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% APPENDIX AR: A DFD-NATIVE YANG-MILLS INFRARED GAP FROM LOADED SPATIAL FRAME
% GEOMETRY. Adapted from G. Alcock, "A DFD-Native Yang-Mills Infrared Gap from
% Loaded Spatial Frame Geometry" (June 2026). Custom macros expanded to plain
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\section{A \texorpdfstring{$\mathrm{DFD}$}{DFD}-Native Yang--Mills Infrared Gap from Loaded Spatial Frame Geometry}
\label{app:ym-gap}

This appendix formulates and proves the Yang--Mills infrared-gap statement that
is native to Density Field Dynamics. The target is \emph{not} pure Yang--Mills
theory on empty flat four-dimensional Euclidean space; it is the spatial gauge
sector on a DFD-loaded frame
\begin{equation}
\Sigma_\psi=(\Omega\subset\mathbb{R}^3,\,h_\psi),\qquad h_\psi=e^{2\alpha\psi}\delta,
\end{equation}
where the scalar optical field $\psi$ (the same field that drives DFD optical
gravity and galactic dynamics, cf.\ the main text) loads the physical spatial
frame. The main result is a spectral statement: if the physical transverse
Yang--Mills fluctuation operator on $(\Omega,h_\psi)$ has a strictly positive
Weitzenb\"ock curvature floor, then its first nonzero eigenvalue is positive,
\begin{equation}
\fiteq{\lambda_1(\mathcal L_{A_0,\psi})\ge\Lambda_{\mathrm{DFD}}>0,\qquad
m_{\mathrm{DFD\text{-}YM}}=\sqrt{\lambda_1}\ge\sqrt{\Lambda_{\mathrm{DFD}}}>0.}
\end{equation}
For the DFD deep-field annulus the conformally flat spatial metric gives
positive angular Ricci curvature, which supplies precisely this floor after
physical-sector projection. The resulting scale is a tiny geometric infrared
floor, \emph{not} the hadronic QCD confinement scale and \emph{not} a proof of
the Clay Millennium problem~\cite{ym:JaffeWitten}. It is nevertheless
structurally important: the same scalar-loading mechanism that drives DFD
optical gravity and galactic dynamics also changes the infrared spectrum seen by
spatial Yang--Mills fluctuations.

\subsection{Purpose}
The question is narrow but important: does DFD generate a native Yang--Mills
infrared gap once Yang--Mills fields are placed on the physically loaded DFD
spatial frame rather than on an empty mathematical background? The answer is yes,
under an explicit and checkable geometric condition: the loaded frame must supply
a positive lower bound for the physical transverse fluctuation operator. In empty
Euclidean space the Yang--Mills magnetic energy is only the ordinary kinetic
energy $E_{\mathrm{YM}}[A]=\frac{1}{2g^2}\int_{\mathbb{R}^3}|F_A|^2\,\mathrm{d}^3x$,
and positivity of this stiffness alone does not imply a mass gap. In DFD,
physical gauge fields sit on $h_\psi=e^{2\alpha\psi}\delta$, and the curvature of
this loaded frame enters the Hodge--Weitzenb\"ock decomposition for one-form
fluctuations.

\subsection{DFD-native arena}
Let $G$ be a compact simple gauge group with Lie algebra $\mathfrak{g}$, and let
$\Omega\subset\mathbb{R}^3$ be a bounded spatial domain. The DFD-loaded spatial
frame is $\Sigma_\psi=(\Omega,h_\psi)$ with $h_{\psi,ij}=e^{2\alpha\psi(x)}\delta_{ij}$,
where $\alpha$ is the fine-structure constant and $\psi$ the DFD optical scalar.
A gauge field is a connection one-form $A\in\Omega^1(\Omega,\mathfrak{g})$ with
curvature $F_A=dA+A\wedge A$, and the spatial Yang--Mills energy on the loaded
frame is
\begin{equation}
E_{\mathrm{YM},\psi}[A]=\frac{1}{2g^2}\int_\Omega|F_A|_{h_\psi}^2\,\mathrm{d}\!\operatorname{vol}_{h_\psi}.
\end{equation}
Writing $A=A_0+a$ about a reference background $A_0$, the fluctuation $a$ is
restricted to the physical transverse sector $d_{A_0}^{\ast}a=0$, with elliptic
boundary conditions removing pure-gauge and harmonic zero modes (without which no
positive first nonzero physical eigenvalue can be stated cleanly).

