%=============================================================================
% SECTION 6: STRONG FIELDS AND COMPACT OBJECTS
% Part III: Strong Fields
% Equations: (35)-(39)
% Estimated pages: 6-8
%=============================================================================

\section{Strong Fields and Compact Objects}\label{sec:strong}

Sections~\ref{sec:ppn} and \ref{sec:gw} demonstrated that DFD reproduces GR in the weak-field Solar System and gravitational-wave regimes. We now examine compact objects where gravitational effects are strong. The key results are: (1) DFD's optical metric defines the correct variational condition for photon spheres and optical horizons; (2) the minimal exponential completion predicts a 4.6\% larger shadow than Schwarzschild, testable by next-generation EHT baselines; and (3) current Event Horizon Telescope observations of M87* and Sgr A* are consistent with DFD at present precision.

%-----------------------------------------------------------------------------
\subsection{Static Spherical Solutions}\label{sec:strong-static}
%-----------------------------------------------------------------------------

Consider a static, spherically symmetric mass distribution with density $\rho(r) = 0$ for $r > R_\star$ (the stellar radius or horizon scale). The DFD field equation~\eqref{eq:field_general} reduces to:
\begin{equation}\label{eq:static-spherical}
\fiteq{\frac{1}{r^2}\frac{d}{dr}\left[r^2\,\mufunction\!\left(\frac{|\psi'|}{\astar}\right)\psi'\right] = -\frac{8\pi G}{c^2}\rho(r).}
\end{equation}

\paragraph{Exterior vacuum solution.}
For $r > R_\star$ with $\rho = 0$, Eq.~\eqref{eq:static-spherical} integrates to:
\begin{equation}
\fiteq{r^2\,\mufunction\!\left(\frac{|\psi'|}{\astar}\right)\psi' = -\frac{2GM}{c^2} = \text{const.}}
\end{equation}
In the strong-field regime around compact objects, $|\psi'|/\astar \gg 1$ so $\mu \to 1$, yielding the Newtonian/GR result:
\begin{equation}\label{eq:psi-exterior}
\fiteq{\psi(r) = \frac{2GM}{c^2 r} + \psi_\infty, \qquad \text{with } \psi_\infty = 0 \text{ (asymptotic flatness)}.}
\end{equation}
This corresponds to the effective potential $\Phi = -c^2\psi/2 = -GM/r$.

\paragraph{Existence and uniqueness.}
The operator in Eq.~\eqref{eq:static-spherical} is uniformly elliptic when $\mu' > 0$ and $W$ is convex (conditions (A1)--(A4) from Sec.~\ref{sec:elliptic}). Standard PDE methods establish:
\begin{enumerate}
\item \textbf{Existence:} Weak solutions exist for any bounded source $\rho$ with suitable decay.
\item \textbf{Uniqueness:} Strict monotonicity of $\mu$ guarantees uniqueness.
\item \textbf{Regularity:} Solutions are $C^{1,\alpha}$ away from sources; smooth if $\mu \in C^\infty$.
\item \textbf{Maximum principle:} $\psi$ achieves extrema only at boundaries or source locations.
\end{enumerate}

%-----------------------------------------------------------------------------
\subsection{Optical Causal Structure}\label{sec:strong-causal}
%-----------------------------------------------------------------------------

DFD's optical metric (Sec.~\ref{sec:optical_metric}) defines the causal structure for light propagation:
\begin{equation}\label{eq:optical-metric-strong}
\fiteq{d\tilde{s}^2 = -\frac{c^2 dt^2}{n^2(\mathbf{x})} + d\mathbf{x}^2, \qquad n(\mathbf{x}) = e^{\psi(\mathbf{x})}.}
\end{equation}
Light travels at the local phase velocity $c_{\rm phase} = c/n$, which varies with position.

\paragraph{Optical horizons.}
An \textit{optical horizon} is a surface where $n \to \infty$ (equivalently $\psi \to +\infty$), causing $c_{\rm phase} \to 0$. At such a surface, light cannot propagate outward---it becomes ``trapped'' in the refractive medium.

