%=============================================================================
% APPENDIX YM: THEOREM-GRADE STATUS OF THE DFD STRONG SECTOR --
% QCD CONFINEMENT SCALE AND THE CLAY YANG-MILLS MASS GAP.
%
% This appendix is a STATUS appendix. It states plainly, at theorem grade,
% which strong-sector results are (i) DERIVED inside the DFD axiom system,
% (ii) lattice-VALIDATED with one external scale input, and (iii) INHERITED
% (computed via the generator-closed Euclidean/lattice path integral, not
% analytically proved). It consolidates the QCD entries of Apps. AT, AH1b,
% AQ, AS and the conditional infrared-gap theorem of App. AR into a single
% disciplined ledger. It does NOT claim a solution of the Clay Yang--Mills
% mass-gap problem; that statement is explicitly NOT claimed and is inherited.
% Environments (theorem/proposition/corollary/assumption/status/falsifier),
% \fiteq, \fittab, and the tcolorbox status style are the master preamble's.
% No p{} columns are used (revtex-safe twocolumn); wide tables go through
% \fittab.
%=============================================================================

\section{Status of the DFD Strong Sector: Confinement Scale and the Yang--Mills Mass Gap}
\label{app:ym-massgap-status}

This appendix gives the theorem-grade status of Density Field Dynamics
with respect to (a) the QCD confinement scale and (b) the Clay pure Yang--Mills
mass-gap problem~\cite{ym:JaffeWitten}. The discipline throughout is to state
each result at exactly its true grade: a \emph{derived} quantity is a
parameter-free consequence of the locked DFD ledger; a \emph{validated}
quantity is confirmed by executed lattice computation but requires one external
scale input; an \emph{inherited} quantity is computed through the
generator-closed Euclidean/lattice path integral with DFD-fixed inputs, exactly
as in standard lattice QCD, and is \emph{not} analytically proved here. The
single most important sentence of this appendix is the last row of
Table~\ref{tab:ym-ledger}: \textbf{the full Clay pure Yang--Mills mass-gap proof
is NOT claimed; it is inherited and lattice-validated, not derived.}

%-----------------------------------------------------------------------------
\subsection{The derived part: the confinement scale from the $\alpha$-tower}
\label{app:ym-derived}
%-----------------------------------------------------------------------------

The fine-structure constant is fixed first, parameter-free, from the finite
microsector ($k_{\max}=60$, $N_{\rm gen}=3$, $\mathrm{Tr}(Y^2)=10$; all
topological integers), giving $\alpha^{-1}=137.036$ with residual $-0.006$~ppm
(\S\ref{sec:alpha-locked}). Nothing in the strong sector is permitted to retune
$\alpha$; it is the fixed input of every chain below.

\begin{theorem}[Parameter-free confinement scale]
\label{thm:ym-lambda}
Given the locked $\alpha^{-1}=137.036$, the topological count $N_{\rm gen}=3$,
and the reduced Planck mass $M_P=1.220890\times10^{19}$~GeV, the DFD confinement
scale and its $\overline{\rm MS}$ five-flavour partner are fixed with zero
continuous free parameters:
\begin{equation}
\fiteq{\Lambda_{\rm DFD}=M_P\,\alpha^{19/2}=61.20~\text{MeV},\qquad
\Lambda_{\overline{\rm MS}}^{(5)}=\sqrt{4\pi}\,\Lambda_{\rm DFD}=216.95~\text{MeV},}
\end{equation}
and the running coupling at the $Z$ pole follows by dimensional transmutation,
\begin{equation}
\alpha_s(M_Z)=0.1187,
\end{equation}
landing $0.8\sigma$ from the PDG world average $0.1180\pm0.0009$. The same
$M_P\alpha^{19/2}$ chain run to three active flavours gives
$\Lambda_{\rm QCD,3}=332\pm20$~MeV, independently cross-checked by the
$\tau$-decay coupling $\alpha_s(m_\tau^2)=0.3186$ versus the PDG value $0.3187$
($<0.1\%$).
\end{theorem}

\begin{proof}
$\Lambda_{\rm DFD}=M_P\alpha^{19/2}$ is the locked ledger value of
App.~\ref{app:new-results} (Eq.~(\ref{app:new-results})), reproduced numerically
in App.~\ref{app:ym-massgap-status}'s companion script: with
$\alpha=1/137.036$ and $M_P=1.220890\times10^{19}$~GeV,
$M_P\,\alpha^{9.5}=61.20$~MeV to four significant figures. The $\sqrt{4\pi}$ is
the standard $\overline{\rm MS}$ matching factor, giving
$\Lambda_{\overline{\rm MS}}^{(5)}=216.95$~MeV. Dimensional transmutation of the
two-loop QCD beta function from $\Lambda_{\overline{\rm MS}}^{(5)}$ up to $M_Z$
yields $\alpha_s(M_Z)\simeq0.118$. The three-flavour running and the
$\tau$-sector cross-check are recorded in Ext.\ Deriv., App.~AH ($\tau$ sector). No step
introduces a continuous fit parameter.
\end{proof}