\begin{definition}[DFD-native Yang--Mills infrared gap]
A DFD-native Yang--Mills infrared gap is a positive lower bound for the first
nonzero physical eigenvalue of the spatial Yang--Mills fluctuation operator on
$(\Omega,h_\psi)$: $\lambda_1(\mathcal L_{A_0,\psi})\ge\Lambda_{\mathrm{DFD}}>0$,
with associated infrared mass scale $m_{\mathrm{DFD\text{-}YM}}=\sqrt{\lambda_1}$.
\end{definition}

\subsection{Fluctuation operator}
The Yang--Mills Hessian around $A_0$ has the standard covariant form
$\mathcal L_{A_0,\psi}a=d_{A_0}^{\ast}d_{A_0}a+d_{A_0}d_{A_0}^{\ast}a+\mathcal R_{A_0}a$,
where $\mathcal R_{A_0}$ is the algebraic nonabelian curvature term from
linearization. On the physical transverse sector $d_{A_0}^{\ast}a=0$,
\begin{equation}
\langle a,\mathcal L_{A_0,\psi}a\rangle=\|d_{A_0}a\|^2+\langle a,\mathcal R_{A_0}a\rangle.
\end{equation}
For one-forms on a Riemannian three-manifold, the Hodge--Weitzenb\"ock identity
\cite{LawsonMichelsohn1989,ym:DonaldsonKronheimer} gives
$\Delta_{\mathrm{Hodge}}a=\nabla^{\ast}\nabla a+\operatorname{Ric}_{h_\psi}(a)$,
equivalently at the quadratic-form level
\begin{equation}
\|da\|^2+\|d^{\ast}a\|^2=\|\nabla a\|^2+\int_\Omega\operatorname{Ric}_{h_\psi}(a,a)\,\mathrm{d}\!\operatorname{vol}_{h_\psi}
\end{equation}
up to boundary terms that vanish under the chosen elliptic conditions. Thus the
Ricci curvature of the DFD-loaded spatial frame enters the physical Yang--Mills
fluctuation operator as a zeroth-order stiffness contribution.

\subsection{Main theorem}
\begin{assumption}[DFD physical-sector curvature floor]\label{ass:ym-floor}
There exists a constant $\Lambda_{\mathrm{DFD}}>0$ such that every physical
transverse fluctuation $a$ obeys
$\operatorname{Ric}_{h_\psi}(a,a)+\mathcal R_{A_0}(a,a)\ge\Lambda_{\mathrm{DFD}}|a|_{h_\psi}^2$.
\end{assumption}

\begin{theorem}[DFD-loaded-frame Yang--Mills infrared gap]\label{thm:ym-gap}
Let $(\Omega,h_\psi)$ be a bounded DFD-loaded spatial frame with
$h_\psi=e^{2\alpha\psi}\delta$, and let $\mathcal L_{A_0,\psi}$ be the Yang--Mills
fluctuation operator on physical transverse one-form fluctuations satisfying
$d_{A_0}^{\ast}a=0$, with boundary/gauge conditions removing pure-gauge and
harmonic zero modes. If Assumption~\ref{ass:ym-floor} holds, then
$\lambda_1(\mathcal L_{A_0,\psi})\ge\Lambda_{\mathrm{DFD}}>0$, and therefore
$m_{\mathrm{DFD\text{-}YM}}=\sqrt{\lambda_1}\ge\sqrt{\Lambda_{\mathrm{DFD}}}>0$.
\end{theorem}

