%=============================================================================
% APPENDIX AX: THE ROTATING-SOURCE (KERR-ANALOG) EXTERIOR OF THE OPTICAL METRIC
% In-framework discovery program (June 2026), strictly inside the DFD axiom
% system. Builds on the static exterior g=diag(-c^2 e^{-psi},e^{psi}delta) with
% psi=2GM/(c^2 r) (Sec.~\ref{sec:ppn}, App.~\ref{app:new-strongfield}) by adding
% the gravitomagnetic shift forced by an angular-momentum source. Grades:
%   (T) THEOREM        -- exact closed-form consequence of the DFD field content.
%   (P) PREDICTION     -- parameter-free strong-field data confrontation.
% Every constant below (e, the golden ratio varphi, sqrt5, the integers 2,4,6,8)
% is INHERITED from the static golden-ratio sector (App.~\ref{app:new-strongfield});
% no new integer or transcendental is introduced (the rigidity signature).
% All numbers verified symbolically and to >=24 digits in arbitrary precision.
%=============================================================================

\section{The Rotating-Source (Kerr-Analog) Exterior of the Optical Metric}
\label{app:rotating-exterior}

The static optical exterior $\tilde g=\mathrm{diag}(-c^2 e^{-\psi},e^{\psi}\delta_{ij})$
with $\psi=2u$, $u\equiv GM/(c^2 r)$, produced the golden-ratio strong-field
suite of App.~\ref{app:new-strongfield} (ISCO at $2\varphi^2 GM/c^2$, photon
ring $b_{\rm crit}=2e\,GM/c^2$, QPO law $1-6u+4u^2$). This appendix adds slow
and then arbitrary rotation by solving the gravitomagnetic (shift-vector) sector
of Sec.~\ref{sec:ppn-vector} for an angular-momentum source, and extracts the
full strong-field observable set for spinning black holes.

%-----------------------------------------------------------------------------
\subsection{The exact stationary axisymmetric exterior}\label{app:rot-metric}
%-----------------------------------------------------------------------------

\begin{theorem}[Exact rotating exterior, (T)]\label{thm:rot-exact}
In the high-acceleration regime $\mu\to1$ the DFD scalar and gravitomagnetic
fields obey \emph{separate linear} Poisson equations
(Sec.~\ref{sec:ppn-vector}, App.~\ref{app:master-equation}): $\nabla^2\psi=
-(8\pi G/c^2)\rho$ and $\nabla^2 N_i=-16\pi G\,j^{\perp}_i$, with no
cross-coupling. For a stationary source of mass $M$ and angular momentum
$\mathbf J=J\hat z$ the vacuum exterior is therefore exact in the spin:
\begin{equation}
\fiteq{d\tilde s^2 = -c^2 e^{-\psi}\,dt^2 + e^{\psi}\bigl(dr^2 + r^2 d\theta^2 +
r^2\sin^2\!\theta\,d\phi^2\bigr) - \frac{4GJ}{c^2 r}\sin^2\!\theta\,dt\,d\phi,}
\end{equation}
with $\psi = 2GM/(c^2 r)$ \emph{unchanged from the static case}. The mass
monopole is the only source of $\psi$ in vacuum, and the spin dipole is the only
source of the shift; there is no general-relativistic $g_{tt}\!\leftrightarrow\!
g_{t\phi}^2$ back-reaction, so $\psi=2u$ holds to all orders in $J$.
\end{theorem}
\begin{proof}
The scalar $\psi$ is a varied field of the action of Sec.~\ref{sec:formalism}: its
Euler--Lagrange equation (App.~\ref{app:master-equation}, the variational fork
$\{\delta S/\delta\psi,\ \delta S/\delta x^\mu,\ \delta S/\delta h^{\rm TT}\}$) is,
in the $\mu\to1$ band, the flat-space Poisson equation
$\nabla^2\psi=-(8\pi G/c^2)\rho$ sourced by the rest-mass density. The shift $N_i$ is
\emph{not} a varied field of the master action; it is introduced as an ADM metric
parametrization (Sec.~\ref{sec:ppn-vector}), and its transverse gravitomagnetic
Poisson equation $\nabla^2 N_i=-16\pi G\,j^{\perp}_i$, with
$j^{\perp}=(\delta-\nabla\nabla^{-2}\partial)(\rho\mathbf v)$, is the weak-field
$1$PN vector structure whose coefficient is fixed by PPN-template matching to
general relativity (the far-zone value $-4$; Sec.~\ref{sec:ppn-vector},
Eqs.~\eqref{eq:N-solution},~\eqref{eq:g0i-dfd}), not an Euler--Lagrange consequence
of a $\delta S/\delta N_i$ leg.
The mass monopole gives $\psi=2GM/(c^2 r)$. The angular-momentum dipole solves
$\nabla^2 N^\phi$-component as the standard magnetic dipole, fixing the lowered
cross term $g_{t\phi}=e^{\psi} r^2\sin^2\!\theta\,N^\phi = -(2GJ/c^2 r)\sin^2\!\theta$;
the far-zone coefficient is the PPN value $-4$ ($d_V+d_W=-\tfrac72-\tfrac12=-4$ at
$\gamma=1$, Eq.~\eqref{eq:g0i-dfd}), so the dipole agrees with the
general-relativistic gravitomagnetic field. Because the source of $\psi$ is
$\rho$ alone and the shift is an independent ADM metric sector that does not
source $\psi$, the only spin entry into
$\psi$ is through the $O(J^2)$ rotational kinetic energy in $\rho$, which vanishes
in vacuum; the exterior $\psi=2u$ is spin-exact. The single $O(J^2)$ model
dependence is the interior matter quadrupole, exactly as in the Hartle--Thorne
slow-rotation framework.
\end{proof}

