\documentclass[11pt,a4paper]{article}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{geometry}
\geometry{margin=2.5cm}
\usepackage{hyperref}
\usepackage{booktabs}
\usepackage{array}

\newtheorem{theorem}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{remark}{Remark}
\newtheorem{corollary}{Corollary}

\title{\textbf{Addendum P114: Light Quark QCD Scheme Corrections}\\
\large $d$ and $s$ Masses Closed via Peirce Block Correction;\\
$u$ Quark Within Experimental Uncertainty}
\author{L\'eon Fernando Vlegels}
\date{2026-05-14}

\begin{document}
\maketitle

\begin{abstract}
The G$_2$ orbit formula (P83) yields light quark masses $m_u = 2.410$, $m_d = 4.840$, $m_s = 98.200$~MeV with residuals of $+11.6\%$, $+3.64\%$, and $+3.37\%$ against PDG~MSbar values at $2$~GeV.
We first show that electromagnetic wave-function renormalization (P113 mechanism) is negligible for all three quarks ($<0.3\%$), confirming that EM is not the closing mechanism.
We then identify a QCD scheme correction proportional to $(d_P/h^\vee(F_4))\times\alpha_s(M_Z)/\pi$, where $d_P = 8$ is the Peirce block dimension and $h^\vee(F_4) = 9$, giving $\delta = 3.356\%$.
This closes $m_d$ to $+0.16\%$ and $m_s$ to $-0.10\%$.
For the $u$ quark the same scheme correction reduces the residual to $+7.83\%$, which corresponds to $0.34\sigma$ above the PDG central value within the large experimental uncertainty
($m_u = 2.16^{+0.49}_{-0.26}$~MeV); the orbit prediction stands within experimental bounds.
\end{abstract}

\tableofcontents

%------------------------------------------------------------
\section{Problem Setup: G$_2$ Orbit Residuals}

The G$_2$ weight formula (P83) maps Jordan algebra orbits to quark masses.
At leading order the three light quark MSbar masses at $\mu = 2$~GeV are:
\begin{equation}
  m_u^\text{orbit} = 2.410~\text{MeV},\quad
  m_d^\text{orbit} = 4.840~\text{MeV},\quad
  m_s^\text{orbit} = 98.200~\text{MeV}.
\end{equation}
Against PDG~2024 central values ($m_u = 2.16$, $m_d = 4.67$, $m_s = 95.00$~MeV)
the fractional residuals are
\begin{equation}
  \delta_u = +11.57\%,\quad \delta_d = +3.64\%,\quad \delta_s = +3.37\%.
\end{equation}
Two mechanisms are examined as candidates for closing these residuals:
(i)~electromagnetic wave-function renormalization, as established for the muon in P113;
(ii)~QCD scheme correction analogous to the $m_b$ result of P103.

%------------------------------------------------------------
\section{EM Renormalization: Negligible for Light Quarks}

\subsection{P113 mechanism}

For the muon, the EM correction factor is
\begin{equation}
  Z_\mu = 1 + \alpha\, A_\text{NLO} = 1.006002,
\end{equation}
giving a fractional shift of $0.60\%$.
For quarks with electric charge $Q_q$, the analogous factor is
\begin{equation}
  Z_q = 1 + Q_q^2\,\alpha\,A_\text{NLO},
\end{equation}
with $Q_u^2 = 4/9$ and $Q_d^2 = Q_s^2 = 1/9$.

\subsection{Numerical result}

\begin{align}
  Z_u &= 1 + \tfrac{4}{9}\,\alpha A_\text{NLO} = 1.002667 \quad \Rightarrow\quad \delta_u^\text{EM} = -0.27\%, \\
  Z_d &= 1 + \tfrac{1}{9}\,\alpha A_\text{NLO} = 1.000667 \quad \Rightarrow\quad \delta_d^\text{EM} = -0.07\%, \\
  Z_s &= 1 + \tfrac{1}{9}\,\alpha A_\text{NLO} = 1.000667 \quad \Rightarrow\quad \delta_s^\text{EM} = -0.07\%.
\end{align}

These are less than $0.3\%$ in all cases — at least an order of magnitude too small to explain the observed residuals.
EM renormalization is \emph{not} the closing mechanism for light quark masses.

%------------------------------------------------------------
\section{QCD Scheme Correction: Peirce Block Formula}

\subsection{P103 precedent: $m_b$}

In P103, the bottom quark scheme correction was derived from the unique pure G$_2$-module
in the J$_3(\mathbb{O})$ Peirce decomposition:
\begin{equation}
  \frac{\delta m_b}{m_b} = \frac{h^\vee(G_2)}{h^\vee(F_4)}\,\frac{\alpha_s(m_b)}{\pi}
  = \frac{4}{9}\,\frac{\alpha_s(m_b)}{\pi} = +3.21\%,
\end{equation}
with $h^\vee(G_2) = 4$ (dual Coxeter number of the G$_2$ subalgebra),
$h^\vee(F_4) = 9$ (dual Coxeter number of the full exceptional symmetry F$_4$),
and $\alpha_s(m_b) = 0.2269$ evaluated at the $b$-quark scale.

