho,\\
&\quad\mu(x)=\frac{x}{1+x},\qquad a_0=2\sqrt{\alpha}\,cH(z),
\end{aligned}}
\end{equation}
the same interpolation function and acceleration scale fixed in
Appendix~\ref{app:alpha57-closure} and used for the optical $\sigma_8$ analysis of
Appendix~\ref{app:sigma8-optical}. Matter raises $\psi$ (since the source $\rho>0$ and the
operator is positive), so along a line of sight that crosses a foreground overdensity a photon
accumulates a strictly \emph{positive} excess optical density
\begin{equation}\label{eq:sl-dpsi}
\fiteq{\begin{aligned}
\Delta\psi_{\rm LOS}&=\int d\chi\,W(\chi)\,\frac{\kappa_{\rm struct}(\chi)}{\mu\!\bigl(g/a_0\bigr)}
\;=\;g_{\rm geo}\,Q\,\kappa_{\rm fg}>0,\\
&\quad Q\equiv\frac{1}{\mu(\bar x)},
\end{aligned}}
\end{equation}
where $\kappa_{\rm fg}$ is the foreground convergence the line of sight samples, $W$ is the
optical window, $g_{\rm geo}$ is the order-unity geometric kernel factor (the optical-to-lensing
window overlap, \ref{app:sl-kernel}), and $Q=1/\mu(\bar x)$ is the deep-field $\mu$-law boost
evaluated at the screen acceleration ratio $\bar x=g_\psi/a_0$.

\begin{theorem}[DFD SN-lensing sign, (T)]\label{thm:sl-dfd}
Under the DFD optical postulate $n=e^{\psi}$, the slope of the standardized SN~Ia distance-modulus
residual with respect to the foreground convergence is strictly positive,
\begin{equation}\label{eq:sl-sdfd}
\boxed{\;s_{\rm DFD}\equiv\frac{d(\delta\mu)}{d\kappa_{\rm fg}}
=\frac{5}{\ln10}\,g_{\rm geo}\,Q>0\;,}
\end{equation}
with central value $s_{\rm DFD}\simeq+11$~mag per unit $\kappa$ and band $[+7,+28]$. The sign is
independent of the magnitude inputs $g_{\rm geo},Q$, both strictly positive.
\end{theorem}
\begin{proof}
The distance modulus is $\mu=5\log_{10}(D_L)+25$. The optical metric stretches the photon path
uniformly along the line of sight, $D_L\to D_L\,e^{\Delta\psi_{\rm LOS}}$, so the residual against
an unscreened (matter-only) model is
\begin{equation}\label{eq:sl-prefactor}
\fiteq{\begin{aligned}
&\delta\mu=5\log_{10}\!\bigl(e^{\Delta\psi_{\rm LOS}}\bigr)=\frac{5}{\ln10}\,\Delta\psi_{\rm LOS},\\
&\quad \frac{5}{\ln10}=2.17147\ (\text{exact, the magnitude system}).
\end{aligned}}
\end{equation}
Inserting Eq.~\eqref{eq:sl-dpsi}, $\delta\mu=(5/\ln10)\,g_{\rm geo}\,Q\,\kappa_{\rm fg}$;
differentiating in $\kappa_{\rm fg}$ gives Eq.~\eqref{eq:sl-sdfd}. Since the magnitude prefactor
$5/\ln10>0$, the field-equation operator is positive ($\mu>0$ for $\bar x>0$, hence $Q>0$), and
the kernel overlap $g_{\rm geo}\in[1,2]>0$, the product is positive: $\Delta\psi_{\rm LOS}>0$ for
an overdense line of sight, so $\delta\mu>0$. The sign is therefore forced by the postulate, not
fitted; only the magnitude depends on the inputs.
\end{proof}

\begin{remark}[Sign is theorem-grade \emph{given} the optical-distance axiom]
The positivity of $s_{\rm DFD}$ follows by definition of the magnitude scale once
$D_L\to D_L\,e^{\Delta\psi_{\rm LOS}}$ is granted. This optical-distance dressing is the DFD axiom
under test: standard achromatic lensing of the \emph{same} $n>1$ clump gives the opposite sign
(solid-angle focusing brightens; surface brightness is conserved), so ``object looks
farther/dimmer'' is \emph{not} derivable from $n=e^\psi$ alone---it is the founding DFD optical
postulate. The wording ``theorem'' is therefore exact \emph{given} that postulate, and it is
precisely this postulate the sign test probes. This is the correct posture for a discriminator:
the experiment adjudicates the axiom.
\end{remark}

%-----------------------------------------------------------------------------
\subsection{The $\mu$-law boost and the geometric kernel}\label{app:sl-kernel}
%-----------------------------------------------------------------------------

The magnitude of $s_{\rm DFD}$ is set by two order-unity-to-few factors.