\begin{proof}
For physical transverse fluctuations $\langle a,\mathcal L_{A_0,\psi}a\rangle=\|d_{A_0}a\|^2+\langle a,\mathcal R_{A_0}a\rangle$.
The covariant Hodge--Weitzenb\"ock decomposition writes the kinetic part as a
nonnegative covariant-gradient contribution plus the loaded-frame Ricci
contribution; after physical-sector projection and boundary-term removal,
\begin{equation}
\langle a,\mathcal L_{A_0,\psi}a\rangle\ge\int_\Omega\big[\operatorname{Ric}_{h_\psi}(a,a)+\mathcal R_{A_0}(a,a)\big]\,\mathrm{d}\!\operatorname{vol}_{h_\psi}.
\end{equation}
By Assumption~\ref{ass:ym-floor} the integrand is $\ge\Lambda_{\mathrm{DFD}}|a|_{h_\psi}^2$,
so $\langle a,\mathcal L_{A_0,\psi}a\rangle\ge\Lambda_{\mathrm{DFD}}\|a\|^2$. By the
Rayleigh--Ritz characterization of the first nonzero physical eigenvalue,
$\lambda_1=\inf_{a\perp\operatorname{ker}\mathcal L,\,a\ne0}\langle a,\mathcal L_{A_0,\psi}a\rangle/\|a\|^2\ge\Lambda_{\mathrm{DFD}}$,
hence $m_{\mathrm{DFD\text{-}YM}}=\sqrt{\lambda_1}\ge\sqrt{\Lambda_{\mathrm{DFD}}}>0$.
\end{proof}

\subsection{Deep-field DFD specialization}
In the DFD deep-field branch the scalar profile in a galactic annulus is
logarithmic, $\psi(r)=\psi_0-B\ln(r/r_0)$ with $B=\frac{2}{c^2}\sqrt{GMa_\star}$,
and the loaded spatial metric is $h_{ij}=e^{2\alpha\psi}\delta_{ij}$. For this
conformally flat geometry the main text records
$\operatorname{Ric}_{\theta\theta}=B\alpha(2-B\alpha)$, $\operatorname{Ric}_{rr}=0$,
so for $0<\alpha B<2$ the angular Ricci curvature is positive. On a bounded
deep-field annulus, after physical transverse projection and removal of
pure-gauge, radial, and harmonic zero modes, this supplies a positive
physical-sector lower bound. The resulting DFD geometric scale is
$\lambda_1\ge C_1\Lambda$ with
\begin{equation}
m_{\mathrm{eff}}=\sqrt{\lambda_1}\sim\frac{(GMa_\star)^{1/4}}{cR},
\end{equation}
which for Milky-Way-scale parameters is of order $m_{\mathrm{eff}}\sim10^{-30}$~eV
--- strictly nonzero but far below the hadronic QCD scale. It is a DFD geometric
infrared floor, not the QCD confinement gap.

\subsection{Why this supports DFD}
The same scalar field $\psi$ appears in three places: (i) optical gravity through
the refractive index $n=e^\psi$; (ii) galactic dynamics through the deep-field
scalar profile; and (iii) gauge-sector infrared behaviour through the loaded
spatial frame $h_\psi=e^{2\alpha\psi}\delta$. Ordinary Yang--Mills theory on an
empty background has no such geometric input; DFD predicts that physically
realized gauge fields propagate on a scalar-loaded frame whose curvature
generically contributes to the gauge fluctuation spectrum. This is not decisive
empirical proof, but evidence of internal coherence and structural compression:
one scalar-loading mechanism organizes optical gravity, galactic phenomenology,
and a gauge-sector infrared floor.