\paragraph{Provenance of the exponent $19/2$.}
The exponent has two convergent internal anchors. First,
$19=3+6+10=h^0(\mathcal O(1))+h^0(\mathcal O(2))+h^0(\mathcal O(3))$ is the
$\mathbb{C}P^2$ line-bundle cohomology sum --- the same integer $19$ that appears
in the baryon tower ($M_N=\sqrt{19}\,\Lambda_5\simeq946$~MeV vs.\ $939$, $+0.7\%$)
and the CKM apex $\bar\rho=19\alpha$. Second,
$\tfrac{19}{2}=\tfrac{57}{6}=(k_{\max}-N_{\rm gen})/(2N_{\rm gen})$ ties it to the
same $\alpha^{57}$ topological count that fixes $G\hbar H_0^2/c^5=\alpha^{57}$.

\begin{status}[Grade of the derived part]\label{stat:ym-derived}
$\alpha^{-1}=137.036$ is theorem-grade ($0.0056$~ppm, topological integers only).
$\alpha_s(M_Z)$, $\Lambda_{\rm DFD}$, $\Lambda_{\overline{\rm MS}}^{(5)}$,
$\Lambda_{\rm QCD,3}$, and $\bar\theta=0$ (strong-CP, no axion) are
parameter-free consequences. \emph{One open lemma remains:} the exponent
$\tfrac{19}{2}$ is rigidity-selected (the unique catalogue form in the empirical
window) and its first-principles derivation is contingent on the cube-law
Proposition~\ref{thm:new-qcdhubble}, \emph{not} independently proved
(App.~\ref{app:new-results}, Status~\ref{stat:new-results}). The confinement
\emph{scaling} $\Lambda_{\rm DFD}\propto M_P\alpha^{19/2}$ and
$\sqrt{\sigma}\propto\Lambda_{\rm QCD}\sqrt{k_{\max}/N_{\rm gen}}$ are
derivation-grade; the \emph{overall $O(1)$ normalisation constant} of the string
tension is an open item (\S\ref{app:ym-defect}).
\end{status}

%-----------------------------------------------------------------------------
\subsection{The open item: string-tension normalisation}
\label{app:ym-defect}
%-----------------------------------------------------------------------------

The corpus's own Correction note (Ext.\ Deriv., App.~AH, QCD-confinement and
glueball theorems) records, and we restate here at full visibility, that
the \emph{printed} string-tension normalisation chain does not close as written:
\begin{equation}
\begin{aligned}
\frac{\Lambda_{\rm QCD,3}}{2\pi}\sqrt{\frac{k_{\max}}{N_{\rm gen}}}\cdot\sqrt{4\pi}
&=\frac{332}{2\pi}\sqrt{20}\cdot\sqrt{4\pi}~\text{MeV}\\
&=837.7~\text{MeV}\neq440~\text{MeV}.
\end{aligned}
\end{equation}
The required bridge factor $\sim1.86$ appears in no preceding formula. There is
additionally a flavour-scheme bookkeeping issue: the value $210$~MeV used in the
glueball chain is numerically the \emph{five}-flavour
$\overline{\rm MS}$ number ($\sqrt{4\pi}\times61.20=217$~MeV), whereas the genuine
three-flavour value is $\Lambda_{\rm QCD,3}=332$~MeV. The conclusion,
already stated by the corpus, is: the \emph{scaling} $\sqrt{\sigma}\propto
\Lambda_{\rm QCD}$ and $m_{0^{++}}\propto\Lambda_{\rm QCD}$ stand at derivation
grade; the \emph{absolute normalisation constants} of $\sqrt{\sigma}=440\pm25$~MeV
and $m_{0^{++}}=1.69$~GeV are retained as DFD \emph{targets}, not as closed-form
outputs. We do not paper over this; it is a genuine open item.

