%=============================================================================
% APPENDIX AQ: EXECUTED LATTICE-QCD VALIDATION OF THE DFD QCD LEDGER (PROGRAM II)
% Adapted from "DFD-QCD Lattice Boundary Program II: Executed Pure-Gauge SU(3)
% String Tension, 0++ Glueball Consistency, and Validated Nf=2 Wilson-HMC
% Screening Checks" (G. Alcock, June 2026). Macros expanded to plain LaTeX
% (\sqsig -> \sqrt{\sigma}); sections demoted to subsections; floats converted
% to revtex twocolumn (table*/figure with [t]); references routed to the master
% DFD_references.bib with latqcd:* keys. Figures live in figures/.
%=============================================================================

\section{Executed Lattice-QCD Validation of the DFD QCD Ledger (Program II)}
\label{app:qcd-lattice}

DFD-QFT treats nonperturbative QCD as a generator-closed sector: the Euclidean
lattice path integral defines the correlation functions and masses follow from
exponential decay, $C(t)\sim e^{-Mt}$. This appendix executes that propagation
at modest but real scale on commodity hardware, then states each claim at its
actual status. The governing discipline is: scale-free ratios are the primary
outputs; MeV values require an external scale such as the Sommer parameter
$r_0$; and finite-volume, laptop-scale calculations are consistency checks, not
physical-point determinations.

\subsection{Executive summary: what this appendix verifies}
This appendix is a numerical bridge between the DFD QCD ledger and standard
nonperturbative lattice gauge theory. It does not introduce a new lattice-QCD
algorithm, and it does not claim a physical-point full-QCD spectrum. Its purpose
is narrower: take the DFD QCD-sector ledger targets, run standard lattice
machinery in the corresponding pure-gauge and exploratory dynamical sectors, and
determine whether the resulting scale-free lattice observables are consistent
with the ledger. The answer is yes, within the limits of the computation.

The primary pure-gauge check is the $SU(3)$ Wilson-action string tension. The
DFD ledger quotes $\sqrt{\sigma}_{\rm DFD}=440\pm25$~MeV. The executed
calculation does not treat this MeV number as the fundamental object, because
converting a lattice observable to MeV requires external scale setting; the
scale-free quantity is $r_0\sqrt{\sigma}$. The largest-volume box-size check,
$20^4$ at $\beta=6.2$, gives $r_0\sqrt{\sigma}=1.178$, while the standard
quenched reference region is about $1.18$--$1.19$~\cite{latqcd:NeccoSommer,latqcd:LuciniTeper}.
With the canonical Sommer convention $r_0=0.5$~fm this corresponds to roughly
$465$~MeV, at the upper edge of the DFD band; a phenomenological larger-$r_0$
convention quotes the same dimensionless output closer to $440$~MeV. The
conclusion is therefore scale-free consistency, not independent fixation of the
MeV central value.

The second pure-gauge check is the scalar glueball, with DFD ledger target
$m_{0^{++}}\simeq1.69$~GeV. The $16^4$, $\beta=6.0$ run uses a zero-momentum
$A_1^+$ operator built from smeared spatial plaquettes, with vacuum subtraction
and an arccosh effective-mass plateau. After conservative error analysis,
including autocorrelation inflation, the defensible estimate is
$a\,m_{0^{++}}=0.81(8)$, i.e.\ $m_{0^{++}}\simeq1.71\pm0.17$~GeV under
$r_0=0.5$~fm. The scale-free ratios $m_{0^{++}}/\sqrt{\sigma}\simeq3.7$ and
$r_0 m_{0^{++}}\simeq4.3$ sit in the standard quenched range~\cite{latqcd:MP}
and overlap the DFD ledger; because the plateau is short and the effective
sample is reduced by autocorrelation, this is a successful consistency check,
not precision spectroscopy.

The third check is dynamical-fermion validation. A from-scratch $N_f=2$ Wilson
HMC engine passes gamma-matrix algebra, $\gamma_5$-Hermiticity, conjugate-gradient
residual control, reversibility, the HMC identity $\langle e^{-\Delta H}\rangle
\simeq1$, and a direct finite-difference check of the fermion force. A $6^4$
$\kappa$-scan resolves the expected sea-quark screening trend: as $\kappa$ rises
from $0.148$ to $0.156$, the dynamical plaquette rises monotonically from
$0.5703$ to $0.5808$, well above the matched quenched reference. This validates
the dynamical machinery at Program-II level; it is not a physical-point ensemble.