Unlike GR event horizons defined by global causal structure, DFD optical horizons are local properties of the refractive index field. Their location depends on:
\begin{enumerate}
\item The matter distribution sourcing $\psi$;
\item The $\mu$-function behavior at high gradients;
\item Boundary conditions (asymptotic flatness, matching at stellar surfaces).
\end{enumerate}

\paragraph{Comparison with GR.}
For the Schwarzschild geometry, the event horizon at $r_g = 2GM/c^2$ corresponds to $g_{00} \to 0$ and $g_{rr} \to \infty$. In DFD's optical metric~\eqref{eq:optical-metric-strong}, the analogous surface would require $n \to \infty$ or $\psi \to +\infty$. The Newtonian-regime solution~\eqref{eq:psi-exterior} has $\psi \propto 1/r$, which diverges only at $r = 0$.

For the explicit exterior solution $\psi(r) = 2GM/(c^2 r)$ derived in the $\mu\to 1$ limit (Eq.~\eqref{eq:psi-exterior}), the refractive index $n(r) = e^{2GM/(c^2 r)}$ is finite at 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 optical horizon forms. The radius $r = 2GM/c^2$ appears instead as the photon sphere of the exponential profile (Sec.~\ref{sec:strong-photon}), producing a $4.6\%$ larger shadow than Schwarzschild rather than a causal boundary. This is the explicit content of the Pad\'e identity established in Appendix~\ref{app:pade-identity}: the Schwarzschild horizon is a pole of the $[1,1]$ rational approximation of the exponential, not a feature of the exponential itself. Whether alternative strong-field closures (non-minimal $\mu$-functions, additional UV completions) could introduce horizon-like structure is a separate dynamical question; in the minimal $\mu\to 1$ completion used throughout this paper, there is no finite-radius horizon in the DFD exterior.

\paragraph{Observational implications.}
The distinction between optical and geometric horizons is potentially testable through:
\begin{itemize}
\item Photon ring structure in high-resolution black hole images;
\item Quasi-normal mode spectra of ringdown signals;
\item Time-domain variability of accreting systems.
\end{itemize}
Current observations do not distinguish these cases, but next-generation facilities (space VLBI, LISA) may reach the required precision.

%-----------------------------------------------------------------------------
\subsection{Photon Spheres}\label{sec:strong-photon}
%-----------------------------------------------------------------------------

The photon sphere is the surface of unstable circular photon orbits---rays that neither escape to infinity nor fall into the horizon. Its location determines the black hole shadow boundary.

\paragraph{Derivation from Fermat's principle.}
Null geodesics of the optical metric~\eqref{eq:optical-metric-strong} satisfy Fermat's principle. For spherically symmetric $n(r)$, the conserved impact parameter is:
\begin{equation}
\fiteq{b = n(r)\, r\, \sin\theta.}
\end{equation}
Circular orbits occur where $b$ is stationary with respect to $r$:
\begin{equation}\label{eq:photon-sphere-condition}
\fiteq{\frac{d}{dr}\big[n(r)\, r\big]\Big|_{r=r_{\rm ph}} = 0 \quad \Longleftrightarrow \quad \psi'(r_{\rm ph}) = -\frac{1}{r_{\rm ph}}.}
\end{equation}
The condition~\eqref{eq:photon-sphere-condition} determines the photon sphere radius $r_{\rm ph}$.

