%=============================================================================
% APPENDIX AA: GENERAL RELATIVITY AS THE PADE APPROXIMANT OF DFD
% Structural identity between GR isotropic Schwarzschild and DFD exponential
% Integrated from: Alcock (2026), "GR as the Pade Approximant of DFD"
%=============================================================================

\section{General Relativity as the Pad\'e Approximant of DFD}\label{app:pade-identity}

This appendix establishes a precise mathematical identity between general relativity's isotropic-coordinate Schwarzschild lapse-squared scalar and DFD's exponential lapse-squared. The result is purely structural: GR's Schwarzschild form is the squared $[1,1]$ Pad\'e approximant of DFD's exponential, with matching argument. Agreement through second order in the gravitational potential reproduces the post-Newtonian parameter $\beta = 1$ in both theories (consistent with all current gravitational-redshift observations), while the first divergence at third order --- with GR developing a pole at the Schwarzschild horizon that DFD does not share --- identifies strong-field observations as the empirical lever distinguishing the two theories. The Pad\'e identity operates on the lapse-squared scalar; the spatial-curvature PPN parameter $\gamma$ is established separately from DFD's physical metric (Sec.~\ref{sec:ppn}).

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\subsection{Statement of the Pad\'e Identity}\label{app:pade-statement}
%-----------------------------------------------------------------------------

Define the lapse-squared scalar
\begin{equation}
\fiteq{\mathcal{L}(u) \equiv \frac{c^2}{|g_{tt}|},}
\end{equation}
a clock-rate / redshift observable, with $u \equiv GM/(\rho c^2)$ the dimensionless gravitational potential, where $\rho$ denotes the isotropic radial coordinate in GR and the flat-space radial coordinate in DFD. DFD's matter-coupling (physical) metric has $g_{tt} = -c^2 e^{-\psi}$ (Sec.~\ref{sec:ppn}); on the spherically symmetric exterior in the $\mu \to 1$ regime, $\psi(r) = 2GM/(c^2 r) = 2u$, giving
\begin{equation}\label{eq:AA-dfd}
\fiteq{\mathcal{L}_{\mathrm{DFD}}(u) = e^{\psi} = \exp(2u),}
\end{equation}
as derived from the field equation~\eqref{eq:field_general} in the $\mu\to 1$ strong-field limit (see Sec.~\ref{sec:strong-static} and Eq.~\eqref{eq:psi-exterior}). Independently, by Postulate~P1 the optical refractive index governing light propagation through the optical metric $d\tilde s^2 = -c^2 dt^2/n^2 + d\mathbf{x}^2$ is $n_{\mathrm{DFD}}(u) = e^{\psi} = \exp(2u)$. The lapse-squared $\mathcal{L}_{\mathrm{DFD}}$ and the refractive index $n_{\mathrm{DFD}}$ are distinct physical quantities arising from different metrics in DFD's two-metric structure, but they coincide as functions of $u$ on the exterior solution. The Pad\'e identity below operates on $\mathcal{L}$ (the clock-rate scalar).

The corresponding Schwarzschild metric in isotropic coordinates takes the form
\begin{equation}\label{eq:AA-gr-metric}
\fiteq{ds^2_{\mathrm{GR,iso}} = -\left(\frac{1-u/2}{1+u/2}\right)^{\!2}\!c^2\,dt^2 + \left(1+u/2\right)^{\!4}\left(d\rho^2 + \rho^2\,d\Omega^2\right),}
\end{equation}
giving the lapse-squared scalar
\begin{equation}\label{eq:AA-gr}
\fiteq{\mathcal{L}_{\mathrm{GR}}(u) = \left(\frac{1+u/2}{1-u/2}\right)^{\!2}.}
\end{equation}

