%=============================================================================
% APPENDIX AN: THE OPTICAL-METRIC MASTER EQUATION
% Unifies light propagation (c_1 = c/n) and matter acceleration
% (a = (c^2/2) grad psi) as the null and timelike sectors of one wave
% equation on the optical metric. Master compile via DFD_Main.tex.
%=============================================================================

\section{The Optical-Metric Master Equation: Unified Light and Matter Motion}
\label{app:master-equation}

The two foundational motion laws of \DFD, the one-way light speed
$c_1 = c\,e^{-\psi} = c/n$ and the matter acceleration
$\mathbf{a} = (c^2/2)\nabla\psi$, are not
independent postulates. This appendix shows that both are limits of a single
equation of motion: the massive wave equation on one optical metric, with light
as its null sector and matter as the slow limit of its timelike sector. The
factor $1/2$ in the acceleration law, which might appear fitted, is forced by the
rest-energy structure of that equation.

\subsection{The isotropic optical metric}
\label{app:me-metric}

Light propagation fixes only the conformal class of a metric: null cones encode
the local speed but not an overall scale. Within that class we adopt the
isotropic representative
\begin{equation}
d\tilde s^2 \;=\; -\,c^2 e^{-\psi}\,dt^2 \;+\; e^{\psi}\,\delta_{ij}\,dx^i dx^j ,
\label{eq:me-metric}
\end{equation}
with $\tilde g_{00} = -c^2 e^{-\psi}$, $\tilde g_{ij} = e^{\psi}\delta_{ij}$, and
inverse $\tilde g^{00} = -e^{\psi}/c^2$, $\tilde g^{ij} = e^{-\psi}\delta^{ij}$.
Its null condition $d\tilde s^2 = 0$ gives $|d\mathbf{x}/dt| = c\,e^{-\psi} =
c/n$, so Eq.~\eqref{eq:me-metric} agrees with the eikonal light metric
$d\tilde s^2 = -c^2 dt^2/n^2 + d\mathbf{x}^2$ on null structure; the two are
conformally related and yield identical light cones.

Equation~\eqref{eq:me-metric} is distinguished within its conformal class by its
timelike sector. In the weak field $\tilde g_{00}/c^2 \approx -(1-\psi) =
-(1+2\Phi/c^2)$ identifies the potential $\Phi = -\tfrac{1}{2}c^2\psi$,
reproducing the effective Newtonian potential of the main text and the matter
acceleration. The spatial part gives $\tilde g_{ij} \approx (1+\psi)\delta_{ij} =
(1+2\gamma\Phi/c^2)\delta_{ij}$ with $\Phi = \tfrac12 c^2\psi$, hence the
parametrized post-Newtonian value $\gamma = 1$ and the \GR\ light-deflection
coefficient $4GM/c^2 b$. Equation~\eqref{eq:me-metric} is therefore the unique
representative whose \emph{full} geodesic structure, null and timelike,
reproduces simultaneously the refractive index, the matter acceleration, and
$\gamma=1$. (The bare eikonal form $-c^2 dt^2/n^2 + d\mathbf{x}^2$ encodes only
the light sector; its timelike geodesics do not by themselves yield
$\Phi = -c^2\psi/2$, which is why the isotropic representative
Eq.~\eqref{eq:me-metric} is the natural carrier of the unified motion law.)

\subsection{The master wave equation and its eikonal limit}
\label{app:me-wave}

The fundamental object for motion is the wave equation on
Eq.~\eqref{eq:me-metric},
\begin{equation}
\left(\tilde\Box_{\tilde g} - \frac{m^2 c^2}{\hbar^2}\right)\Psi \;=\; 0,
\qquad
\tilde\Box_{\tilde g} = \frac{1}{\sqrt{-\tilde g}}\,\partial_\mu\!\big(
\sqrt{-\tilde g}\,\tilde g^{\mu\nu}\partial_\nu\big),
\label{eq:me-wave}
\end{equation}
which covers the full wave regime. The ray, or particle, regime follows from the
eikonal substitution $\Psi = A\,e^{iS/\hbar}$, whose leading order in $\hbar$ is
the mass shell $\tilde g^{\mu\nu}\partial_\mu S\,\partial_\nu S + m^2 c^2 = 0$.
With the inverse metric of Eq.~\eqref{eq:me-metric} this becomes the single
dispersion relation
\begin{equation}
\boxed{\;E^2 \;=\; c^2 e^{-2\psi}\,|\mathbf{p}|^2 \;+\; m^2 c^4 e^{-\psi}\;}
\label{eq:me-dispersion}
\end{equation}
This relation contains both sectors and is the eikonal of
Eq.~\eqref{eq:me-wave}, not an independent ansatz.