\paragraph{The $\mu$-law boost $Q=1/\mu(\bar x)$.}
The screened operator in Eq.~\eqref{eq:sl-fieldeq} enhances the response to a fixed source by
$Q=1/\mu(\bar x)$ in the low-acceleration regime. The screen acceleration ratio
$\bar x=0.24$ is the value that closes the BAO differential in the anisotropic-screen analysis
(it solves $x_{\rm need}\simeq1/(Q_{\rm need}-1)$), sitting between the deep-MOND limit
($\bar x\to0$, $Q\to\infty$) and the Hubble--EFE / $\sigma_8$ scale
($\bar x=1/(2\sqrt\alpha)=5.85$, $Q=1.17$):
\begin{equation}
\fiteq{\begin{aligned}
&\mu(0.24)=\frac{0.24}{1.24}=0.19355
\;\Longrightarrow\;
Q=5.167\ (\text{central}),\\
&\quad\bar x\in[0.18,0.45]\Rightarrow Q\in[3.22,6.56].
\end{aligned}}
\end{equation}
\emph{Status:} the gate $a_0=2\sqrt\alpha\,cH$ is derived; $\bar x\simeq0.24$ is selected by the
void-dominated line-of-sight within that derived deep-MOND$\to$EFE range---the single residual
axiom of the magnitude (not the sign).

\paragraph{The geometric kernel $g_{\rm geo}$.}
The convergence the optical screen actually samples, $\kappa_{\rm struct}$, and the
\emph{measured} convergence $\kappa_{\rm fg}$ both integrate the \emph{same} foreground density
field and so are strictly proportional, with an order-unity coefficient set by the overlap of the
optical window $W_{\rm opt}(\chi)$ and the lensing efficiency
$W_{\rm lens}(\chi)=\chi(\chi_s-\chi)/\chi_s$:
\begin{itemize}\itemsep2pt
\item kernel-matched limit ($W_{\rm opt}\!\approx\!W_{\rm lens}$): $g_{\rm geo}=1$;
\item full PPN-$\gamma=1$ doubling ($\psi=2\Phi/c^2$, the 1PN light-bending coupling DFD inherits):
$g_{\rm geo}=2$.
\end{itemize}

\paragraph{Final slope.}
Combining, $s_{\rm DFD}=(5/\ln10)\,Q\,g_{\rm geo}=2.1715\times5.167\times g_{\rm geo}$:

\begin{center}
\begin{tabular}{lccc}
\hline\hline
case & $Q\,(\bar x)$ & $g_{\rm geo}$ & $s_{\rm DFD}$ (mag$/\kappa$)\\
\hline
kernel-matched, central & $5.17\,(0.24)$ & $1.0$ & $+11.2$\\
nominal                 & $5.17\,(0.24)$ & $1.5$ & $+16.8$\\
full $\gamma=1$ doubling & $5.17\,(0.24)$ & $2.0$ & $+22.4$\\
weak boost ($\bar x=0.45$) & $3.22$ & $1.0$ & $+7.0$\\
strong boost ($\bar x=0.18$) & $6.56$ & $2.0$ & $+28.5$\\
\hline\hline
\end{tabular}
\end{center}

\noindent The central value $s_{\rm DFD}\simeq+11$~mag$/\kappa$ (kernel-matched, $\bar x=0.24$)
matches the conservative slope $b_D=10$ hard-coded in the power forecast
(\texttt{preflight\_forecast.py}); the plausible band is $s_{\rm DFD}\in[+7,+28]$, dominated by
the $\mu$-law boost $Q$ and the kernel factor $g_{\rm geo}\in[1,2]$.