\subsection{Frozen DFD QCD ledger}
The QCD-sector ledger targets considered here are
\begin{align}
\alpha_s(M_Z)&=0.1187, & \bar\theta&=0,\\
\sqrt{\sigma}_{\rm DFD}&=440\pm25~{\rm MeV}, &
m_{0^{++}}^{\rm DFD}&=1.69\pm0.10~{\rm GeV},\\
f_\pi^{\rm DFD}&=92.0\pm4.6~{\rm MeV}. &&
\end{align}
Only the pure-gauge string tension and scalar glueball are evaluated as
numerical outputs; the dynamical Wilson-HMC results validate the execution
engine and the sea-quark screening response, not a physical-point $f_\pi$.

\subsection{Methods}
\paragraph{Pure gauge.} We use the $SU(3)$ Wilson action $S_W=\beta\sum_p(1-\frac13\mathrm{Re}\,\mathrm{Tr}\,U_p)$.
Configurations are generated by multi-hit Metropolis updates on random $SU(2)$
subgroups, interleaved with Cabibbo--Marinari microcanonical overrelaxation. The
static potential uses APE-smeared spatial links ($\alpha=0.5$, maximal-trace
$SU(3)$ projection). Wilson loops $W(R,T)$ give
\begin{equation}
V(R)=\left\langle\log\frac{W(R,T)}{W(R,T+1)}\right\rangle_{T\in[2,4]},
\end{equation}
fit to the Cornell form $V(R)=V_0+\sigma R-e/R$ ($e$ the Coulomb coefficient).
Statistical errors are jackknife over measured configurations.

\paragraph{Dynamical fermions.} We implement two degenerate flavours of
unimproved Wilson fermions,
\begin{equation}
D=1-\kappa\sum_\mu\big[(1-\gamma_\mu)U_\mu(x)\delta_{x+\hat\mu}+(1+\gamma_\mu)U_\mu^\dagger(x-\hat\mu)\delta_{x-\hat\mu}\big],
\end{equation}
with pseudofermion action $S_{\rm pf}=\phi^\dagger(M^\dagger M)^{-1}\phi$ and
$M^\dagger=\gamma_5 M\gamma_5$. Molecular dynamics is reversible leapfrog with
Metropolis accept/reject; the fermion force uses a loose conjugate-gradient
tolerance and the Hamiltonian a tight one.

\subsection{Validation: average plaquette}
The plaquette is UV-dominated and is the first unambiguous sanity check.
Table~\ref{tab:aq-plaq} compares measured values with standard $SU(3)$ Wilson
references; agreement is at the sub-$0.1\%$ level.
\begin{table}[t]\centering
\caption{Average plaquette validation. The $20^4$ row is the final $\beta=6.2$
box-size check.}
\label{tab:aq-plaq}
\begin{tabular}{ccccc}\toprule
volume & $\beta$ & measured $\langle P\rangle$ & reference & deviation\\\midrule
$16^4$ & 5.8 & $0.56750(6)$ & $0.5670$ & $+0.09\%$\\
$16^4$ & 6.0 & $0.59405(5)$ & $0.5937$ & $+0.06\%$\\
$16^4$ & 6.2 & $0.61381(4)$ & $0.6136$ & $+0.03\%$\\
$20^4$ & 6.2 & $0.61376(4)$ & $0.6136$ & $+0.03\%$\\\bottomrule
\end{tabular}
\end{table}
\begin{figure}[t]\centering
\includegraphics[width=\linewidth]{plaquette_validation.png}
\caption{Measured plaquettes against the published $SU(3)$ Wilson curve.}
\label{fig:aq-plaq}
\end{figure}

\subsection{Static potential and string tension}
The static potential exhibits the textbook Coulomb-plus-linear (confining) form
(Fig.~\ref{fig:aq-pot}); the slope is the string tension in lattice units. The
extraction (Table~\ref{tab:aq-string}) is essentially exact at $\beta=6.0$
($a\sqrt{\sigma}=0.217$ vs.\ literature $0.2189$, a $0.8\%$ difference) and
carries a $1$--$5\%$ fit-window systematic at the coarse and fine ends.
\begin{figure}[t]\centering
\includegraphics[width=\linewidth]{static_potential.png}
\caption{Cornell-form static potentials reconstructed from the fitted string
tensions, using APE-smeared Wilson loops and plateau fits.}
\label{fig:aq-pot}
\end{figure}
\begin{table*}[t]\centering
\caption{String tension in lattice units and the scale-free $r_0\sqrt{\sigma}$.
The $20^4$, $\beta=6.2$ row is the final box-size check; literature
$a\sqrt{\sigma}$ from EHK~\cite{latqcd:EHK}, Teper~\cite{latqcd:Teper}, and
Bali--Schilling~\cite{latqcd:Bali}, with $r_0/a$ from Necco--Sommer~\cite{latqcd:NeccoSommer}.}
\label{tab:aq-string}
\begin{tabular}{cccccc}\toprule
$\beta$ & volume & $a\sqrt{\sigma}$ & reference $a\sqrt{\sigma}$ & $r_0\sqrt{\sigma}$ & note\\\midrule
5.8 & $16^4$ & $0.329(10)$ & $0.313$ & $1.206$ & coarse, high\\
5.9 & $16^4$ & $0.244(6)$ & $\sim0.26$ & $1.093$ & interpolation\\
6.0 & $16^4$ & $0.217(3)$ & $0.2189$ & $1.165$ & cleanest point\\
6.1 & $16^4$ & $0.177(4)$ & $\sim0.18$ & $1.123$ & interpolation\\
6.2 & $16^4$ & $0.1623(30)$ & $0.1610$ & $1.198$ & small box, high\\
6.2 & $20^4$ & $0.1596(30)$ & $0.1610$ & $1.178$ & larger box\\\bottomrule
\end{tabular}
\end{table*}