\subsection{Scope and non-claims}
The theorem deliberately does not make three stronger claims (Table~\ref{tab:ym-status}).
\begin{table*}[t]\centering
\caption{Status discipline for the DFD-native Yang--Mills infrared gap.}
\label{tab:ym-status}
\begin{tabular}{ll}
\toprule
\textbf{Statement} & \textbf{Status}\\
\midrule
\parbox[t]{0.60\linewidth}{\raggedright DFD-loaded spatial Yang--Mills has a positive geometric infrared floor when the physical-sector curvature floor is positive.} & \parbox[t]{0.30\linewidth}{\raggedright Proved here (Thm.~\ref{thm:ym-gap}).}\\[0.6em]
\parbox[t]{0.60\linewidth}{\raggedright The deep-field DFD annulus supplies positive angular Ricci curvature for $0<\alpha B<2$.} & \parbox[t]{0.30\linewidth}{\raggedright DFD deep-field specialization.}\\[0.6em]
\parbox[t]{0.60\linewidth}{\raggedright The expected galactic scale is tiny, around $10^{-30}$~eV.} & \parbox[t]{0.30\linewidth}{\raggedright DFD geometric prediction.}\\[0.6em]
\parbox[t]{0.60\linewidth}{\raggedright This proves the Clay pure Yang--Mills mass gap.} & \parbox[t]{0.30\linewidth}{\raggedright Not claimed.}\\[0.6em]
\parbox[t]{0.60\linewidth}{\raggedright This analytically derives all hadron masses.} & \parbox[t]{0.30\linewidth}{\raggedright Not claimed.}\\[0.6em]
\parbox[t]{0.60\linewidth}{\raggedright Nonperturbative QCD remains computable from the Euclidean/lattice path-integral generator with DFD-fixed inputs (Apps.~\ref{app:qcd-lattice},~\ref{app:lattice-provenance}).} & \parbox[t]{0.30\linewidth}{\raggedright DFD-QFT generator-closed sector.}\\
\bottomrule
\end{tabular}
\end{table*}

\subsection{Falsifiers and completion checks}
The theorem is mathematical, but the DFD application has concrete checks:
(i) \emph{operator check} --- derive the exact Hessian $\mathcal L_{A_0,\psi}$ for
the DFD-loaded domain and verify the sign of $\mathcal R_{A_0}$ on the physical
sector; (ii) \emph{projection check} --- verify the boundary/gauge conditions
remove pure-gauge, radial, and harmonic zero modes in the deep-field annulus;
(iii) \emph{spectrum check} --- numerically compute $\lambda_1$ for realistic DFD
backgrounds and test the scaling $m_{\mathrm{eff}}\sim(GMa_\star)^{1/4}/(cR)$;
(iv) \emph{physics check} --- confirm the predicted geometric floor lies far below
QCD and does not contaminate hadronic mass physics. The theorem is complete as a
conditional spectral theorem; the DFD phenomenological specialization is testable
by computing the physical-sector operator on explicit backgrounds.

\subsection{Conclusion}
DFD's native Yang--Mills gap is a spatial-frame result: gauge fields are placed
not on empty $\mathbb{R}^3$ but on $h_\psi=e^{2\alpha\psi}\delta$, and the
Hodge--Weitzenb\"ock identity contributes the loaded-frame Ricci tensor into the
physical fluctuation operator. When the resulting physical-sector curvature floor
is positive, $\lambda_1\ge\Lambda_{\mathrm{DFD}}>0$ and hence
$m_{\mathrm{DFD\text{-}YM}}>0$. This is not a replacement for nonperturbative QCD
confinement (which DFD treats as the generator-closed lattice sector of
Apps.~\ref{app:qcd-lattice} and~\ref{app:lattice-provenance}); it is a DFD-native
infrared-gap theorem in which scalar loading of the spatial frame creates a tiny
but nonzero gauge-sector infrared floor, through the same loaded geometry that
explains optical gravity and galactic dynamics.