%-----------------------------------------------------------------------------
\subsection{The standard ($\Lambda$CDM/GR) baseline}\label{app:sl-lcdm}
%-----------------------------------------------------------------------------

\begin{proposition}[$\Lambda$CDM lensing-magnification slope, (T)]\label{prop:sl-lcdm}
In GR/$\Lambda$CDM the slope of the standardized SN~Ia residual with respect to the foreground
convergence is robustly non-positive,
\begin{equation}\label{eq:sl-slcdm}
\boxed{\;s_{\Lambda{\rm CDM}}=\frac{d(\delta\mu)}{d\kappa_{\rm fg}}
=-\frac{5}{\ln10}=-2.17\ \text{mag per unit }\kappa\;,}
\end{equation}
and no physical $(\kappa,\gamma)$ makes it positive.
\end{proposition}
\begin{proof}
Weak-lensing magnification of a standard candle is
$\mu_{\rm mag}=1/[(1-\kappa)^2-|\gamma|^2]\approx1+2\kappa$ for $\kappa,\gamma\ll1$. A SN behind a
foreground overdensity ($\kappa>0$) is magnified, hence brighter; standardized as a fixed-luminosity
candle it is read as ``too close,'' giving a \emph{negative} distance-modulus residual
\begin{equation}
\fiteq{\begin{aligned}
&\delta\mu_{\Lambda{\rm CDM}}=-2.5\log_{10}\mu_{\rm mag}=-2.5\log_{10}(1+2\kappa),\\
&\quad\frac{d(\delta\mu)}{d\kappa}\Big|_{0}=-\frac{2.5\cdot2}{\ln10}=-\frac{5}{\ln10}=-2.17 .
\end{aligned}}
\end{equation}
This is exact and convention-fixed; it matches the value $b_L=-2.17$ used in the forecast pipeline.
Keeping the shear term and going to second order only makes the local slope \emph{more} negative
($-2.18$ at $\kappa=0.005$, $-2.29$ at $\kappa=0.05$). Flux conservation $\langle\mu_{\rm mag}\rangle=1$
fixes the \emph{mean} residual, not the \emph{slope}; Malmquist/magnification-bias selection can dilute
$|s_{\Lambda{\rm CDM}}|$ toward zero but never flip its sign, since every selected SN still has
$\delta\mu=-2.5\log_{10}\mu_{\rm mag}<0$. A coherent foreground monopole is removed by the residual-mean
marginalization in the pipeline, leaving pure line-of-sight lensing.
\end{proof}

\begin{corollary}[The slope sign is a clean DFD$/\Lambda$CDM discriminator]\label{cor:sl-discriminator}
The DFD slope (Theorem~\ref{thm:sl-dfd}) and the $\Lambda$CDM slope (Proposition~\ref{prop:sl-lcdm})
are \emph{genuinely opposite-signed}, with physical separation
\begin{equation}
\fiteq{\begin{aligned}
&s_{\rm DFD}-s_{\Lambda{\rm CDM}}=10-(-2.17)=12.17\ \text{mag per unit }\kappa\\
&\qquad(\text{conservative }b_D=10).
\end{aligned}}
\end{equation}
A measured slope $d(\delta\mu)/d\kappa_{\rm fg}>0$ falsifies the $\Lambda$CDM lensing prediction and
confirms the DFD optical postulate; a slope $\leq0$ falsifies DFD's optical-distance dressing. The sign
is the discriminator; the magnitude band only sets the available statistical power.
\end{corollary}

\begin{remark}[The one genuine confound: correlated foreground dust]
The $\Lambda$CDM \emph{lensing} sign has no escape---but a \emph{different} effect shares the DFD
sign: correlated foreground dust/extinction. Foreground galaxies that produce $\kappa>0$ also redden
and dim the SN, giving a \emph{positive} $\delta\mu$ of order $O(1)$ mag$/\kappa$, comparable to the
$-2.17$ lensing term. This does not flip the GR \emph{lensing} prediction, so it is a
detection-cleanliness requirement (reddening-corrected/NIR samples, host-color cuts), not a sign error.
A positive measured slope is a clean DFD win only \emph{after} dust is controlled---this is the real
systematic threat, openly disclosed.
\end{remark}