The reviewer-predicted systematic is the $16^4$, $\beta=6.2$ box: its physical
size is about $1.08$~fm, so the fit window sits near the onset of the linear
regime and can bias $\sigma$ upward. The larger $20^4$ box ($\approx1.35$~fm)
shifts $a\sqrt{\sigma}:0.1623\to0.1596$ and $r_0\sqrt{\sigma}:1.198\to1.178$, a
$1.7\%$ downward move that straddles the literature value
$a\sqrt{\sigma}\simeq0.1610$ and lands in the standard quenched reference region
(Figs.~\ref{fig:aq-r0sig}--\ref{fig:aq-box}). This is a directional finite-volume
check, not a continuum extrapolation.
\begin{figure}[t]\centering
\includegraphics[width=\linewidth]{r0_sqrt_sigma.png}
\caption{Scale-free string-tension trend. The $20^4$, $\beta=6.2$ point moves
the high $16^4$ value down into the standard reference region.}
\label{fig:aq-r0sig}
\end{figure}
\begin{figure}[t]\centering
\includegraphics[width=0.92\linewidth]{box_size_test_beta62.png}
\caption{Box-size test at $\beta=6.2$: increasing the box from $16^4$ to $20^4$
reduces the positive bias in $r_0\sqrt{\sigma}$.}
\label{fig:aq-box}
\end{figure}

\subsection{Scale convention and the DFD string-tension ledger}
The MeV value of $\sqrt{\sigma}$ depends on the external physical value assigned
to $r_0$. The scale-free $20^4$ result $r_0\sqrt{\sigma}=1.178$ gives
\begin{equation}
\sqrt{\sigma}\simeq\frac{1.178}{0.5~{\rm fm}}\,(197.327~{\rm MeV\,fm})\simeq465~{\rm MeV}
\end{equation}
under canonical $r_0=0.5$~fm scale setting, meeting the DFD band $440\pm25$~MeV
at its upper edge; a larger phenomenological $r_0$ convention quotes the same
dimensionless output closer to $440$~MeV~\cite{latqcd:LuciniTeper}. The correct
conclusion is scale-free consistency, not independent fixation of the MeV value.

\subsection{Scalar $0^{++}$ glueball}
The glueball operator is the zero-momentum cubic-scalar ($A_1^+$) sum of
APE-smeared spatial plaquettes on each timeslice, with vacuum subtraction. The
arccosh effective-mass plateau is short and the scalar channel is
autocorrelation-limited; the conservatively widened estimate is
$a\,m_{0^{++}}=0.81(8)$ (Fig.~\ref{fig:aq-glue}). Using $r_0=0.5$~fm,
\begin{equation}
\fiteq{m_{0^{++}}=1.71(17)~{\rm GeV},\quad
\frac{m_{0^{++}}}{\sqrt{\sigma}}=3.7(4),\quad
r_0 m_{0^{++}}=4.3(4),}
\end{equation}
which overlaps the standard quenched reference~\cite{latqcd:MP} and the DFD
ledger value $1.69\pm0.10$~GeV. Being quenched, single-spacing, single-volume,
and based on a short plateau, this is a consistency check, not precision
spectroscopy.
\begin{figure}[t]\centering
\includegraphics[width=\linewidth]{glueball_effmass.png}
\caption{$0^{++}$ glueball arccosh effective mass. The shaded band is the
autocorrelation-widened estimate $a\,m_{0^{++}}=0.81(8)$.}
\label{fig:aq-glue}
\end{figure}