\begin{theorem}[Unified light--matter motion; \textsc{Derived}]
\label{thm:me-unified}
Light and matter are the null ($m=0$) and timelike ($m>0$) sectors of the single
dispersion relation Eq.~\eqref{eq:me-dispersion}. The massless sector gives the
group velocity $v_g = \partial E/\partial|\mathbf{p}| = c\,e^{-\psi} = c/n$; the
slow massive sector gives $\mathbf{a} = (c^2/2)\nabla\psi$, with the factor $1/2$
arising from the $e^{-\psi/2}$ rest-energy term rather than being assumed.
\end{theorem}

\begin{proof}
Setting $m=0$ in Eq.~\eqref{eq:me-dispersion} gives
$E = c\,e^{-\psi}|\mathbf{p}|$ and $v_g = c\,e^{-\psi} = c/n$. For $m>0$,
\begin{equation}
E = m c^2 e^{-\psi/2}\sqrt{1 + \frac{e^{-\psi}|\mathbf{p}|^2}{m^2 c^2}}
\;\approx\; m c^2 e^{-\psi/2} + \frac{|\mathbf{p}|^2}{2m}\,e^{-3\psi/2}.
\end{equation}
In the weak field $e^{-\psi/2}\approx 1-\psi/2$, so to leading order
$E \approx m c^2 - \tfrac{1}{2} m c^2\psi + |\mathbf{p}|^2/2m$, giving the
nonrelativistic Hamiltonian $H_{\rm NR} = |\mathbf{p}|^2/2m + U$ with
$U = -\tfrac{1}{2} m c^2\psi$. Hence
$\mathbf{a} = -\nabla U/m = (c^2/2)\nabla\psi$, the \DFD\ matter
acceleration law.
\end{proof}

\subsection{Gravity as the gradient of the squared light speed}
\label{app:me-identity}

The two sectors share one exact statement. For a particle momentarily at rest the
all-orders acceleration is
\begin{equation}
\boxed{\;\mathbf{a} \;=\; -\tfrac{1}{4}\,\nabla\!\big(c_{\rm light}^2\big),
\qquad c_{\rm light} = c\,e^{-\psi}\;}
\label{eq:me-optid}
\end{equation}
since $-\tfrac14\,\partial_\psi(c^2 e^{-2\psi}) = \tfrac12 c^2 e^{-2\psi}$, which
reduces to $(c^2/2)\nabla\psi$ in the weak field. Equation~\eqref{eq:me-optid} is
exact and mechanism-agnostic: matter falls down the gradient of the squared local
speed of light. It is the optical statement of gravitation, and it makes the
unification of the two sectors manifest.

\subsection{Relation to the action, and the boundary on matter}
\label{app:me-boundary}

Equations~\eqref{eq:me-wave}--\eqref{eq:me-optid} descend, together with the
field equation for $\psi$ and the radiative sector, from a single action by
varying distinct fields, a fork rather than a linear chain:
\begin{align}
\frac{\delta S}{\delta\psi}=0 &\;\Rightarrow\;
\nabla\!\cdot\!\big[\mu(|\nabla\psi|/a_*)\nabla\psi\big] = -\frac{8\pi G}{c^2}\rho
\nonumber\\
&\qquad \text{[Eq.~\eqref{eq:field_general}, App.~\ref{app:mu-derivation}]},
\nonumber\\
\frac{\delta S}{\delta x^\mu}=0 &\;\Rightarrow\;
\text{geodesics of }\tilde g \text{ (Theorem~\ref{thm:me-unified})},
\nonumber\\
\frac{\delta S}{\delta h^{\rm TT}}=0 &\;\Rightarrow\;
\Box\,h^{\rm TT}=\text{source}, \quad c_T=c .
\nonumber
\end{align}
The gravitational field equation comes from varying $\psi$; the matter motion law
comes from varying the trajectory. These are different variations of different
fields, and the matter law is not obtained from $\delta S/\delta\psi$.

We are explicit about the one input Eq.~\eqref{eq:me-wave} does not supply: the
origin of the mass $m$. The dispersion relation and the geodesics are identical
for any value of $m$, whatever its source; $m$ enters through the matter action.
Three statements bound the options. First, the appealing reading of matter
as a static finite-energy soliton of $\psi$ itself is obstructed by Derrick's
theorem in three spatial dimensions, where the deep-field nonlinearity is absent
($\mu\to1$) and the field is canonical. Second, a scalar Kaluza--Klein tower on
the internal manifold is excluded by Theorem~T177 (every such mode has mass
$\ge\sqrt{\alpha}\,M_P\approx 10^{18}$~GeV) and is moreover non-chiral. Third, the
fermion-mass content of \DFD\ resides instead in the spectral-action
Dirac/Yukawa sector on $\mathbb{C}P^2\times S^3$ (Appendices~\ref{app:Yfinite}
and~\ref{app:neutrino}), where the light fermions are the chiral zero modes
acquiring mass through Higgs overlap, dressed by powers of $\alpha$. That sector
is chiral and sub-electroweak precisely because it is not a Laplacian tower.
Equation~\eqref{eq:me-wave} is thus the master equation of \DFD's
gravitational-optical sector, complete within that domain and deliberately silent
on the origin of matter.

\subsection{Status and falsifiability}
\label{app:me-status}

The unification carries empirical content. Because light and matter share one
metric but couple to different invariants of it (null versus timelike), \DFD\
predicts a species- and altitude-dependent local-position-invariance signature in
co-located optical-clock comparisons (Sec.~\ref{sec:clocks}) and a modified
distance duality on cosmological scales (Sec.~\ref{sec:cosmology}). The
light-bending coefficient is the \GR\ value $4GM/c^2 b$ by construction
($\gamma=1$), so solar-system tests are passed, while the deep-field sector
reproduces galaxy rotation curves through Eq.~\eqref{eq:field_general}. The single
most distinctive statement is Eq.~\eqref{eq:me-optid}: gravitation is the gradient
of the squared local speed of light.