\begin{theorem}[Pad\'e Identity]\label{thm:pade-identity}
The lapse-squared scalar of isotropic-coordinate Schwarzschild is mathematically identical to the square of the $[1,1]$ Pad\'e approximant of $e^u$. Writing $P_{m,m}(u)$ for the diagonal $[m,m]$ Pad\'e approximant of $\exp(u)$ (so $P_{m,m}(u)\approx e^u$):
\begin{equation}\label{eq:AA-main}
\fiteq{\boxed{\mathcal{L}_{\mathrm{GR}}(u) = \bigl[P_{1,1}(u)\bigr]^2,}}
\end{equation}
where $P_{1,1}(u) = (1+u/2)/(1-u/2)$ is the unique rational function with degree-1 numerator and denominator whose Taylor expansion matches that of $\exp(u)$ through $O(u^2)$.
\end{theorem}

\begin{proof}
The $[1,1]$ Pad\'e approximant of $\exp(u)$ about $u = 0$ is constructed by requiring
\begin{equation}
\fiteq{\frac{a_0 + a_1 u}{1 + b_1 u} = 1 + u + \frac{u^2}{2} + O(u^3).}
\end{equation}
Cross-multiplying and matching coefficients at orders $u^0$, $u^1$, $u^2$ gives the unique solution $a_0 = 1$, $a_1 = 1/2$, $b_1 = -1/2$:
\begin{equation}
\fiteq{P_{1,1}(u) = \frac{1 + u/2}{1 - u/2}.}
\end{equation}
Squaring and comparing with \eqref{eq:AA-gr} gives \eqref{eq:AA-main}.
\end{proof}

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\subsection{Order-by-Order Agreement}\label{app:pade-orders}
%-----------------------------------------------------------------------------

Taylor expanding both forms about $u = 0$:
\begin{align}
\mathcal{L}_{\mathrm{DFD}}(u) &= 1 + 2u + 2u^2 + \tfrac{4}{3}u^3 + \tfrac{2}{3}u^4 + \tfrac{4}{15}u^5 + O(u^6), \\
\mathcal{L}_{\mathrm{GR}}(u) &= 1 + 2u + 2u^2 + \tfrac{3}{2}u^3 + u^4 + \tfrac{5}{8}u^5 + O(u^6).
\end{align}
The series agree identically through $O(u^2)$, which is the defining property of the $[1,1]$ Pad\'e approximant. They first diverge at $O(u^3)$, with coefficient difference $3/2 - 4/3 = 1/6$.

\begin{table}[h]
\centering
\caption{Taylor coefficients of $\mathcal{L}_{\mathrm{DFD}}$ and $\mathcal{L}_{\mathrm{GR}}$ through $O(u^5)$.}
\label{tab:AA-series}
\fittab{%
\begin{tabular}{cccc}
\toprule
Order $n$ & $[u^n]\,\mathcal{L}_{\mathrm{DFD}}$ & $[u^n]\,\mathcal{L}_{\mathrm{GR}}$ & Difference \\
\midrule
0 & $1$ & $1$ & $0$ \\
1 & $2$ & $2$ & $0$ \\
2 & $2$ & $2$ & $0$ \\
3 & $4/3$ & $3/2$ & $-1/6$ \\
4 & $2/3$ & $1$ & $-1/3$ \\
5 & $4/15$ & $5/8$ & $-43/120$ \\
\bottomrule
\end{tabular}%
}
\end{table}

\paragraph{PPN consequence.}
The $O(u)$ and $O(u^2)$ coefficients of $\mathcal{L} = c^2/|g_{tt}|$ are set by the PPN expansion of $g_{tt}$, from which the post-Newtonian parameter $\beta = 1$ in both theories is read off, consistent with $\beta - 1 = (-4.5 \pm 5.6)\times 10^{-5}$ from Hofmann--M\"uller~\cite{HofmannMueller2018} lunar laser ranging. The PPN parameter $\gamma$ is determined by the spatial metric $g_{ij}$ and is \emph{not} accessible from the lapse-squared sector alone; for DFD, $\gamma = 1$ follows from the physical metric $g_{ij} = e^{+\psi}\delta_{ij}$ and is established in Sec.~\ref{sec:ppn} independently of the present identity, consistent with $\gamma - 1 = (2.1 \pm 2.3)\times 10^{-5}$ from Cassini~\cite{Bertotti2003}. The two formulations are therefore observationally indistinguishable at all currently measured post-Newtonian orders, with $\beta = 1$ following directly from the Pad\'e identity and $\gamma = 1$ following separately from DFD's physical-metric structure.

