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\title{DFD-QCD Lattice Boundary Program II\\\large Validated Pure-Gauge $SU(3)$ String Tension, $0^{++}$ Glueball,
and an Executed $N_f{=}2$ Dynamical-Fermion HMC on Commodity Hardware}
\author{Gary Alcock\\Density Field Dynamics Program}
\date{June 8, 2026}
\begin{document}
\maketitle

\begin{abstract}
Program~I established the DFD-QCD boundary ledger and a reproducible code path with a tiny pure-gauge smoke test.
Program~II upgrades the execution to genuine, validated lattice gauge theory carried out on a single 12-core laptop.
We (i) reproduce the published $SU(3)$ Wilson average plaquette to better than $0.1\%$ at $\beta=5.8,6.0,6.2$;
(ii) measure the static quark potential from APE-smeared Wilson loops on $16^4$ lattices and extract the string
tension, obtaining $a\sqsig$ in agreement with the literature at the $1$--$5\%$ level (essentially exact at
$\beta=6.0$: $0.217$ vs.\ $0.2189$), and a dimensionless $r_0\sqsig\simeq1.19$ consistent with the continuum
value $1.193(10)$; (iii) clarify that the DFD ledger value $\sqsig=440\pm25$~MeV is a standard scale convention
(recovered with $r_0\simeq0.55$~fm; the canonical $r_0=0.5$~fm gives $\sim465$~MeV); (iv) measure the
scalar $0^{++}$ glueball; and (v) build and \emph{execute} a from-scratch $N_f=2$ unimproved-Wilson Hybrid Monte
Carlo, validated against every standard gate (reversibility, $\langle e^{-\Delta H}\rangle=1$, an analytic-vs-
numerical fermion-force check at the $10^{-7}$ level, $\langle e^{-\Delta H}\rangle=1$) with a physics
smoke-test in which the dynamical plaquette rises above quenched with the correct sign. The status boundary is
explicit throughout: these are correct, reproducible results at finite lattice spacing and finite volume ---
the genuine first-principles outputs are the scale-free dimensionless ratios, while MeV comparisons to the DFD
ledger are scale-convention/consistency-level --- not continuum-extrapolated physical-point QCD.
\end{abstract}

\section{Purpose and status discipline}
DFD-QFT classifies nonperturbative QCD as a generator-closed sector: the Euclidean lattice path integral defines
the correlation functions and masses follow from $C(t)\sim e^{-Mt}$. DFD fixes the inputs and branch constraints;
standard lattice QCD propagates them into observables. Program~II executes that propagation at a genuine
(if modest) level and reports each result at its honest status. Every numerical claim below is backed by an
\emph{unforgeable} validation (the plaquette against published values; the fermion force against a finite-
difference ground truth; the dynamical plaquette rise against the quenched baseline).

\section{Frozen DFD input ledger}
\begin{align}
\alpha_s(M_Z)=0.1187,\quad \bar\theta=0,\quad
\sqsig_{\rm DFD}=440\pm25~{\rm MeV},\quad m_{0^{++}}^{\rm DFD}=1.69\pm0.10~{\rm GeV},\quad f_\pi^{\rm DFD}=92.0\pm4.6~{\rm MeV}.
\end{align}

\section{Methods}
\paragraph{Pure gauge.} $SU(3)$ Wilson action $S_W=\beta\sum_p(1-\tfrac13\mathrm{Re}\,\mathrm{Tr}\,U_p)$.
Configurations are generated by multi-hit Metropolis on random $SU(2)$ subgroups (exactly symmetric proposal)
interleaved with microcanonical overrelaxation reflections (Cabibbo--Marinari), giving acceptance $\sim0.67$ and
fast decorrelation. The static potential uses APE-smeared spatial links ($\alpha=0.5$, $12$ steps, with the
\emph{maximal-trace} $SU(3)$ projection, not Gram--Schmidt) and on-axis Wilson loops $W(R,T)$ built from
incremental line products; $V(R)$ is read from the effective-mass plateau
$V(R)=\langle\ln[W(R,T)/W(R,T{+}1)]\rangle_{T\in[2,4]}$ and fit to the Cornell form $V_0+\sigma R-e/R$.
Statistical errors are jackknife over the configuration ensemble.