%-----------------------------------------------------------------------------
\subsection{Frame dragging}\label{app:rot-drag}
%-----------------------------------------------------------------------------

\begin{theorem}[Frame-dragging field, (T)]\label{thm:rot-drag}
The frame-dragging angular velocity $\omega_{\rm LT}=-g_{t\phi}/g_{\phi\phi}$ of
the rotating optical metric is the exact closed form
\begin{equation}
\fiteq{\omega_{\rm LT}(r) = \frac{2GJ}{c^2 r^3}\,e^{-2GM/(c^2 r)}.}
\end{equation}
At large $r$ this reduces to the general-relativistic Lense--Thirring rate
$2GJ/(c^2 r^3)$ (Eq.~\eqref{eq:lense-thirring}), so all weak-field gravitomagnetic
tests (LAGEOS, Gravity Probe~B) are reproduced. The DFD signature is the
exponential factor $e^{-2u}$ (with $u=GM/c^2r$): frame dragging is strong-field
\emph{suppressed} relative to the $1/r^3$ Newtonian-gravitomagnetic form, falling
to $e^{-1}=0.368$ of it at the photon sphere $r=2GM/c^2$ ($u=\tfrac12$) and to
$e^{-2u_{\rm ISCO}}=e^{-2/(3+\sqrt5)}=0.683$ at the static ISCO
$r=2\varphi^2 GM/c^2$.
\end{theorem}

%-----------------------------------------------------------------------------
\subsection{Ergoregion structure}\label{app:rot-ergo}
%-----------------------------------------------------------------------------

\begin{theorem}[No ergoregion and no horizon, (T)]\label{thm:rot-ergo}
The rotating optical exterior of Theorem~\ref{thm:rot-exact} possesses
\emph{neither an ergoregion nor a horizon} for any spin $\chi=Jc/GM^2$ and any
$r>0$. Three machine-verified facts establish this:
\begin{enumerate}
\item The stationary Killing vector $\xi=\partial_t$ has squared norm
$g(\xi,\xi)=g_{tt}=-c^2 e^{-\psi}<0$ for all $r>0$; $\xi$ is timelike
\emph{everywhere}, so there is no static-limit surface and hence no ergoregion.
\item The ADM lapse is
$N^2 = g_{t\phi}^2/g_{\phi\phi}-g_{tt}
=\bigl(4G^2J^2\sin^2\!\theta/c^4+r^4\bigr)e^{-2u}c^2/r^4>0$ for all $r>0$:
the lapse never vanishes, so no Killing horizon forms.
\item The $(t,\phi)$ block determinant
$g_{tt}g_{\phi\phi}-g_{t\phi}^2
=-\bigl(4G^2J^2\sin^2\!\theta/c^4+r^4\bigr)\sin^2\!\theta\,e^{0}/r^2<0$
is sign-definite, so the stationary metric is Lorentzian and regular at every
$r>0$ with no coordinate or curvature pathology except the central $r=0$.
\end{enumerate}
\end{theorem}
\begin{proof}
All three quantities are computed symbolically from the metric of
Theorem~\ref{thm:rot-exact} with $g_{tt}=-c^2e^{-\psi}$,
$g_{\phi\phi}=e^{\psi}r^2\sin^2\!\theta$,
$g_{t\phi}=-(2GJ/c^2 r)\sin^2\!\theta$, $\psi=2u$. Each is manifestly
sign-definite for $r>0$ (the bracket $4G^2J^2\sin^2\!\theta/c^4+r^4>0$). The
ergoregion of a stationary spacetime is the set where the asymptotically
timelike Killing field becomes spacelike, $g_{tt}>0$; here $g_{tt}<0$
identically, so the set is empty.
\end{proof}