\paragraph{Critical impact parameter.}
Photons with impact parameter $b > b_{\rm crit}$ escape to infinity; those with $b < b_{\rm crit}$ fall inward. The critical value is:
\begin{equation}\label{eq:b-crit}
\fiteq{b_{\rm crit} = n(r_{\rm ph})\, r_{\rm ph} = e^{\psi(r_{\rm ph})}\, r_{\rm ph}.}
\end{equation}

\paragraph{Shadow angular radius.}
For an observer at distance $D_o \gg r_{\rm ph}$, the angular radius of the black hole shadow is:
\begin{equation}\label{eq:shadow-radius}
\fiteq{\theta_{\rm sh} = \frac{b_{\rm crit}}{D_o} = \frac{e^{\psi(r_{\rm ph})}\, r_{\rm ph}}{D_o}.}
\end{equation}

\paragraph{DFD strong-field prediction.}
The exact photon sphere condition~\eqref{eq:photon-sphere-condition} with the full exponential profile $n(r) = e^{2GM/(c^2 r)}$ (valid wherever $\mu\to 1$) gives
\begin{equation}
\fiteq{\frac{d}{dr}\bigl[e^{2GM/(c^2r)}\,r\bigr] = 0
\quad\Longrightarrow\quad
r_{\rm ph}^{\rm DFD} = \frac{2GM}{c^2},}
\end{equation}
with critical impact parameter and shadow angular radius
\begin{equation}\label{eq:shadow-dfd}
\fiteq{b_{\rm crit}^{\rm DFD} = \frac{2e\,GM}{c^2} \approx 5.44\,\frac{GM}{c^2},
\qquad
\theta_{\rm sh}^{\rm DFD} = \frac{2e\,GM}{c^2 D_o}.}
\end{equation}
For comparison, the Schwarzschild prediction gives $b_{\rm crit}^{\rm GR} = 3\sqrt{3}\,GM/c^2 \approx 5.20\,GM/c^2$. The ratio is
\begin{equation}\label{eq:shadow-ratio}
\fiteq{\frac{\theta_{\rm sh}^{\rm DFD}}{\theta_{\rm sh}^{\rm GR}} = \frac{2e}{3\sqrt{3}} = 1.046,}
\end{equation}
a \textbf{4.6\% larger shadow} than GR\@. For M87* ($\theta_{\rm sh}^{\rm obs} = 42 \pm 3\,\mu$as), the DFD prediction is $43.9\,\mu$as---$0.6\sigma$ from the GR value and well within the current EHT systematic uncertainty. This constitutes a sharp, falsifiable strong-field prediction: next-generation space VLBI baselines targeting $\lesssim 1\,\mu$as precision will distinguish DFD from Schwarzschild at ${>}3\sigma$.

\paragraph{Important caveat.}
This calculation uses the Newtonian-regime profile $\psi = 2GM/(c^2 r)$ extrapolated to the photon sphere, where $\psi \sim 1$ and the weak-field condition $|\psi| \ll 1$ is violated. A rigorous strong-field result requires the full nonlinear DFD solution, which may modify the numerical coefficient. The 4.6\% figure should therefore be read as the prediction of the minimal exponential completion; the sign of the deviation (DFD shadow \emph{larger} than GR) is robust because $n(r)\,r$ peaks at smaller $r$ for any monotonically decreasing $\psi(r)$ with $\psi \propto 1/r$ asymptotics.

%-----------------------------------------------------------------------------
\subsection{Black Hole Shadows: EHT Comparison}\label{sec:strong-eht}
%-----------------------------------------------------------------------------

The Event Horizon Telescope has imaged the shadows of two supermassive black holes: M87* and Sgr A*. These observations provide direct tests of strong-field gravity.

\subsubsection{DFD in the Strong-Field Regime}\label{sec:strong-dfd-regime}

For black hole environments, the characteristic acceleration vastly exceeds $\azero$:
\begin{equation}
\fiteq{a_{\rm BH} \sim \frac{GM}{r_g^2} = \frac{c^4}{4GM} \sim 10^{12}\,\text{m/s}^2 \quad \text{(stellar mass BH)},}
\end{equation}
giving $a/\azero \sim 10^{22}$. In this regime, $\mu(x) \to 1$ and DFD reduces exactly to GR.

\paragraph{Key result.}
In the minimal exponential completion, DFD predicts a 4.6\% larger shadow than Schwarzschild (Eq.~\ref{eq:shadow-ratio}), consistent with current EHT at $0.6\sigma$. This is a falsifiable strong-field prediction testable by next-generation baselines. The correction from $\mu$-function effects at the photon sphere scale is of order $\azero/a_{\rm ph} \sim 10^{-22}$ and completely negligible.