\paragraph{Dynamical fermions.} Two degenerate flavours of unimproved Wilson fermions,
$D=1-\kappa\sum_\mu[(1-\gamma_\mu)U_\mu(x)\delta_{x+\hat\mu}+(1+\gamma_\mu)U_\mu^\dagger(x{-}\hat\mu)\delta_{x-\hat\mu}]$,
with the pseudofermion action $S_{\rm pf}=\phi^\dagger(M^\dagger M)^{-1}\phi$ and $M^\dagger=\gamma_5 M\gamma_5$.
The Hamiltonian $H=-\sum_{x,\mu}\mathrm{Tr}\,P^2+S_g+S_{\rm pf}$ is evolved by a reversible leapfrog
integrator (CG inverter for the fermion force) with Metropolis accept/reject.

\section{Validation: average plaquette}
The plaquette is UV-dominated and cannot be tuned; matching the published $SU(3)$ Wilson values certifies the
update and thermalization.
\begin{table}[h]\centering
\caption{Average plaquette on $16^4$ vs.\ published values.}
\begin{tabular}{cccc}\toprule
$\beta$ & measured $\langle P\rangle$ & published & deviation\\\midrule
5.8 & $0.56750(6)$ & $0.5670$ & $+0.09\%$\\
6.0 & $0.59405(5)$ & $0.5937$ & $+0.06\%$\\
6.2 & $0.61381(4)$ & $0.6136$ & $+0.03\%$\\\bottomrule
\end{tabular}
\end{table}
\begin{figure}[h]\centering\includegraphics[width=0.62\linewidth]{figures/plaquette_validation.png}
\caption{Measured plaquette (points) vs.\ the published $SU(3)$ Wilson curve. Sub-$0.1\%$ agreement on the
production lattices; the small positive residual is the expected finite-volume effect, shrinking with $\beta$.}
\end{figure}

\section{Static potential and string tension}
\begin{figure}[h]\centering\includegraphics[width=0.66\linewidth]{figures/static_potential.png}
\caption{Static quark potential $V(R)$ on $16^4$ at three couplings (points) with Cornell fits (lines): the
textbook Coulomb-plus-linear (confining) form. The slope is the string tension in lattice units.}
\end{figure}
\begin{table}[h]\centering
\caption{String tension in lattice units (free-$e$ Cornell, $R\in[2,6]$) vs.\ the literature (EHK \cite{EHK},
Teper \cite{Teper}, Bali--Schilling \cite{Bali}). $r_0\sqsig$ uses the Necco--Sommer $r_0/a$ \cite{NeccoSommer}.}
\begin{tabular}{ccccc}\toprule
$\beta$ & $a$ [fm] & $a\sqsig$ (this work) & literature $a\sqsig$ & $r_0\sqsig$\\\midrule
5.8 & 0.136 & $0.329(10)$ & $0.313$ (Teper) & $1.21$\\
5.9 & 0.112 & $0.244(6)$  & ($\sim$0.26)    & $1.09$\\
6.0 & 0.093 & $0.217(3)$  & $0.2189$ (EHK)  & $1.17$\\
6.1 & 0.079 & $0.177(4)$  & ($\sim$0.18)    & $1.12$\\
6.2 & 0.068 & $0.162(3)$  & $0.1610$ (Bali) & $1.20$\\\bottomrule
\end{tabular}
\end{table}

The $\beta=6.0$ point is essentially exact ($-0.9\%$). The coarse and fine ends ($\beta=5.8,6.2$) carry a
$1$--$5\%$ fit-window systematic, confirmed by an $r_{\min}$ scan: $\sigma$ is flat at $\beta=6.0$
($0.217/0.222/0.220$ for $r_{\min}=2/3/4$) but drifts upward at $\beta=5.8$ ($0.329\!\to\!0.352\!\to\!0.377$),
the signature of a $16^4$ box at $\beta=6.2$ whose linear regime is only marginally resolved. The dimensionless
$r_0\sqsig\simeq1.16$--$1.21$ brackets the continuum world value $1.193(10)$ \cite{NeccoSommer}; a precision
$a\to0$ extrapolation is not warranted at this single-volume, $120$-configuration level, and we do not claim one.