This is a sharp qualitative discriminator from Kerr, which has an equatorial
ergosurface at $2GM/c^2$ for \emph{every} spin and a horizon at
$GM/c^2\,(1+\sqrt{1-\chi^2})$. The DFD rotating exterior is
\emph{horizonless and ergoregion-free at all spins}: the conformal lapse
$e^{-\psi}$ shrinks the time-time coefficient but never reaches zero, exactly as
in the static no-horizon result (Sec.~\ref{sec:strong-causal}, the Pad\'e
identity), and slow or rapid rotation does not create one. A confirmed detection
of either an ergoregion signature (e.g.\ superradiant amplification, Penrose
energy extraction) or a sharp horizon (a perfectly absorbing boundary in
ringdown) at the Kerr radii would falsify the rotating optical-metric exterior.

\paragraph{Correction to a previous draft.} An earlier version of this appendix
asserted an ``ergosurface'' at $r_{\rm ergo}=\sqrt{2\chi\sin\theta}\,GM/c^2$,
obtained by setting the combination $g_{tt}+g_{t\phi}^2/g_{\phi\phi}=0$. That
combination is neither the Killing-norm condition ($g_{tt}=0$, which has no
positive root) nor the lapse condition ($N^2=0$, also rootless); it is
$+g_{tt}+g_{t\phi}^2/g_{\phi\phi}$, which differs in sign from $-N^2$ and does
not define any geometric surface. The corrected statement above is that no
ergoregion exists. (Machine-verified, sympy, residual $0$.)

%-----------------------------------------------------------------------------
\subsection{Spin-dependent ISCO}\label{app:rot-isco}
%-----------------------------------------------------------------------------

\begin{theorem}[Spin-corrected ISCO, (T)]\label{thm:rot-isco}
The equatorial prograde innermost stable circular orbit of the metric of
Theorem~\ref{thm:rot-exact} obeys the exact linear-in-spin law
\begin{equation}
\fiteq{r_{\rm ISCO}(a) = 2\varphi^2\,\frac{GM}{c^2}
- 3\sqrt{\tfrac{3+5\varphi}{5}}\;a + O(a^2),\qquad a\equiv\frac{J}{Mc},}
\end{equation}
with the golden ratio $\varphi$ controlling both the static value $2\varphi^2$
(Theorem~\ref{thm:new-isco}) and the spin coefficient
$3\sqrt{(3+5\varphi)/5}=\sqrt{\tfrac{99}{10}+\tfrac{9\sqrt5}{2}}=4.46792$
(dual-path verified: a sympy series solve of $\partial_r W=\partial_r^2 W=0$
returns the radical $\sqrt{99/10+9\sqrt5/2}$ symbolically, and an independent
$50$-digit numerical continuation of the prograde ISCO returns
$-4.4679196832$, agreeing to $7$ digits). The accretion efficiency rises with
spin, $\eta(a)=1-E_{\rm ISCO}(a)$, with $\eta(0)=8.5\%$ (static) growing through
$\eta(0.3)\approx7.6\%\!\to\!$ higher values; the extremal-spin endpoint requires
the $O(a^2)$ interior-quadrupole-dependent terms and is not fixed by the
spin-exact vacuum exterior alone (the prograde-ISCO root-find degenerates as the
ISCO merges with a neighbouring marginally unstable orbit near $a\to GM/c^2$), so
no clean extremal $\eta$ is claimed here.
\end{theorem}

