%=============================================================================
% APPENDIX AS: LATTICE BOUNDARY PROGRAM PROVENANCE (PROGRAMS I AND I-A).
% Consolidates G. Alcock, "DFD-QCD Lattice Boundary Program I: Frozen Inputs,
% Pure-Gauge SU(3) Smoke Test, and Unquenched Execution Scaffold" and
% "...Program I-A: Executed Quenched SU(3) String-Tension Runs from DFD-Fixed
% Inputs" (June 2026). The final, validated results live in Appendix
% \ref{app:qcd-lattice} (Program II); this appendix records the staged-execution
% lineage and discipline without re-printing the consolidated tables.
%=============================================================================

\section{Lattice Boundary Program: Staged Execution and Provenance (Programs I--II)}
\label{app:lattice-provenance}

The validated lattice results of Appendix~\ref{app:qcd-lattice} (``Program~II'')
were reached through a deliberately staged execution, recorded here for
provenance and to document the status discipline applied at each rung. The
lineage is Program~I (frozen inputs, smoke test, unquenched scaffold)
$\to$ Program~I-A (first executed quenched $16^4$ string-tension runs)
$\to$ Program~II (consolidated $16^4$ and $20^4$ string tension, $0^{++}$
glueball, and validated $N_f=2$ Wilson-HMC). At no stage was a laptop-scale
calculation presented as continuum-extrapolated, physical-point QCD.

\subsection{Program~I: frozen inputs, smoke test, unquenched scaffold}
Program~I established the four-stage methodology that the later rungs inherit.
\emph{Stage~1} froze the DFD QCD ledger (the same targets used throughout
Apps.~\ref{app:qcd-lattice}--\ref{app:lattice-provenance}: $\alpha_s(M_Z)=0.1187$,
$\bar\theta=0$, $\sqrt{\sigma}_{\rm DFD}=440\pm25$~MeV,
$m_{0^{++}}^{\rm DFD}=1.69\pm0.10$~GeV, $f_\pi^{\rm DFD}=92.0\pm4.6$~MeV).
\emph{Stage~2} audited the analytic nonperturbative targets (string tension,
lightest scalar glueball, low-energy chiral scales) against the
generator-closed-sector statement. \emph{Stage~3} executed a small pure-gauge
$SU(3)$ Wilson-action Metropolis smoke test on $4^4$ lattices at
$\beta=5.4,5.6,5.8$, measuring plaquettes, small Wilson loops, an effective
potential, and a crude string-slope proxy. \emph{Stage~4} specified --- without
executing --- the unquenched dynamical-fermion HMC bridge, enumerating the
Dirac-operator, pseudofermion, molecular-dynamics, and accept/reject steps
required for a production run.

The Stage-3 $4^4$ smoke test was explicitly a code-path verification, not a
physics result: at such a tiny volume and with limited thermalization the
plaquette values were not at their infinite-volume benchmarks, and the
``string-slope proxy'' $V_{\rm eff}(2)-V_{\rm eff}(1)$ was a placeholder, not a
fitted string tension. Program~I's conclusion was therefore that the DFD
QCD sector had been moved from ``generator exists'' to ``generator has an
executable boundary-program prototype'' --- with the production rung still to
come. The Stage-4 scaffold is the exact specification that was later implemented
and validated as the $N_f=2$ Wilson-HMC engine of Appendix~\ref{app:qcd-lattice}.

\subsection{Program~I-A: first executed quenched \texorpdfstring{$16^4$}{16^4} string tension}
Program~I-A replaced the $4^4$ smoke-test headline with executed, production-grade
quenched pure-gauge $SU(3)$ runs on $16^4$ lattices at $\beta=5.8,6.0,6.2$ (the
$4^4$ smoke test was archived, not used as physics). The plaquette benchmark was
reproduced to better than $0.1\%$ at all three couplings, and APE-smeared
Wilson loops with effective-potential plateaus and Cornell fits gave
\begin{align}
\sqrt{\sigma}\,a&=(0.3292,\,0.2171,\,0.1623),\\
r_0\sqrt{\sigma}&=(1.206,\,1.165,\,1.198),
\end{align}
i.e.\ $\sqrt{\sigma}=476\pm15$, $460\pm7$, and $473\pm7$~MeV at $r_0=0.5$~fm, for a
best quenched estimate $\sqrt{\sigma}=470\pm10$~MeV. Program~I-A already drew the
correct conclusion: this overlaps the DFD ledger target
$\sqrt{\sigma}_{\rm DFD}=440\pm25$~MeV at its upper edge, with the residual
assigned to quenching and scale-setting convention rather than to a DFD failure.

These three $16^4$ points are precisely the $\beta=5.8,\,6.0,\,6.2$ $16^4$ rows subsequently consolidated
in Table~\ref{tab:aq-string} of Appendix~\ref{app:qcd-lattice}; they are not
re-tabulated here. Program~II then added the decisive refinements that
Program~I-A flagged as ``next rung'': the $20^4$ box-size check at $\beta=6.2$
(which confirmed the expected positive finite-volume bias of the $16^4$ fine-end
point and moved $r_0\sqrt{\sigma}$ from $1.198$ to $1.178$), the autocorrelation
re-analysis of the error budget, the $0^{++}$ glueball estimate, and the executed
$N_f=2$ Wilson-HMC $\kappa$-scan.

\subsection{Status across the program}
The staged lineage illustrates the discipline applied throughout the DFD QCD
sector: scale-free dimensionless ratios are the primary outputs; MeV values are
quoted only with an explicit external scale; finite-volume, single-spacing
calculations are consistency checks rather than physical-point determinations;
and superseded results (the $4^4$ smoke test, the un-error-inflated glueball) are
archived and corrected rather than retained. The consolidated, validated
outcome of this program is the content of Appendix~\ref{app:qcd-lattice}, and its
analytic gauge-sector companion is the infrared-gap theorem of
Appendix~\ref{app:ym-gap}.