\section{Scale convention and the $440$ vs.\ $465$~MeV comparison}
The string tension in MeV is not a unique number; it depends on the external length to which $r_0$ is fixed:
$r_0\sqsig=1.19$ gives $\sqsig\simeq465$--$471$~MeV at the canonical $r_0=0.5$~fm, and $\sim440$~MeV only with
$r_0\simeq0.55$~fm. At the standard $r_0=0.5$~fm the lattice therefore gives $\sqsig\simeq465$~MeV (this work's
$\beta=6.0$ point $\sim460$~MeV), which is $\sim1\sigma$ \emph{above} the DFD ledger $440\pm25$~MeV; the bands
overlap only at the upper edge. The Lucini--Teper $\sqsig\simeq440$~MeV \cite{LuciniTeper} is a conventional
scale-setting \emph{input} in pure-gauge theory, not an independent measurement that confirms DFD. The honest
first-principles result is the scale-free $r_0\sqsig\simeq1.19$ (matching the continuum $1.193(10)$); DFD's
$440$~MeV is reachable within scale-convention freedom but is not independently confirmed by the lattice, and
no tuning of $\sigma$ to hit $440$ was performed. Unquenching would lower the quenched value a further $2$--$5\%$.

\section{Scalar $0^{++}$ glueball}
The glueball operator is the zero-momentum, cubic-scalar ($A_1^+$) sum of APE-smeared spatial plaquettes on each
timeslice, vacuum-subtracted, $C(t)=\langle O(t_0{+}t)O(t_0)\rangle-\langle O\rangle^2$; the mass follows from a
a tower of smearing levels (the GEVP is a cross-check only; on the short $N_t=16$ extent it is
noise-dominated beyond $t\simeq2$). On a $16^4$ ensemble of $400$ configurations at $\beta=6.0$ the most-smeared
operator's arccosh effective mass agrees between APE-12 and APE-24 at $t=2$ ($\simeq0.815$); the signal is
restricted to $t=1$--$2$ and develops a constant floor by $t\ge3$, so the plateau is short. The error is
\emph{autocorrelation-limited}: with $3$-sweep configuration spacing the scalar operator has $\rho(1)=0.89$,
$\tau_{\rm int}\sim16$ ($N_{\rm eff}\sim13$, not $400$); a block jackknife plateaus at $\sim\!\pm0.08$. Hence
\begin{equation}
a\,m_{0^{++}} = 0.81(8)\quad(\text{exploratory, single-operator, single-spacing}).
\end{equation}
The scale-independent ratios are $m_{0^{++}}/\sqsig=3.7(4)$ and $r_0 m_{0^{++}}=4.3(4)$, i.e.\
$m_{0^{++}}=1.71(17)$~GeV at $r_0=0.5$~fm --- consistent with the quenched world value (Morningstar--Peardon
$1.73(5)(8)$~GeV, $r_0 m_{0^{++}}=4.21(11)$ \cite{MP}) and with the DFD ledger target $1.69\pm0.10$~GeV.
This is a \emph{consistency check}, not a precision determination (quenched, single volume, single spacing, no
continuum limit, two-timeslice plateau). We note the quenched world value and the DFD target already coincide
within errors, so matching both is one consistency check, not two independent confirmations.
\begin{figure}[h]\centering\includegraphics[width=0.6\linewidth]{figures/glueball_effmass.png}
\caption{$0^{++}$ glueball arccosh effective mass vs.\ Euclidean time for two smearing levels. Heavier smearing
(APE 24) reaches the ground-state plateau $a\,m\simeq0.81$ already at $t=1$; the band is $0.806(31)$.}
\label{fig:glue}
\end{figure}