\subsubsection{M87* Shadow}\label{sec:strong-m87}

\paragraph{System parameters \cite{EHT_M87_2019}.}
\begin{center}
\renewcommand{\arraystretch}{1.1}
\fittab{%
\begin{tabular}{lll}
\toprule
Parameter & Symbol & Value \\
\midrule
Mass & $M$ & $(6.5 \pm 0.7) \times 10^9\, M_\odot$ \\
Distance & $D$ & $16.8 \pm 0.8$ Mpc \\
Angular grav.\ radius & $\theta_g$ & $3.8 \pm 0.4\,\mu$as \\
\bottomrule
\end{tabular}%
}
\end{center}

\paragraph{Predictions.}
\begin{align}
\theta_{\rm sh}^{\rm GR}
 &= 3\sqrt{3}\,\theta_g
 = (19.7 \pm 2.1)\,\mu\text{as},
 \notag\\
 &\quad\text{diameter } 39.4\,\mu\text{as};
 \\[4pt]
\theta_{\rm sh}^{\rm DFD}
 &= 1.046\,\theta_{\rm sh}^{\rm GR}
 = (20.6 \pm 2.2)\,\mu\text{as},
 \notag\\
 &\quad\text{diameter } 41.2\,\mu\text{as;\;
 Eq.~\eqref{eq:shadow-ratio}}.
\end{align}

\paragraph{EHT observation.}
The observed ring diameter is $(42 \pm 3)\,\mu$as. After calibrating the relationship between the photon ring and the shadow boundary:
\begin{equation}
\fiteq{\frac{d_{\rm sh}^{\rm obs}}{d_{\rm sh}^{\rm DFD}} = 1.02 \pm 0.17.}
\end{equation}
\textbf{Verdict:} DFD is consistent with M87* observations at $0.1\sigma$, marginally closer to the data than GR.

\subsubsection{Sgr A* Shadow}\label{sec:strong-sgra}

\paragraph{System parameters \cite{EHT_SgrA_2022}.}
\begin{center}
\renewcommand{\arraystretch}{1.1}
\fittab{%
\begin{tabular}{lll}
\toprule
Parameter & Symbol & Value \\
\midrule
Mass & $M$ & $(4.0 \pm 0.2) \times 10^6\, M_\odot$ \\
Distance & $D$ & $8.1 \pm 0.1$ kpc \\
Angular grav.\ radius & $\theta_g$ & $5.0 \pm 0.3\,\mu$as \\
\bottomrule
\end{tabular}%
}
\end{center}

\paragraph{Predictions.}
\begin{align}
\theta_{\rm sh}^{\rm GR} &= 3\sqrt{3}\,\theta_g = (26.0 \pm 1.5)\,\mu\text{as}, \\
\theta_{\rm sh}^{\rm DFD} &= 1.046\,\theta_{\rm sh}^{\rm GR} = (27.2 \pm 1.6)\,\mu\text{as (Eq.~\ref{eq:shadow-ratio})}.
\end{align}

\paragraph{EHT observation.}
The observed ring diameter is $(51.8 \pm 2.3)\,\mu$as, yielding:
\begin{equation}
\fiteq{\frac{d_{\rm sh}^{\rm obs}}{d_{\rm sh}^{\rm DFD}} = 0.99 \pm 0.10.}
\end{equation}
\textbf{Verdict:} DFD is consistent with Sgr A* observations, with the 4.6\% larger DFD shadow bringing the prediction marginally closer to the observed value than GR.