\subsection{Executed $N_f=2$ Wilson-HMC validation}
The HMC implementation passes the core algorithmic gates in
Table~\ref{tab:aq-hmc}. The finite-difference force check is the most important,
since it directly tests the fermion-force gradient used in molecular dynamics.
\begin{table}[t]\centering
\caption{Validation gates for the $N_f=2$ Wilson-HMC implementation.}
\label{tab:aq-hmc}
\fittab{%
\begin{tabular}{ll}\toprule
gate & result\\\midrule
$\{\gamma_\mu,\gamma_\nu\}=2\delta_{\mu\nu}$ and $\gamma_5$ & $<10^{-13}$\\
free-field Wilson Dirac operator & exact\\
$\gamma_5$-Hermiticity & $4\times10^{-13}$\\
CG residual & $<10^{-9}$\\
analytic vs.\ numerical fermion force & $6.5\times10^{-7}$ (run-recorded)\\
MD reversibility & $10^{-12}$\\
$\langle e^{-\Delta H}\rangle$ & $\simeq0.997$\\\bottomrule
\end{tabular}%
}
\end{table}

The final dynamical validation is the $6^4$, $\beta=5.6$ $\kappa$-scan
(Table~\ref{tab:aq-kappa}). The matched quenched reference is
$\langle P\rangle=0.52200$. Turning on the $N_f=2$ Wilson determinant raises the
plaquette well above quenched, monotonically in $\kappa$
(Fig.~\ref{fig:aq-kappa}).
\begin{table*}[t]\centering
\caption{$6^4$ dynamical $\kappa$-scan at $\beta=5.6$. Statistical errors are
about $10^{-3}$; the matched quenched reference is $\langle P\rangle=0.52200$.}
\label{tab:aq-kappa}
\begin{tabular}{ccccccc}\toprule
$\kappa$ & 0.148 & 0.150 & 0.152 & 0.154 & 0.155 & 0.156\\\midrule
$\langle P\rangle_{\rm dyn}$ & 0.5703 & 0.5745 & 0.5776 & 0.5790 & 0.5801 & 0.5808\\\bottomrule
\end{tabular}
\end{table*}
\begin{figure}[t]\centering
\includegraphics[width=\linewidth]{hmc_kappa_scan.png}
\caption{Resolved dynamical plaquette $\kappa$-scan. The monotonic increase with
$\kappa$ is the expected sea-quark screening response; the heavy end lands in the
expected SESAM/TXL Wilson-HMC range~\cite{latqcd:SESAM1,latqcd:SESAM2}, while the
light end retains finite-volume excess.}
\label{fig:aq-kappa}
\end{figure}
This directly addresses the limitations of the earlier $4^4$ smoke test: the
magnitude is pulled toward the expected Wilson-HMC range on the larger box, and
the sea-quark mass dependence is resolved. It remains a validation scan, not a
physical-point dynamical ensemble.

\subsection{Verification corrections and claim discipline}
A separate critical re-analysis tightened the claims. The corrections adopted here
are: (i) the glueball error is autocorrelation-inflated to
$a\,m_{0^{++}}=0.81(8)$; (ii) the HMC plaquette rise is an engine and
sea-quark-screening validation, not a precision determinant benchmark; (iii) the
$440$ vs.\ $465$~MeV comparison is a scale-convention issue, with the lattice
sitting $\sim1\sigma$ above $440$ at the canonical $r_0=0.5$~fm; and
(iv) ``first-principles'' is reserved for dimensionless lattice ratios, not MeV
values that require external scale setting.

\subsection{Status boundary}
The pure-gauge string-tension and glueball sectors are quenched. The dynamical
sector is an HMC validation and $\kappa$-scan, not physical-point QCD. No
continuum or infinite-volume extrapolation is claimed. What is established is
that the DFD QCD-ledger targets survive executed lattice-QCD validation checks at
the scale-free, finite-volume consistency level: pure-gauge string tension,
scalar-glueball spectroscopy consistency, dynamical sea-quark screening, and a
finite-volume box-size test. This upgrades the DFD QCD sector from a purely
generator-closed statement to an executed numerical consistency check.

\subsection{Reproducibility}
The companion code package (\texttt{su3\_fast.py}, \texttt{wilson\_hmc.py}, and
the analysis scripts), included as the \texttt{supplement/} directory of this
archive together with the executed run outputs (\texttt{results/}), curated CSV
tables (\texttt{data/}), provenance logs, and the standalone Program~II paper,
reproduces every number above; representative invocations:
\begin{widetext}
\begin{verbatim}
python3 src/su3_fast.py  --L 16 --beta 6.0 --potential   # string tension
python3 src/su3_fast.py  --L 20 --beta 6.2 --potential   # box-size check
python3 src/su3_fast.py  --L 16 --beta 6.0 --glueball    # 0++ glueball
python3 src/wilson_hmc.py --test                         # HMC validation gates
python3 src/wilson_hmc.py --L 6 --beta 5.6 --kappa 0.152 # dynamical kappa point
python3 src/analyze_potential.py ; python3 src/analyze_glueball.py
\end{verbatim}
\end{widetext}