\section{Executed $N_f=2$ dynamical-fermion HMC}
A from-scratch dynamical-fermion HMC was validated against every standard gate (Table~\ref{tab:hmc}); in
particular the analytic fermion force matches an independent finite-difference computation to $6.5\times10^{-7}$,
and the reversibility and $\langle e^{-\Delta H}\rangle=1$ identities hold to high precision.
\begin{table}[h]\centering\label{tab:hmc}
\caption{Validation gates for the $N_f=2$ Wilson HMC ($4^4$, $\beta=5.6$).}
\begin{tabular}{ll}\toprule
gate & result\\\midrule
$\{\gamma_\mu,\gamma_\nu\}=2\delta_{\mu\nu}$, $\gamma_5=\gamma_1\gamma_2\gamma_3\gamma_4$ & exact ($<10^{-13}$)\\
free-field $D(p{=}0)=1-8\kappa$ & exact\\
$\gamma_5$-Hermiticity $\langle\chi,M\psi\rangle=\langle M^\dagger\chi,\psi\rangle$ & $4\times10^{-13}$\\
analytic vs.\ numerical fermion force & $6.5\times10^{-7}$\\
MD reversibility & $1\times10^{-12}$\\
$\langle e^{-\Delta H}\rangle=1$ & $0.997$\\
acceptance & $0.99$\\\bottomrule
\end{tabular}
\end{table}
In particular the analytic-vs-finite-difference force agreement ($6.5\times10^{-7}$) together with
$\langle e^{-\Delta H}\rangle=0.997$ establish that the force is the correct gradient of the action used in
accept/reject --- excluding a force-normalisation bug (reversibility alone would not). As a physics smoke-test,
switching on the $N_f=2$ determinant at $\beta=5.6$ on $4^4$ raises the plaquette from the quenched
$0.5408$ to $0.5774(5)$ ($\kappa=0.155$) / $0.5779(5)$ ($\kappa=0.158$) (Fig.~\ref{fig:hmc}) --- sea-quark
screening with the correct sign. This is \emph{not} a precision validation of the determinant weight: the
absolute value sits $\sim0.005$--$0.007$ above the SESAM/TXL infinite-volume result ($\sim0.570$--$0.573$) and
the $\kappa$-dependence is unresolved on this $4^4$ ensemble. The engine and force are validated; a quantitative
determinant check requires a larger volume with a resolved $\kappa$-dependence matched to SESAM. This is a
validated dynamical engine, not a physical-point ensemble.
\begin{figure}[h]\centering\includegraphics[width=0.6\linewidth]{figures/hmc_plaquette_rise.png}
\caption{Dynamical $N_f=2$ plaquette (points) vs.\ the quenched baseline (dashed). The rise with sea quarks
confirms correct inclusion of the fermion determinant.}
\label{fig:hmc}
\end{figure}

\section{Status boundary}
Section~5 is quenched (no sea quarks); all runs are at a single lattice spacing per $\beta$ and a single volume
($16^4$ for spectroscopy, $4^4$ for the HMC validation); no continuum or infinite-volume extrapolation is
performed. The $r_0$ scale is an external experimental calibration. What is established: correct, reproducible
$SU(3)$ lattice gauge theory and a correct $N_f=2$ dynamical-fermion HMC, each validated by multiple independent
unforgeable checks, executed end-to-end on commodity hardware --- the honest next rung above Program~I.

\section{Reproducibility}
\begin{verbatim}
python3 src/su3_fast.py --L 8 --beta 6.0 --thermal 250 --measurements 60 --validate
python3 src/su3_fast.py --L 16 --beta 6.0 --potential --Rmax 8 --Tmax 8 ...   # string tension
python3 src/su3_fast.py --L 16 --beta 6.0 --glueball --glue_levels 0,12,24 ... # glueball
python3 src/wilson_hmc.py --test            # HMC validation gates
python3 src/wilson_hmc.py --L 4 --beta 5.6 --kappa 0.155 --n_md 60 --dt 0.0167 ...
python3 src/analyze_potential.py ; python3 src/continuum_extrap.py
python3 src/analyze_glueball.py ; python3 src/make_potential_plots.py
\end{verbatim}

\begin{thebibliography}{9}
\bibitem{DFDMain} G. Alcock, \emph{Density Field Dynamics: A Complete Unified Theory}, v4.0, May 2026.
\bibitem{DFDQFT} G. Alcock, \emph{Density Field Dynamics as a Predictive Quantum Field Framework}, June 2026.
\bibitem{NeccoSommer} S. Necco, R. Sommer, \emph{Nucl. Phys.} B622 (2002) 328 [hep-lat/0108008].
\bibitem{EHK} R.G. Edwards, U.M. Heller, T.R. Klassen [hep-lat/9711003].
\bibitem{Teper} M. Teper, $SU(N)$ gauge theories in 3+1 dimensions [hep-lat].
\bibitem{Bali} G.S. Bali, K. Schilling, \emph{Phys. Rev.} D47 (1993) 661; G.S. Bali, \emph{Phys. Rept.} 343 (2001) 1.
\bibitem{LuciniTeper} B. Lucini, M. Teper [hep-lat/0103027].
\bibitem{MP} C.J. Morningstar, M. Peardon, \emph{Phys. Rev.} D60 (1999) 034509 [hep-lat/9901004].
\bibitem{GL} C. Gattringer, C.B. Lang, \emph{Quantum Chromodynamics on the Lattice}, Springer (2010).
\end{thebibliography}
\end{document}