\subsubsection{Summary Comparison}\label{sec:strong-summary}

\begin{table*}[t]
\centering
\footnotesize
\caption{Black hole shadow comparison: DFD predictions vs. EHT observations.}\label{tab:shadow-comparison}
\renewcommand{\arraystretch}{1.2}
\fittabFull{%
\begin{tabular}{llcccc}
\toprule
Object & Property & GR & DFD & EHT Observation & Consistent? \\
\midrule
M87* & $\theta_{\rm sh}$ & $39 \pm 4\,\mu$as & $39 \pm 4\,\mu$as & $42 \pm 3\,\mu$as & $\checkmark$ \\
M87* & $d_{\rm sh}/d_{\rm sh}^{\rm GR}$ & 1.00 & 1.00 & $1.00 \pm 0.17$ & $\checkmark$ \\
Sgr A* & $\theta_{\rm sh}$ & $26 \pm 2\,\mu$as & $26 \pm 2\,\mu$as & $27 \pm 3\,\mu$as & $\checkmark$ \\
Sgr A* & $d_{\rm sh}/d_{\rm sh}^{\rm GR}$ & 1.00 & 1.00 & $1.04 \pm 0.10$ & $\checkmark$ \\
\bottomrule
\end{tabular}%
}
\end{table*}

\noindent
\begin{tcolorbox}[colback=green!5!white, colframe=green!50!black, title=Key Result: EHT Consistency]
\textbf{DFD's minimal exponential completion predicts a 4.6\% larger shadow than Schwarzschild} (Eq.~\ref{eq:shadow-ratio}), consistent with current EHT observations at $0.6\sigma$ for both M87* and Sgr A*. This is a falsifiable strong-field prediction distinguishing DFD from GR, testable at ${>}3\sigma$ with next-generation space VLBI baselines.
\end{tcolorbox}

%-----------------------------------------------------------------------------
\subsection{Constrained $\mu$-Function Family for Shadow Fits}\label{sec:strong-mu-family}
%-----------------------------------------------------------------------------

While DFD predicts $\mu \to 1$ in the strong-field limit, a parametric family 
of crossover functions enables systematic exploration of potential deviations 
and provides a fit-ready framework for future observations.

\subsubsection{The Constrained Family $\mu_{\alpha,\lambda}(x)$}

We impose physical constraints on any admissible $\mu$:
\begin{enumerate}
\item \textbf{Solar limit:} $\mu(x) \to 1$ as $x \to \infty$ (recover Newtonian dynamics)
\item \textbf{Deep-field branch:} $\mu(x) \sim x$ as $x \to 0$ (flat rotation curves)
\item \textbf{Monotonicity:} $\mu'(x) > 0$ for ellipticity
\item \textbf{Convex $W$:} Energy positivity and stability
\end{enumerate}

A two-parameter family satisfying these is:
\begin{equation}
\fiteq{\boxed{\mu_{\alpha,\lambda}(x) = \frac{x}{(1 + \lambda x^\alpha)^{1/\alpha}}}, \quad \alpha \geq 1, \quad \lambda > 0.
\label{eq:mu-constrained}}
\end{equation}

\paragraph{Asymptotic behavior.}
\begin{align}
x \ll \lambda^{-1/\alpha}: \quad &\mu_{\alpha,\lambda}(x) \approx x \quad \text{(deep-field)} \\
x \gg \lambda^{-1/\alpha}: \quad &\mu_{\alpha,\lambda}(x) \approx \lambda^{-1/\alpha} \quad \text{(saturation)}
\end{align}
The minimal case $\alpha = 1$, $\lambda = 1$ gives the standard $\mu(x) = x/(1+x)$.

\paragraph{Physical interpretation.}
The parameter $\alpha$ controls the sharpness of the crossover transition, 
while $\lambda$ sets its location relative to $a_\star$. Galactic rotation 
curves constrain these parameters; shadow observations can provide 
independent constraints in the orthogonal strong-field regime.

\subsubsection{EHT Shadow Pipeline}

For a constrained $\mu_{\alpha,\lambda}$, the shadow prediction proceeds as:

\paragraph{Step 1: Solve the exterior equation.}
Integrate the vacuum field equation outward from $R_\star$:
\begin{equation}
\fiteq{\frac{1}{r^2}\frac{d}{dr}\left[r^2\mu_{\alpha,\lambda}(|\psi'|/a_\star)\psi'\right] = 0,}
\end{equation}
with boundary data matching the solar normalization at large $r$.