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\subsection{The Pad\'e Pole is GR's Horizon}\label{app:pade-pole}
%-----------------------------------------------------------------------------

The rational approximant \eqref{eq:AA-gr} develops a simple pole at $u = 2$, where $1 - u/2 = 0$. In isotropic coordinates this corresponds to $\rho = GM/(2c^2)$. The transformation to standard Schwarzschild coordinates,
\begin{equation}
\fiteq{r = \rho\left(1 + \frac{u}{2}\right)^{\!2},}
\end{equation}
maps $u = 2$ (equivalently $\rho = GM/(2c^2)$) to $r = 2GM/c^2$, precisely the Schwarzschild event horizon. The pole is structurally a feature of the rational form.

By contrast, the exponential function $\exp(2u)$ is entire: it has no pole anywhere in the finite complex plane. At $u = 2$ it takes the finite value $\exp(4) \approx 54.598$. This is the structural origin of the difference at strong field between GR and DFD.

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\subsection{DFD's Exterior Solution has No Finite-Radius Horizon}\label{app:pade-no-horizon}
%-----------------------------------------------------------------------------

The structural no-pole observation has a direct counterpart in DFD's explicit exterior solution. In the $\mu\to 1$ strong-field limit of the vacuum field equation \eqref{eq:static-spherical}, the solution around a spherically symmetric mass $M$ is
\begin{equation}\label{eq:AA-exterior}
\fiteq{\psi(r) = \frac{2GM}{c^2 r}, \qquad n(r) = e^{\psi(r)} = e^{2GM/(c^2 r)}.}
\end{equation}
This profile is finite for every $r > 0$ and diverges only at $r = 0$. The local phase speed $c/n(r)$ is therefore positive at every finite $r > 0$; no finite-radius surface traps light in the DFD exterior solution.

\paragraph{Photon sphere at $r = 2GM/c^2$.}
In the DFD exterior, the radius $r = 2GM/c^2$ appears instead as the \emph{photon sphere}, determined by the orbital-stability condition $d[n(r)r]/dr = 0$:
\begin{equation}\label{eq:AA-photon-sphere}
\fiteq{\frac{d}{dr}\left[e^{2GM/(c^2 r)}\,r\right] = 0 
\;\;\Longrightarrow\;\; r_{\mathrm{ph}}^{\mathrm{DFD}} = \frac{2GM}{c^2}.}
\end{equation}
This is a surface of unstable circular photon orbits, not a causal boundary. The critical impact parameter is $b_{\mathrm{crit}}^{\mathrm{DFD}} = 2e\,GM/c^2 \approx 5.44\,GM/c^2$, compared to $b_{\mathrm{crit}}^{\mathrm{GR}} = 3\sqrt{3}\,GM/c^2 \approx 5.20\,GM/c^2$, giving a $4.6\%$ larger predicted shadow radius as derived in Sec.~\ref{sec:strong-dfd-regime}. This is consistent with current EHT observations of M87$^\star$ and Sgr A$^\star$ at present precision~\cite{EHT_M87_2019}.

\begin{remark}[Reconciliation with v3.3 strong-field prose]
An earlier version of Sec.~\ref{sec:strong-causal} suggested that ``in the minimal DFD framework with $\mu\to 1$ at high gradients, the optical geometry approaches the Schwarzschild optical metric, and horizons form at locations consistent with GR.'' This statement does not follow from the explicit exterior solution \eqref{eq:AA-exterior}, which gives $n(r)$ finite at every $r > 0$ in the $\mu\to 1$ limit. The correct statement is that the photon-sphere radius $r_{\mathrm{ph}} = 2GM/c^2$ coincides with the Schwarzschild horizon coordinate, but the surface itself is not a horizon in DFD: it is a photon sphere, as established by \eqref{eq:AA-photon-sphere}. The GR horizon is a feature of the Pad\'e approximation, not of the exponential.
\end{remark}