%-----------------------------------------------------------------------------
\subsection{Pre-registered power forecast}\label{app:sl-forecast}
%-----------------------------------------------------------------------------

\begin{theorem}[Sample size for a $3\sigma$ sign test, (P)]\label{thm:sl-forecast}
For $N$ SN~Ia with intrinsic scatter $\sigma_{\rm int}$ and an effective convergence-map noise
$\sigma_{\kappa,\rm eff}$ (the rms spread of the regressor $\kappa_{\rm fg}$ degraded by map noise),
the standard error of the fitted slope and the DFD$/\Lambda$CDM separation are
\begin{equation}\label{eq:sl-sigmas}
\fiteq{\begin{aligned}
\sigma_s&=\frac{\sigma_{\rm int}}{\sqrt{N}\,\sigma_{\kappa,\rm eff}},\\
Z&=\frac{s_{\rm DFD}-s_{\Lambda{\rm CDM}}}{\sigma_s}
=\frac{(s_{\rm DFD}-s_{\Lambda{\rm CDM}})\,\sqrt{N}\,\sigma_{\kappa,\rm eff}}{\sigma_{\rm int}} ,
\end{aligned}}
\end{equation}
so the number of SNe for a $3\sigma$ test is
\[\fiteq{N_{3\sigma}=\bigl[3\,\sigma_{\rm int}/((s_{\rm DFD}-s_{\Lambda{\rm CDM}})\,\sigma_{\kappa,\rm eff})\bigr]^2.}\]
With $\sigma_{\rm int}=0.13$, the physical gap $12.17$, and the anchor that $648$ SNe with a
\emph{true} map give $6\sigma$ (which pins $\sigma_{\kappa,\rm eff}^{\rm true}=0.0025$):
\end{theorem}

\begin{center}
\fittab{\begin{tabular}{lccccc}
\hline\hline
survey ($\kappa$ map) & $N$ & $\sigma_{\kappa,\rm eff}$ & $\sigma_s$ & $Z$ & $N_{3\sigma}$\\
\hline
DES-era proxy (current)        & $1{,}378$ & $0.0002$ & $16.4$ & $0.74\sigma$ & $22{,}400$\\
DES-era proxy (best case)      & $4{,}000$ & $0.0002$ & $9.6$  & $1.27\sigma$ & $22{,}400$\\
DES-era, \emph{if} true map now & $1{,}378$ & $0.0025$ & $1.39$ & $8.8\sigma$  & $162$\\
LSST Y1 (good wl $\kappa$)      & $30{,}000$ & $0.0013$ & $0.60$ & $20.4\sigma$ & $648$\\
LSST Y1 (deep $\kappa$)         & $30{,}000$ & $0.0019$ & $0.40$ & $30.6\sigma$ & $288$\\
Roman (deep, low-noise $\kappa$) & $20{,}000$ & $0.0025$ & $0.37$ & $33.3\sigma$ & $162$\\
\hline\hline
\end{tabular}}
\end{center}

\noindent\textbf{Read-out.} The bottleneck is the \emph{map}, not the SNe. Map quality enters
\emph{linearly} ($Z\propto\sigma_{\kappa,\rm eff}$) while the SN count enters only as $\sqrt{N}$:
the $\sim$$12\times$ true-vs-proxy map gap would need $\sim$$150\times$ more SNe to compensate. With
the current noisy proxy one would need $\sim$$22{,}000$ SNe to reach $3\sigma$; with a true convergence
map, only $\sim$$160$. Both LSST~Y1 and Roman clear $3\sigma$ ($20$--$33\sigma$ on paper) the moment a
real deep convergence map covers the SN footprint. The forecast is the hard-rule pre-flight: no
sky-map run is launched without this $\geq3\sigma$ projection.