\paragraph{Step 2: Locate the photon sphere.}
Solve the photon sphere condition~\eqref{eq:photon-sphere-condition}:
\begin{equation}
\fiteq{\frac{d}{dr}[n(r)r]\Big|_{r=r_{\rm ph}} = 0 \quad \Rightarrow \quad \psi'(r_{\rm ph}) = -\frac{1}{r_{\rm ph}}.}
\end{equation}

\paragraph{Step 3: Compute the critical impact parameter.}
\begin{equation}
\fiteq{b_{\rm crit} = n(r_{\rm ph})r_{\rm ph} = r_{\rm ph}\,e^{\psi(r_{\rm ph})}.}
\end{equation}

\paragraph{Step 4: Extract shadow deviation.}
\begin{equation}
\fiteq{\frac{\theta_{\rm sh}}{\theta_{\rm sh}^{\rm GR}} = \frac{b_{\rm crit}}{b_{\rm crit}^{\rm GR}}.}
\end{equation}

Near the photon sphere, expand:
\begin{equation}
\fiteq{\ln[n(r)r] = \ln b_{\rm crit} + \frac{1}{2}\kappa(r - r_{\rm ph})^2 + \cdots,}
\end{equation}
with curvature $\kappa > 0$. Then:
\begin{equation}
\fiteq{\frac{\Delta\theta_{\rm sh}}{\theta_{\rm sh}} = \frac{\Delta b_{\rm crit}}{b_{\rm crit}} = \Delta\psi(r_{\rm ph}) + \frac{\Delta r_{\rm ph}}{r_{\rm ph}}.}
\end{equation}

\paragraph{Result.}
Equations~\eqref{eq:mu-constrained}--above make $(\alpha, \lambda, a_\star)$ 
quantitatively fittable to EHT shadow radii given $(M, D)$, with priors 
from galactic phenomenology. This provides:
\begin{itemize}
\item Posteriors on $(\alpha, \lambda)$ from shadow data alone
\item Consistency check with galactic $\mu$-function fits
\item Falsifiability if shadow and galactic constraints are incompatible
\end{itemize}

%-----------------------------------------------------------------------------
\subsection{Compact Star Structure}\label{sec:strong-tov}
%-----------------------------------------------------------------------------

Neutron stars provide additional tests of strong-field gravity through their mass-radius relation and maximum mass.

\paragraph{DFD-TOV equations.}
The structure of a spherically symmetric, static star in hydrostatic equilibrium is governed by the Tolman-Oppenheimer-Volkoff (TOV) equations. In DFD, the modified TOV system reads:
\begin{equation}\label{eq:dfd-tov}
\fiteq{\frac{dP}{dr} = -\frac{G(\rho + P/c^2)(m + 4\pi r^3 P/c^2)}{r^2(1 - 2Gm/(c^2 r))}\left[1 + \mathcal{O}\left(\frac{\astar}{a}\right)\right],}
\end{equation}
where $m(r) = 4\pi \int_0^r \rho(r') r'^2\, dr'$ is the enclosed mass and $P(r)$, $\rho(r)$ are the pressure and density profiles.

\paragraph{Strong-field limit.}
Inside neutron stars, the characteristic acceleration is:
\begin{equation}
\fiteq{\begin{aligned}
a_{\rm NS} &\sim \frac{GM_{\rm NS}}{R_{\rm NS}^2} \\
&\sim \frac{(1.4 \times 2 \times 10^{30}\,\text{kg}) \cdot 6.67 \times 10^{-11}}{(10^4\,\text{m})^2} \\
&\sim 10^{12}\,\text{m/s}^2.
\end{aligned}}
\end{equation}
With $\azero \sim 10^{-10}$ m/s$^2$, the correction factor in Eq.~\eqref{eq:dfd-tov} is $\mathcal{O}(\azero/a_{\rm NS}) \sim \mathcal{O}(10^{-22})$---utterly negligible.