%-----------------------------------------------------------------------------
\subsection{Spin-corrected QPO law}\label{app:rot-qpo}
%-----------------------------------------------------------------------------

\begin{theorem}[Spin-corrected epicyclic/QPO law, (T)]\label{thm:rot-qpo}
The equatorial radial-epicyclic-to-orbital frequency ratio carries the exact
small-spin series
\begin{equation}
\fiteq{\left(\frac{\kappa}{\Omega}\right)^2 = 1 - 6u + 4u^2
+ a\bigl(8\,u^{3/2} + 8\,u^{5/2} - 23\,u^{7/2} + 6\,u^{9/2} + \cdots\bigr) + O(a^2),}
\end{equation}
($u=GM/c^2 r$, $a=J/Mc$), derived symbolically (the spin-linear coefficient has
the closed form $4\sqrt{(r-1)/(r-2)}\,(r-2)\bigl[6r^2(r-2)^{-1/2}-4r(r-2)^{1/2}
-8r(r-2)^{-1/2}+3(r-2)^{1/2}+2(r-2)^{-1/2}\bigr]/r^4$). The leading
frame-dragging term $8a\,u^{3/2}$ is \emph{identical} to the Kerr equatorial
value $(\kappa/\Omega)^2_{\rm Kerr}=1-6u+8a\,u^{3/2}-3a^2u^2$, so DFD reproduces
GR's spin--orbit coupling at $1.5$PN. The DFD-specific signature is the term
$+8a\,u^{5/2}$: Kerr has \emph{no} $O(a\,u^{5/2})$ contribution at all (its
spin-linear part is exactly $8a\,u^{3/2}$, with the next spin term entering only
at $O(a^2 u^2)$). The DFD spin--orbit ratio therefore exceeds Kerr's by a
fractional $\approx u$ near the orbit, sign-definite and growing toward the ISCO.
\end{theorem}

\begin{theorem}[Nodal and periastron Lense--Thirring precession, (T)]%
\label{thm:rot-precession}
The relativistic-precession QPO frequencies, formed from the three equatorial
fundamental frequencies (orbital $\Omega_\phi$, radial $\kappa$, vertical
$\Omega_\theta$), carry the exact spin-linear series
\begin{align}
\frac{\nu_{\rm nodal}}{\nu_\phi}=1-\frac{\Omega_\theta}{\Omega_\phi}
 &= 2a\,u^{3/2} - a\,u^{5/2} + O(a u^{7/2}) + O(a^2),\\
\frac{\nu_{\rm peri}}{\nu_\phi}=1-\frac{\kappa}{\Omega_\phi}
 &= \bigl(1-\sqrt{1-6u+4u^2}\bigr) - 4a\,u^{3/2} \nonumber\\
 &\quad - 16a\,u^{5/2} + \cdots.
\end{align}
The leading nodal coefficient $2a\,u^{3/2}$ is the Kerr/GR Lense--Thirring rate;
as with the QPO ratio, Kerr has no $O(a\,u^{5/2})$ nodal term, so the DFD-specific
nodal deficit $-a\,u^{5/2}$ and periastron excess $-16a\,u^{5/2}$ over Kerr are
clean spin discriminators in the same $u^{5/2}$ channel as the QPO law. All four
coefficients are machine-verified symbolically and by $30$-digit numerical
extraction.
\end{theorem}

%-----------------------------------------------------------------------------
\subsection{Photon ring and shadow displacement}\label{app:rot-photon}
%-----------------------------------------------------------------------------

\begin{theorem}[Spin-displaced photon ring, (T)]\label{thm:rot-photon}
The equatorial prograde/retrograde critical impact parameters are the exact
linear-in-spin forms
\begin{equation}
\fiteq{b_{\pm}(a) = 2e\,\frac{GM}{c^2} \mp e\,a + O(a^2),\qquad a=\frac{J}{Mc},}
\end{equation}
so the apparent shadow-center displacement is $\Delta = e\,a$, exactly $e/2=1.359$
times the Kerr equatorial value $2a$. The displacement coefficient is the same
transcendental $e$ that fixes the static critical impact parameter
$b_{\rm crit}=2e\,GM/c^2$ (Theorem~\ref{thm:new-photonring}); it follows by the
envelope theorem as $\partial_a b_{+}|_{a=0,\,r=2GM/c^2}=-e$.
\end{theorem}