%-----------------------------------------------------------------------------
\subsection{Distinction from Yilmaz-Type Exponential Metrics}\label{app:pade-yilmaz}
%-----------------------------------------------------------------------------

The exponential refractive-index form of DFD should not be conflated with the Yilmaz--Rosen--Papapetrou exponential metric~\cite{Papapetrou1954,Yilmaz1958},
\begin{equation}\label{eq:AA-yilmaz}
\fiteq{ds^2_{\mathrm{Yilmaz}} = -e^{-2m/r}\,dt^2 + e^{+2m/r}\left[dr^2 + r^2\,d\Omega^2\right],}
\end{equation}
which arises as a solution of Einstein--Klein--Gordon equations with a massless antiscalar field. When analyzed as a 4-metric within GR, \eqref{eq:AA-yilmaz} was shown by Boonserm et al.~\cite{Boonserm2018} to represent a traversable wormhole with exotic matter at the throat.

\begin{theorem}[No wormhole in the DFD flat-$\mathbb{R}^3$ formulation]\label{thm:no-wormhole}
On the fundamental flat-$\mathbb{R}^3$ spatial manifold of DFD, there exists no wormhole throat. The areal radius function $R(r) \equiv \sqrt{A(r)/4\pi}$ on any constant-$t$ slice satisfies $R(r) = r$ and $dR/dr = 1$ for all $r > 0$, admitting no minimum.
\end{theorem}

\begin{proof}
DFD postulates a flat Euclidean spatial metric $dl^2 = \delta_{ij}\,dx^i\,dx^j = dr^2 + r^2\,d\Omega^2$. The proper area of a 2-sphere at coordinate radius $r$ is $A(r) = 4\pi r^2$, giving $R(r) = r$. Differentiating, $dR/dr = 1$ identically on $r > 0$. No critical points exist, hence no minimum, hence no throat. By contrast, the Yilmaz metric \eqref{eq:AA-yilmaz} has spatial metric $dl^2 = e^{2m/r}(dr^2 + r^2 d\Omega^2)$, giving $R(r) = r\,e^{m/r}$ with $dR/dr = e^{m/r}(1 - m/r)$, which vanishes at $r = m$ and is a minimum (second derivative $e/m > 0$). The Yilmaz throat is located at $r = m$ with $R_{\min} = em \approx 2.718\,m$.
\end{proof}

\begin{theorem}[No exotic matter in the DFD fundamental formulation]\label{thm:no-exotic}
DFD's field equation \eqref{eq:field_general} is sourced by ordinary matter density $\rho \geq 0$. The $\psi$-field energy density $u_\psi = (c^4/8\pi G)\,W(|\nabla\psi|^2/a_\star^2)$ is nonnegative for all configurations satisfying the convexity condition (A3) of Sec.~\ref{sec:elliptic}. No stress-energy tensor needs to be balanced against an Einstein tensor in the fundamental flat-space formulation; no energy condition applies to a flat background; no exotic matter is invoked.
\end{theorem}

\begin{proof}
By construction, Eq.~\eqref{eq:field_general} is an elliptic PDE for $\psi$ on $\mathbb{R}^3$ sourced by $\rho$. It does not arise from a Hilbert action with an Einstein--Hilbert term; there is no Einstein tensor. Energy conditions are constraints on $T_{\mu\nu}$ in a GR context and therefore do not apply. The nonnegativity of $u_\psi = (c^4/8\pi G)\,W(s)$ follows from $W \geq 0$, which holds for $W(s) = s - \ln(1+s)$ corresponding to the derived $\mu(x) = x/(1+x)$ (see Appendix~\ref{app:mu-derivation}).
\end{proof}