\paragraph{Implications.}
The $\azero/a$ correction to the TOV \emph{pressure gradient} is negligible inside neutron stars, but this does \emph{not} make the compact-star sector identical to GR: the DFD-native closure of the strong-field equation of state shifts the maximum mass. The machine-verified result (App.~\ref{app:new-results}, Theorem~\ref{thm:new-compactstar}) is:
\begin{enumerate}
\item The DFD compact-star sector gives a maximum mass $M_{\max}^{\rm DFD}=1.474\,M_\odot=0.900\times M_{\max}^{\rm GR}$ --- a $\sim$10\% suppression that \emph{persists into the} $\mu\to1$ \emph{regime}, not a return to GR.
\item This is a genuine, falsifiable DFD deviation from GR, with a causal-EOS ceiling $M_{\max}^{\rm DFD}\le 3.03\,M_\odot$ (versus the GR causal ceiling $\approx4.05\,M_\odot$): any neutron star confirmed above $3.03\,M_\odot$ falsifies DFD outright.
\item Observations of massive pulsars (e.g., PSR J0740+6620 at $2.08 \pm 0.07\, M_\odot$) sit below the $3.03\,M_\odot$ causal ceiling and are consistent with DFD.
\end{enumerate}

%-----------------------------------------------------------------------------
\subsection{Potential DFD-Specific Signatures}\label{sec:strong-signatures}
%-----------------------------------------------------------------------------

While DFD matches GR for leading-order strong-field observables, subtle differences could emerge from:

\paragraph{Strong-field $\mu$-closure.}
If the $\mu$-function deviates from unity at extremely high gradients (beyond the parametrized family calibrated on galactic data), shadow sizes would shift. EHT data constrain:
\begin{equation}
\fiteq{\left|\frac{\Delta \theta_{\rm sh}}{\theta_{\rm sh}}\right| = \left|\Delta\psi(r_{\rm ph}) + \frac{\Delta r_{\rm ph}}{r_{\rm ph}}\right| < 0.17 \quad \text{(from M87*)}.}
\end{equation}
This bounds any strong-field modifications at the $\mathcal{O}(10\%)$ level.

\paragraph{Photon ring substructure.}
Higher-order photon rings (light orbiting multiple times before reaching the observer) probe the near-horizon geometry in detail. Next-generation space VLBI could resolve these subrings, potentially distinguishing optical from geometric horizon physics.

\paragraph{Quasi-normal modes.}
The ringdown phase of binary black hole mergers probes the near-horizon potential. DFD modifications to the effective potential would alter quasi-normal mode frequencies. Current LIGO observations constrain deviations at the 10\% level; future detectors (LISA, Cosmic Explorer) will improve this by orders of magnitude.

\noindent
\begin{tcolorbox}[colback=blue!5!white, colframe=blue!50!black, title=Summary: Strong-Field Behavior]
\textbf{DFD passes all strong-field tests:}
\begin{itemize}
\item Photon sphere / shadow: DFD predicts a 4.6\% larger shadow than Schwarzschild (Eq.~\ref{eq:shadow-ratio}), consistent with current EHT at $0.6\sigma$, testable at ${>}3\sigma$ with next-generation baselines
\item Black hole shadows: EHT observations consistent (M87*, Sgr A*)
\item Neutron stars: maximum mass suppressed to $\approx 0.900\times$ GR (App.~\ref{app:new-results}, Thm.~\ref{thm:new-compactstar}), a falsifiable deviation persisting into $\mu\to1$, with a $3.03\,M_\odot$ causal ceiling
\item Constraints: Strong-field modifications bounded at $\lesssim 10\%$
\end{itemize}
The $\mu \to 1$ limit at high accelerations ensures GR recovery. Distinguishing tests require laboratory LPI measurements or galactic-scale dynamics.
\end{tcolorbox}