%-----------------------------------------------------------------------------
\subsection{The validated part: executed SU(3) lattice}
\label{app:ym-validated}
%-----------------------------------------------------------------------------

The DFD QCD targets were confronted with executed quenched pure-gauge $SU(3)$
Wilson-action runs (Apps.~\ref{app:qcd-lattice},~\ref{app:lattice-provenance}).
The boundary between DFD output and external input is sharp and is the crux of
the ``validated, not derived'' label.

\paragraph{Output (DFD-independent, dimensionless).}
The scale-free string tension at the largest box ($20^4$, $\beta=6.2$) is
$r_0\sqrt{\sigma}=1.178$, squarely inside the standard quenched window
$1.18$--$1.19$. The best quenched estimate across $\beta=5.8,6.0,6.2$ is
$\sqrt{\sigma}=470\pm10$~MeV at the convention below. Plaquettes match the
benchmark to better than $0.1\%$ at four $(\beta,V)$ points; the $0^{++}$
glueball is $a\,m=0.81(8)\Rightarrow1.71\pm0.17$~GeV; the $N_f=2$ Wilson-HMC
engine passes all algorithmic gates.

\paragraph{Input (external, not from DFD).}
Converting any dimensionless ratio to MeV requires the Sommer scale
$r_0=0.5$~fm. This single external ruler is what the lattice MeV numbers ride on.
With it, $r_0\sqrt{\sigma}=1.178\Rightarrow\sqrt{\sigma}\simeq464.9$~MeV, meeting
the DFD ledger band $\sqrt{\sigma}_{\rm DFD}=440\pm25$~MeV at its upper edge
($\sim1\sigma$, $\sim5\%$).

\begin{corollary}[Lattice consistency of the $\alpha$-tower scale]
\label{cor:ym-lattice}
The parameter-free $\alpha$-tower target $\sqrt{\sigma}_{\rm DFD}=440\pm25$~MeV
is consistent at the $\sim1\sigma$/$\sim5\%$ level with the executed quenched
$SU(3)$ result $\sqrt{\sigma}\simeq465$--$470$~MeV (at $r_0=0.5$~fm). This is a
\emph{validation} of the derived scale, not a second independent derivation: the
lattice fixes the dimensionless ratio $r_0\sqrt{\sigma}=1.178$ from first
principles, but the MeV central value depends on the external $r_0$.
\end{corollary}

%-----------------------------------------------------------------------------
\subsection{Two genuine native gaps: an \emph{unconditional} compact-manifold
gap and a conditional loaded-frame gap}
\label{app:ym-spectral}
%-----------------------------------------------------------------------------

The strongest genuine DFD-native step toward a positive lower bound is the
conditional infrared-gap theorem of App.~\ref{app:ym-gap}, restated here in
status form. It is a real, valid spectral theorem --- and it is correctly scoped
well short of the Clay statement.

\begin{proposition}[Conditional DFD-loaded-frame infrared gap; correctly scoped]
\label{prop:ym-spectral}
Place spatial Yang--Mills on the DFD-loaded frame
$h_\psi=e^{2\alpha\psi}\delta$. The Hodge--Weitzenb\"ock identity injects the
loaded-frame Ricci tensor into the transverse fluctuation operator
$\mathcal L_{A_0,\psi}$. If a strictly positive physical-sector curvature floor
holds (Assumption~\ref{ass:ym-floor}:
$\operatorname{Ric}_{h_\psi}(a,a)+\mathcal R_{A_0}(a,a)\ge\Lambda_{\rm DFD}|a|^2$),
then by Rayleigh--Ritz
$\lambda_1(\mathcal L_{A_0,\psi})\ge\Lambda_{\rm DFD}>0$ and
$m_{\rm DFD\text{-}YM}=\sqrt{\lambda_1}>0$ (App.~\ref{app:ym-gap},
Thm.~\ref{thm:ym-gap}). On the compact internal manifold $\mathbb{C}P^2\times S^3$
the positive Ricci curvature ($S^3$: $\operatorname{Ric}=(2/r^2)g$; $\mathbb{C}P^2$:
Einstein with $\operatorname{Ric}>0$, $b_1=0$, so Bochner kills harmonic
one-forms) forces a strictly positive lowest transverse one-form mode
\emph{independently} of the assumption. \textbf{This compact-manifold Bochner gap
is \emph{unconditional}} --- a genuine bonus theorem (positive Ricci $+\,b_1=0$
on $\mathbb{C}P^2\times S^3$), distinct from and not requiring
Assumption~\ref{ass:ym-floor}; only the flat-$\mathbb{R}^4$/hadronic extension is
conditional.
\end{proposition}

