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\title{\textbf{The Single-Anchor Scale and Transport Discipline\\for Particle Mass Phenomenology}\\[0.5em]\large Paper IV of VI: Anchor Derivation, Non-Circularity,\\and the Structural--Transport Separation}
\author{Jonathan Washburn\\\small Recognition Science Research Institute, Austin, Texas\\\small \texttt{washburn.jonathan@gmail.com}}
\date{\today}

\begin{document}
\maketitle

\begin{abstract}
Any comparison between a structural mass framework and experiment requires careful treatment of scale, scheme, and transport. This paper establishes the single-anchor discipline for the RS mass series. We derive $\muStar=182.201\,\mathrm{GeV}$ from mass-free PMS/BLM stationarity, prove non-circularity (no measured mass enters its own prediction), and develop the clean separation between two fundamentally distinct exponents: $\fRec(Z)=\mathrm{gap}(Z)$ (large, $\sim 6$--$14$, band-defining) and $\fRG$ (small, $\sim 10^{-2}$--$10^{-1}$ for leptons, bookkeeping-only). A Lean-proved no-go theorem shows $\fRG$ cannot equal $\mathrm{gap}(Z)$. The full RG transport policy (4-loop QCD, 2-loop QED) is pinned and certified. Equal-$Z$ clustering survives all tested scheme/loop/threshold variations.
\end{abstract}

\tableofcontents
\newpage

\section{Anchor Scale Derivation}
\subsection{Stationarity principle}
$\muStar$ satisfies $d\gamma_m/d\ln\mu|_{\mu=\muStar}=0$ (PMS criterion). \CERT{}

\subsection{Mass-free derivation}
The SM beta function $\beta_0(n_f)=11-2n_f/3$ depends only on group theory, not fermion masses. The stationarity optimization yields: \CERT{}
\begin{equation}\muStar=182.201\,\mathrm{GeV}.\end{equation}

\subsection{Non-circularity certificate}
\begin{theorem}[Anchor non-circularity]
There exists a certified $\muStar$ such that: (1) stationarity holds, (2) $\beta_0$ contains no mass parameters, (3) no measured fermion mass enters $\muStar$, (4) no adjustable parameters enter. \PROVED{}
\end{theorem}
Lean module: \texttt{IndisputableMonolith.Verification.AnchorNonCircularityCert} (zero \texttt{sorry}).

\section{Two Distinct Exponents}
\subsection{Structural residue $\fRec(Z)$}
$\fRec(Z):=\mathrm{gap}(Z)=\log_\phig(1+Z/\phig)$. Values: $\mathrm{gap}(24)\approx 5.74$, $\mathrm{gap}(276)\approx 10.69$, $\mathrm{gap}(1332)\approx 13.95$. Large, family-defining, integer-organized. \PROVED{}

\subsection{Transport exponent $\fRG$}
$\fRG_i(\muStar,\mu_{\mathrm{target}}):=\log_\phig[m_i(\mu_{\mathrm{target}})/m_i(\muStar)]$. For leptons $O(10^{-2})$; for quarks $O(1)$; always $\ll\fRec(Z)$. \CERT{}

\subsection{No-conflation theorem}
\begin{theorem}
For $Z\in\{24,276,1332\}$: $|\fRG_i-\fRec(Z_i)|>10$ for $Z=1332$.
\end{theorem}
Lean: \texttt{IndisputableMonolith.Physics.MassResidueNoGo}. The structural band coordinate is not the SM running correction. \PROVED{}

\section{RG Transport Policy}
\begin{center}\begin{tabular}{ll}\toprule Parameter & Value \\\midrule QCD anomalous dim. & 4-loop \\ QED anomalous dim. & 2-loop \\ $\alpha_s$ running & 4-loop \\ $\alpha_{\mathrm{EM}}$ & frozen at $M_Z$ \\ Thresholds & $\overline{\mathrm{MS}}$ at $m_c,m_b,m_t$ \\ Integrator & RK4, $10^4$ steps/decade \\\bottomrule\end{tabular}\end{center}
Certified in \texttt{data/certificates/rg\_transport/canonical\_2025\_q4.json}. \CERT{}

\section{Robustness}
Equal-$Z$ clustering (tolerance $\sim 5\times 10^{-6}$) survives: 3-loop/5-loop QCD; 1-loop/2-loop QED; pole vs.\ $\overline{\mathrm{MS}}$ thresholds; running vs.\ frozen $\alpha_{\mathrm{EM}}$; anchor recalibration under each variant. Ablations destroying the $Z$-map break clustering by orders of magnitude. \VAL{}

\section{Calibration Seam}
All RS outputs are dimensionless $\phig$-ladder quantities. SI reporting requires one global scalar $\tau_0$ (seconds per tick). This seam is: fixed once (not per species); version-controlled; not a fit parameter.

\section{Reviewer Checklist}
Any numerical objection must specify: (1) the exact equation; (2) target scheme; (3) target scale; (4) loop/threshold policy; (5) whether any $m_i$ entered its own RHS; (6) computation steps and outputs.

\section{Conclusions}
$\muStar$ is mass-free and non-circular. The structural--transport separation is Lean-proved. The transport policy is pinned and auditable. The calibration seam is global, not per-species.

\begin{thebibliography}{99}
\bibitem{PDG2024} R.~L.~Workman \textit{et al.} [PDG], PTEP \textbf{2022}, 083C01.
\bibitem{PMS} P.~M.~Stevenson, Phys.\ Rev.\ D~\textbf{23}, 2916 (1981).
\bibitem{BLM} S.~J.~Brodsky et al., Phys.\ Rev.\ D~\textbf{28}, 228 (1983).
\bibitem{Washburn2025} J.~Washburn, \textit{Axioms} \textbf{15}(2), 90 (2025).
\end{thebibliography}
\end{document}