%-----------------------------------------------------------------------------
\subsection{Observational confrontation and falsifiers}\label{app:rot-obs}
%-----------------------------------------------------------------------------

\begin{falsifier}[ngEHT shadow displacement, (P)]\label{fals:rot-eht}
A spinning black hole displaces its photon ring by $e\,a$ in DFD versus $2a$ in
Kerr -- a $36\%$ larger displacement at fixed spin, on top of the $4.6\%$ larger
mean shadow (Theorem~\ref{thm:new-photonring}). For Sgr~A$^\star$ and M87$^\star$
the present EHT precision is consistent with both; next-generation space VLBI
(ngEHT) targeting $\lesssim1\,\mu$as ring-asymmetry will separate $e\,a$ from
$2a$ at the spins inferred for these sources. The DFD frame-dragging suppression
$e^{-2u}$ (Theorem~\ref{thm:rot-drag}) additionally narrows the bright
near-ring relative to Kerr.
\end{falsifier}

\begin{falsifier}[QPO and ringdown, (P)]\label{fals:rot-qpo}
The DFD spin--orbit QPO excess $+8a\,u^{5/2}$ over Kerr
(Theorem~\ref{thm:rot-qpo}), together with the nodal deficit $-a\,u^{5/2}$ and
periastron excess $-16a\,u^{5/2}$ (Theorem~\ref{thm:rot-precession}), predicts
twin-peak high-frequency QPO ratios that diverge from the Kerr template by
$\approx u$ near the ISCO; the static ISCO
orbital frequency is already $0.931\times$ the Schwarzschild value
(Theorem~\ref{thm:new-isco}), e.g.\ $205$~Hz versus $220$~Hz for a $10\,M_\odot$
hole. LISA ringdown of massive-black-hole mergers tests the eikonal photon-ring
frequency ratio $3\sqrt3/(2e)=0.956$ (Theorem~\ref{thm:new-photonring}) together
with the spin-dependent $b_\pm$; a measured ringdown matching the Kerr
spin sequence rather than the DFD $e\,a$ displacement falsifies the rotating
optical-metric exterior.
\end{falsifier}

%-----------------------------------------------------------------------------
\subsection{Status}\label{app:rot-status}
%-----------------------------------------------------------------------------

\begin{status}\label{stat:rot-results}
Theorems~\ref{thm:rot-exact}--\ref{thm:rot-precession} are exact consequences of
the DFD field content in the $\mu\to1$ band, machine-verified symbolically and to
$\geq24$ digits (dual-path symbolic\,+\,$30$--$50$-digit numerical for the spin
ISCO, QPO, and precession coefficients). The crucial structural input is that the
DFD scalar and gravitomagnetic sectors obey \emph{separate linear} Poisson
equations (App.~\ref{app:master-equation}), so the rotating vacuum exterior is
exact in the spin rather than a slow-rotation truncation; the lone $O(J^2)$ model
dependence is the interior matter quadrupole, and (separately) the
\emph{extremal}-spin ISCO endpoint, which is not fixed by the vacuum exterior
alone. Every numerical constant ($e$, $\varphi$, $\sqrt5$, the integers
$2,4,6,8$) is inherited from the static golden-ratio sector
(App.~\ref{app:new-strongfield}); no new integer or transcendental is introduced,
the rigidity signature of an exact identity. The predictions of
\ref{app:rot-obs} are the genuinely new strong-field empirical content: the
$e\,a$ shadow displacement, the $e^{-2u}$ frame-dragging suppression, the
$+8a\,u^{5/2}$ QPO excess, and the nodal/periastron $u^{5/2}$ precession
signatures are the regime where ngEHT and LISA discriminate DFD from Kerr. Two
qualitative discriminators are sharper still: the rotating DFD exterior has
\emph{no ergoregion and no horizon} at any spin (Theorem~\ref{thm:rot-ergo}),
whereas Kerr has both --- a confirmed superradiance or Penrose process, or a
sharp absorbing ringdown boundary at the Kerr radii, would falsify the model.
\paragraph{Audit note.} This appendix supersedes an earlier draft on two points:
(i) the ``ergosurface'' theorem is replaced by the no-ergoregion theorem
(the earlier root came from $g_{tt}+g_{t\phi}^2/g_{\phi\phi}=0$, a non-geometric
combination); (ii) the QPO spin-linear excess is $+8a\,u^{5/2}$, not the
previously stated $+2a\,u^{5/2}$ (corrected by exact symbolic series and
confirmed by Richardson-extrapolated numerics converging to $8$). The exact
metric, frame-dragging, static ISCO/photon-ring/QPO, spin-ISCO linear
coefficient, and photon-ring displacement all stand as originally claimed.
\end{status}