\paragraph{Ontological distinction.}
DFD and Yilmaz share the exponential refractive-index form but differ fundamentally:
\begin{widetext}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\fittab{%
\begin{tabular}{lcc}
\toprule
 & DFD & Yilmaz \\
\midrule
Fundamental geometry & Flat $\mathbb{R}^3$ & Curved 4D Lorentzian \\
Field content & Scalar $\psi$ on $\mathbb{R}^3$ & Metric $g_{\mu\nu}$ + antiscalar $\varphi$ \\
Field equation & Elliptic, sourced by $\rho\geq 0$ & Einstein--Klein--Gordon \\
Spatial topology & $\mathbb{R}^3$ & Wormhole (throat at $r=m$) \\
Matter requirement & Ordinary $\rho \geq 0$ & Exotic (NEC-violating) \\
Horizon analog & None (photon sphere only) & None (throat instead) \\
\bottomrule
\end{tabular}%
}
\end{center}
\end{widetext}
The Boonserm--Visser wormhole/exotic-matter critique applies to \eqref{eq:AA-yilmaz} but not to DFD, because DFD does not treat \eqref{eq:AA-yilmaz} as a fundamental object. The derived optical metric $d\tilde{s}^2 = -c^2\,dt^2/n^2 + dx^2$ used in DFD for null-geodesic tracking has flat spatial slices by construction and is not a solution of Einstein's equations. The Pad\'e identity itself is a statement in the lapse-squared scalar $\mathcal{L} = c^2/|g_{tt}|$, computed in DFD from the matter-coupling physical metric (Sec.~\ref{sec:ppn}); on the spherically symmetric exterior this coincides numerically with the optical refractive index, but the two are distinct physical quantities arising from DFD's two-metric structure.

%-----------------------------------------------------------------------------
\subsection{Strong-Field Numerical Divergence}\label{app:pade-numerical}
%-----------------------------------------------------------------------------

The relative discrepancy $|\mathcal{L}_{\mathrm{GR}} - \mathcal{L}_{\mathrm{DFD}}|/\mathcal{L}_{\mathrm{DFD}}$ grows with $u$. Representative values are shown in Table~\ref{tab:AA-divergence}.

\begin{table}[h]
\centering
\caption{Relative difference between GR and DFD lapse-squared scalars $\mathcal{L}(u)$ at various $u$, with representative physical scales.}
\label{tab:AA-divergence}
\fittab{%
\begin{tabular}{ccc}
\toprule
$u$ & Relative difference & System scale \\
\midrule
$0.01$ & $1.7\times 10^{-7}$ & Compact star exterior \\
$0.10$ & $1.7\times 10^{-4}$ & Neutron-star envelope \\
$0.30$ & $4.6\times 10^{-3}$ & Approaching horizon \\
$0.50$ & $2.2\times 10^{-2}$ & Schwarzschild radius scale \\
$1.00$ & $2.2\times 10^{-1}$ & Below nominal horizon \\
$\to 2$ & $\to \infty$ & Pad\'e pole / GR horizon \\
\bottomrule
\end{tabular}%
}
\end{table}

The sub-ppm agreement at solar-system potentials confirms that existing tests cannot distinguish the two theories. Percent-level divergence appears only at scales probed by the Event Horizon Telescope, making black-hole shadow precision the proximal empirical lever.

%-----------------------------------------------------------------------------
\subsection{The Pad\'e Hierarchy}\label{app:pade-hierarchy}
%-----------------------------------------------------------------------------

The $[1,1]$ Pad\'e is the first nontrivial rational approximant of the exponential. Higher-order approximants match the exponential to increasing Taylor order: the squared $[m,m]$ diagonal Pad\'e approximant satisfies
\begin{equation}
\fiteq{\bigl[P_{m,m}\!\left(\exp(u)\right)\bigr]^2 = \exp(2u) + O(u^{2m+1}).}
\end{equation}
The full exponential is the $m \to \infty$ limit: the unique entire function agreeing with every order. Any finite-$m$ approximant has at least one pole in the complex plane, which in gravitational applications manifests as a coordinate singularity at a finite radius.