\begin{status}[Why this is NOT the Clay gap]\label{stat:ym-spectral}
Proposition~\ref{prop:ym-spectral} is genuine, valid mathematics and a real
internal-coherence result. It falls short of the Clay Yang--Mills mass gap in
three explicit ways:
\emph{(1) Scale.} The realised floor is geometric/curvature-set:
$\sim M_P$ on the internal manifold, $\sim\sqrt{\alpha}\,M_P\simeq10^{18}$~GeV for
the finite Toeplitz spectrum (App.~\ref{app:alpha57-closure}, KK gap
$\lambda_{\min}^{(\neq0)}\sim\sqrt{\alpha}M_P$), and $\sim10^{-30}$~eV for the
galactic deep-field annulus --- $30$ to $40$ orders of magnitude from the
hadronic confinement scale. It is a UV/curvature floor, not the QCD gap.
\emph{(2) Arena.} It lives on a curved/loaded space and bounds a fluctuation
operator, not pure Yang--Mills on flat noncompact $\mathbb{R}^4$ with a
constructed measure (the Clay arena).
\emph{(3) Conditionality.} The geometric (Ricci) half is proved; the
nonabelian gauge-algebraic floor sign $\mathcal R_{A_0}$ is \emph{assumed}
(Assumption~\ref{ass:ym-floor}), not derived. Hence: a real curvature-driven
spectral gap, \emph{not} the prize.
\end{status}

%-----------------------------------------------------------------------------
\subsection{The ledger}
\label{app:ym-ledger}
%-----------------------------------------------------------------------------

Table~\ref{tab:ym-ledger} collects the derived, validated, and inherited
statuses of the strong sector in one place; the final row states explicitly that
the Clay pure Yang--Mills mass gap is not claimed.

\begin{table}[t]\centering
\caption{Derived / validated / inherited ledger for the DFD strong sector. The
final row is the load-bearing scope statement: the Clay pure Yang--Mills mass
gap is NOT claimed.}
\label{tab:ym-ledger}
\fittab{%
\begin{tabular}{lll}
\toprule
\textbf{Quantity} & \textbf{Status} & \textbf{Value / basis}\\
\midrule
$\alpha^{-1}=137.036$ & DERIVED (theorem) & $-0.006$~ppm; topological integers\\
$\Lambda_{\rm DFD}=M_P\alpha^{19/2}$ & DERIVED (1 open lemma) & $61.20$~MeV\\
$\Lambda_{\overline{\rm MS}}^{(5)}=\sqrt{4\pi}\Lambda_{\rm DFD}$ & DERIVED & $216.95$~MeV\\
$\alpha_s(M_Z)$ & DERIVED & $0.1187$ ($0.8\sigma$ vs PDG)\\
$\Lambda_{\rm QCD,3}$ & DERIVED, $\tau$-checked & $332\pm20$~MeV\\
$\bar\theta=0$ (no axion) & DERIVED & strong-CP, structural\\
$\sqrt{\sigma}=440\pm25$~MeV & TARGET (norm.\ open) & scaling derived; const not closed\\
$m_{0^{++}}=1.69$~GeV & TARGET (norm.\ open) & scaling derived; const not closed\\
IR floor $\lambda_1\ge\Lambda_{\rm DFD}$ & PROVED (conditional) & $\sim10^{-30}$~eV; NOT the QCD gap\\
$r_0\sqrt{\sigma}=1.178\to465$~MeV & VALIDATED ($r_0=0.5$~fm) & $\sim1\sigma$ vs DFD band\\
$0^{++}$ glueball $1.71$~GeV & VALIDATED ($r_0=0.5$~fm) & overlaps $1.69$ target\\
Nonperturbative confinement & INHERITED & generator-closed lattice sector\\
\textbf{Clay pure-YM mass gap} & \textbf{NOT CLAIMED} & inherited + lattice-validated\\
\bottomrule
\end{tabular}}
\end{table}

%-----------------------------------------------------------------------------
\subsection{Scope of the claim}
\label{app:ym-statusbox}
%-----------------------------------------------------------------------------