%-----------------------------------------------------------------------------
\subsection{Status and pre-registration}\label{app:sl-status}
%-----------------------------------------------------------------------------

\begin{tcolorbox}[title={Status: a DATA-GATED, pre-registered prediction}]
\textbf{This is not a detection.} The existing-data result is $\sim1$--$1.5\sigma$, under-powered and
limited by convergence-\emph{map} noise, \emph{not} by theory. Five real datasets (Pantheon+$\times$DES-Y3
$\kappa$, BOSS DR12 galaxy proxy, DES-SN5YR$\times$DES-Y3 $\kappa$, Planck CMB-lensing wrong-kernel, DESI
subset) all show the \emph{DFD-positive} sign---which a pure cosmological constant forbids---but each at
$\sim$$0.3$--$1.0\sigma$, combining to $\sim1$--$1.5\sigma$ (and the naive quadrature over-states, because the
samples share foreground structure: the effective-independent SN count is the wall). The slope-\emph{sign}
prediction (Theorem~\ref{thm:sl-dfd}, Corollary~\ref{cor:sl-discriminator}) is theorem-grade given the DFD
optical postulate; its magnitude band is the only place the $\bar x\simeq0.24$ axiom enters. The decisive
$\geq3\sigma$ confirmation is a \emph{pre-registered prediction awaiting LSST/Roman} deep convergence maps
(Theorem~\ref{thm:sl-forecast}). The operative existing-data evidential case for DFD is \emph{not} this
single test but the combined preponderance tabulated in the Confrontation Matrix
(App.~\ref{app:confrontation-matrix}); the present discriminator is the cleanest \emph{future} decisive test,
not a present proof.
\end{tcolorbox}

\begin{falsifier}[Pre-registered SN-lensing sign test]\label{fals:sl-sign}
\emph{Prediction (pre-registered).} Regressing standardized SN~Ia Hubble residuals $\delta\mu$ on the
foreground weak-lensing convergence $\kappa_{\rm fg}$ over a deep, dust-controlled (reddening-corrected /
NIR) SN sample with a true convergence map yields a \emph{positive} slope,
$s_{\rm DFD}\in[+7,+28]$~mag$/\kappa$ (central $+11$), at $\geq3\sigma$ once
$N\,\sigma_{\kappa,\rm eff}^2$ reaches the forecast threshold (Theorem~\ref{thm:sl-forecast}; LSST~Y1
$\sim650$ well-mapped SNe, or Roman $\sim160$). \emph{Falsification.} A measured slope
$\leq0$ at $\geq3\sigma$, after dust control, falsifies the DFD optical-distance dressing
(Theorem~\ref{thm:sl-dfd}); it is the GR/$\Lambda$CDM prediction $s_{\Lambda{\rm CDM}}=-2.17$
(Proposition~\ref{prop:sl-lcdm}). The sign---not the magnitude---is the verdict.
\end{falsifier}

\begin{status}\label{stat:sl}
Theorem~\ref{thm:sl-dfd} (the positive DFD slope) and Proposition~\ref{prop:sl-lcdm} (the
non-positive $\Lambda$CDM slope) are closed consequences of, respectively, the DFD optical postulate
$n=e^\psi$ and the GR magnification relation; the slope \emph{sign} is theorem-grade and the
opposite-sign discriminator (Corollary~\ref{cor:sl-discriminator}) is exact. The slope
\emph{magnitude} carries the $\mu$-law boost band $[+7,+28]$, with the single residual axiom
$\bar x\simeq0.24$ (the void-dominated screen value, data-selected within the derived
deep-MOND$\to$EFE range; the gate $a_0=2\sqrt\alpha\,cH$ is derived). The power forecast
(Theorem~\ref{thm:sl-forecast}) is pre-flight-verified and reproduces both the $6\sigma$ true-map
anchor and the $\sim1\sigma$ proxy reality. \emph{Empirical status is data-gated:} the
present $\sim1$--$1.5\sigma$ existing-data lean is encouraging and uniformly DFD-signed but is
\emph{not} a detection and \emph{not} a falsification---it is map-noise-limited. This appendix
therefore establishes a rigorous, pre-registered prediction (Falsifier~\ref{fals:sl-sign}) whose
$\geq3\sigma$ adjudication awaits LSST/Roman; the operative existing-data case for DFD remains the
combined-evidence preponderance of App.~\ref{app:confrontation-matrix} (trials-factor accounting: Sec.~\ref{subsec:ce-trials-audit}). The one genuine systematic is
correlated foreground dust (same-signed as DFD), a detection-cleanliness requirement, not a sign error.
\end{status}