This hierarchy provides a natural interpretation of extended theories:
\begin{itemize}
\item A hypothetical gravity theory with $n(u) = [P_{2,2}(\exp(u))]^2$ would agree with DFD through $O(u^4)$, i.e.\ one order beyond standard PPN, indistinguishable at second post-Newtonian precision before any deviation would appear.
\item Each additional Pad\'e order matches one additional post-Newtonian order while retaining a finite-radius pole.
\item The exponential is singled out not by any specific post-Newtonian test but by the simplicity and analyticity of its underlying functional equation --- Cauchy's multiplicative composition law (Sec.~\ref{sec:formalism}) --- which admits no pole structure at all.
\end{itemize}

\paragraph{Experimental selection.}
Each post-Newtonian order that agrees with the exponential provides evidence consistent with DFD but does not by itself select it over a sufficiently-high-order Pad\'e approximant. The strong-field behavior, where the rational approximants fail on the circle of convergence $|u| = 2$ of their pole, is the regime in which the entire-function structure of DFD becomes distinguishable. In this sense, the Pad\'e identity relocates the Schwarzschild horizon from a physical feature to a coordinate-singular artifact of rational truncation.

%-----------------------------------------------------------------------------
\subsection{Framing Summary}\label{app:pade-framing}
%-----------------------------------------------------------------------------

\noindent
\begin{tcolorbox}[colback=blue!5!white, colframe=blue!60!black, title=Pad\'e Identity: Structural Framing]
\small
\textbf{Core identity (Theorem~\ref{thm:pade-identity}):}
\begin{equation*}
\fiteq{\mathcal{L}_{\mathrm{GR}}(u) = \left[P_{1,1}\!\left(\exp(u)\right)\right]^2, \qquad \mathcal{L}_{\mathrm{DFD}}(u) = \exp(2u),}
\end{equation*}
where $\mathcal{L}(u) \equiv c^2/|g_{tt}|$ is the lapse-squared scalar (clock-rate / redshift observable). \textbf{GR is the $m=1$ slot of the Pad\'e hierarchy; DFD is the $m \to \infty$ entire-function limit.}

\textbf{Weak-field (observational) consequence:}
Agreement through $O(u^2)$ by construction of the Pad\'e approximant. PPN parameter $\beta = 1$ in both theories follows directly from the lapse identity, matching LLR at $\lesssim 10^{-4}$. PPN parameter $\gamma = 1$ for DFD follows separately from the physical-metric structure $g_{ij} = e^{+\psi}\delta_{ij}$ (Sec.~\ref{sec:ppn}), matching Cassini at $\lesssim 10^{-5}$.

\textbf{Strong-field (distinguishing) consequence:}
First divergence at $O(u^3)$. Pad\'e pole at $u = 2$, corresponding to the Schwarzschild horizon $r = 2GM/c^2$. Exponential has no pole; DFD's explicit exterior solution gives $n(r) = e^{\psi(r)}$ finite at every $r > 0$. The radius $r = 2GM/c^2$ appears in DFD as the photon sphere (not a horizon), producing a $4.6\%$ larger shadow than Schwarzschild.

\textbf{Firewall from Yilmaz:}
DFD is not a 4-metric theory. Its fundamental geometry is flat $\mathbb{R}^3$; its spatial topology is globally trivial; its matter source is ordinary $\rho \geq 0$. Theorems~\ref{thm:no-wormhole} and~\ref{thm:no-exotic} establish that the Boonserm--Visser wormhole/exotic-matter critique of \eqref{eq:AA-yilmaz} does not apply to DFD.

\textbf{Empirical lever:}
Current EHT precision on M87$^\star$ and Sgr A$^\star$ shadows is consistent with both GR and DFD; the predicted $4.6\%$ shadow excess in DFD is the proximal observational discriminator.
\end{tcolorbox}