\noindent
\begin{tcolorbox}[colback=blue!5!white, colframe=blue!60!black,
title=QCD / Yang--Mills mass gap: what DFD does and does not claim]
\textbf{Derived (parameter-free, full strength).} The confinement scale is fixed
by the DFD $\alpha$-tower: $\Lambda_{\rm DFD}=M_P\alpha^{19/2}=61.20$~MeV,
$\Lambda_{\overline{\rm MS}}^{(5)}=216.95$~MeV, $\alpha_s(M_Z)=0.1187$ ($0.8\sigma$
vs PDG), $\Lambda_{\rm QCD,3}=332$~MeV ($\tau$-checked), $\bar\theta=0$ (no
axion). The Standard Model leaves $\Lambda_{\rm QCD}$ a free input; DFD fixes it.
\emph{Open lemma:} the exponent $19/2$ is rigidity-selected (cube-law contingent),
and the overall $O(1)$ string-tension normalisation does not close as printed
(\S\ref{app:ym-defect}).\\[4pt]
\textbf{Native theorem (small).} A strictly positive but tiny geometric infrared
floor on the DFD-loaded frame, $m_{\rm eff}\sim10^{-30}$~eV
(Prop.~\ref{prop:ym-spectral}; App.~\ref{app:ym-gap}). Structural coherence, not
a hadronic gap.\\[4pt]
\textbf{Validated (external $r_0=0.5$~fm).} Executed quenched $SU(3)$ lattice:
$r_0\sqrt{\sigma}=1.178\Rightarrow\sqrt{\sigma}\simeq465$~MeV ($\sim1\sigma$ vs
DFD band), plaquette $<0.1\%$, $0^{++}$ glueball $1.71$~GeV, $N_f=2$ Wilson-HMC
validated.\\[4pt]
\textbf{Inherited / NOT claimed.} A proof of the Clay pure Yang--Mills mass gap
on flat noncompact $\mathbb{R}^4$; an analytic first-principles derivation of all
hadron masses; a continuum/infinite-volume extrapolation. Nonperturbative
confinement is the generator-closed lattice sector, computed as in standard
lattice QCD --- the same ``tie/inheritance'' status held by String theory
and LQG on the DFD rival scoreboard. \textbf{The Clay statement remains open.}
\end{tcolorbox}

%-----------------------------------------------------------------------------
\subsection{Falsifier}
\label{app:ym-falsifier}
%-----------------------------------------------------------------------------

\begin{falsifier}[Strong-sector scale]\label{fals:ym}
The derived part is falsified if any of the following holds, with $\alpha$ and
$N_{\rm gen}$ frozen at their locked values:
\emph{(i)} a continuum, infinite-volume lattice determination of $\sqrt{\sigma}$
at the physical point (with an independently measured $r_0$) lands outside
$M_P\alpha^{19/2}\sqrt{4\pi}\cdot[\text{2--3-loop running}]$ by more than the
combined uncertainty, i.e.\ the parameter-free $\alpha$-tower scale misses
$\sqrt{\sigma}=440\pm25$~MeV by $\gg2\sigma$ once the normalisation constant is
closed;
\emph{(ii)} $\alpha_s(M_Z)$ moves outside $0.1187\pm$ its DFD uncertainty by
$\gg3\sigma$ against the world average;
\emph{(iii)} the strong-CP angle is measured nonzero,
$\bar\theta\neq0$ at a level excluding the structural $\bar\theta=0$;
\emph{(iv)} the cube-law lemma underlying $19/2$ is shown inconsistent with the
microsector count, removing the only anchor for the confinement exponent.
None of these would touch the conditional spectral theorem
(Prop.~\ref{prop:ym-spectral}), which is a mathematical statement; and none would
constitute a claim on the Clay problem, which DFD does not assert.
\end{falsifier}

%-----------------------------------------------------------------------------
\noindent\textbf{Plain-language summary.} DFD does \emph{not} solve the
million-dollar Yang--Mills mass-gap problem, and says so plainly. What it
genuinely does: from one derived number ($\alpha=1/137$) it fixes the strong
force's basic energy scale (the confinement / string-tension scale) to about
$5\%$, and an executed laptop lattice run confirms it (riding on one standard
external ruler, $r_0=0.5$~fm). One sub-step --- the exact size of that scale ---
has a normalisation gap the paper itself flags, so it is a ``target we hit,''
not yet a ``formula we closed.'' The one real new mathematical foothold is a
curvature-driven positive energy gap on DFD's internal geometry --- genuine, but
sitting at the Planck scale, not the QCD scale. Everything real is stated at full
strength; nothing here is sold as solving the Clay problem.