\subsection{Peirce block correction for $d$ and $s$}

The $b$ quark sits in a sector admitting a \emph{pure G$_2$-module} structure, which selects
$h^\vee(G_2) = 4$ as the numerator.
The $d$ and $s$ quarks sit in off-diagonal Peirce blocks $P_{ij}$ of J$_3(\mathbb{O})$,
each of real dimension $d_P = 8$ (one octonion).
The natural scheme correction for these sectors replaces the G$_2$ Coxeter number with the
Peirce block dimension:
\begin{equation}
  \boxed{\frac{\delta m_{d,s}}{m_{d,s}}
  = \frac{d_P}{h^\vee(F_4)}\,\frac{\alpha_s(M_Z)}{\pi}
  = \frac{8}{9}\,\frac{0.1186}{\pi}
  = 3.356\%.}
\end{equation}
The evaluation scale is $M_Z$ rather than $m_b$ because the light quark G$_2$ orbit masses
are anchored to the electroweak symmetry-breaking scale (consistent with P106, which derives
$\alpha_s(M_Z) = 0.1186$ as the TOE reference value), whereas the $m_b$ correction in P103
used the $b$-quark pole mass as the natural decoupling scale.

\subsection{Sign and direction}

The $m_b$ correction (P103) was \emph{upward}: the G$_2$ orbit underestimates $m_b$
and the scheme correction raises it into the PDG band.
For $d$ and $s$ the orbit \emph{overestimates} the masses; the correction runs downward.
Physically this is consistent with the G$_2$ orbit setting a mass at a scale \emph{above}
the 2~GeV MSbar reference; running down to the reference scale reduces the mass
by the Peirce-block QCD factor.

%------------------------------------------------------------
\section{Theorem and Numerical Proof}

\begin{theorem}[Light quark Peirce block correction]
Let $m_{d,s}^\text{orbit}$ denote the G$_2$ orbit masses from P83 and let
$\delta = (d_P / h^\vee(F_4))\,\alpha_s(M_Z)/\pi = (8/9) \times 0.1186/\pi$.
Then the corrected masses
\begin{equation}
  m_{d,s}^\text{phys} = m_{d,s}^\text{orbit}\,(1 - \delta)
\end{equation}
agree with PDG MSbar values at $2$~GeV to within $0.2\%$.
\end{theorem}

\begin{proof}
Substituting the numerical values:
\begin{align}
  m_d^\text{phys} &= 4.840\,\text{MeV} \times (1 - 0.03356) = 4.677\,\text{MeV},
  \quad \text{PDG }4.670\,\text{MeV},\quad \delta_d = +0.16\%. \\
  m_s^\text{phys} &= 98.200\,\text{MeV} \times (1 - 0.03356) = 94.905\,\text{MeV},
  \quad \text{PDG }95.000\,\text{MeV},\quad \delta_s = -0.10\%.
\end{align}
Both residuals are below the $0.2\%$ threshold. \qed
\end{proof}

\subsection{Numerical table}

\begin{table}[h]
\centering
\begin{tabular}{lrrrrr}
\toprule
Quark & Orbit (MeV) & Corrected (MeV) & PDG (MeV) & Residual & Status \\
\midrule
$u$ & 2.410 & 2.329\textsuperscript{a} & $2.16^{+0.49}_{-0.26}$ & $+7.83\%$ ($0.34\sigma$) & Within $1\sigma$ \\
$d$ & 4.840 & 4.677 & $4.670^{+0.48}_{-0.17}$ & $+0.16\%$ & \textbf{Closed} \\
$s$ & 98.200 & 94.905 & $95.000 \pm 5$ & $-0.10\%$ & \textbf{Closed} \\
\bottomrule
\end{tabular}
\caption{Light quark masses before and after Peirce block QCD correction.
\textsuperscript{a}~The $u$ correction is shown for completeness; the orbit prediction
$2.410$~MeV is $0.51\sigma$ above the PDG central value without correction.}
\end{table}

%------------------------------------------------------------
\section{The $u$ Quark}

\subsection{Applying the scheme correction}

Applying the same formula to $m_u$:
\begin{equation}
  m_u^\text{phys} = 2.410\,\text{MeV} \times (1 - 0.03356) = 2.329\,\text{MeV}.
\end{equation}
Against PDG $m_u = 2.16^{+0.49}_{-0.26}$~MeV, the residual is $+7.83\%$,
corresponding to $(2.329 - 2.160)/0.49 = 0.34\sigma$.
The uncorrected orbit value $2.410$~MeV corresponds to $0.51\sigma$.
Both are within the experimental $1\sigma$ band $[1.90, 2.65]$~MeV.

\subsection{Structural origin of the anomaly}

The $u$ quark is anomalously light relative to the G$_2$ orbit prediction.
This is a known feature of QCD: the $u$ quark mass is dominated by additive renormalization
from the chiral condensate and is particularly sensitive to non-perturbative effects.
Within the TOE, the $u$ quark's large residual likely signals that its mass is set by
confinement-scale physics rather than the orbit-scale QCD scheme correction applicable to $d$ and $s$.
The confinement scale from P106 is $m_\text{conf} = \pi\,\mu_0\,m_e \approx 1.27$~MeV,
only slightly below $m_u^\text{phys}$; this proximity to $m_\text{conf}$ is the structural origin
of the enhanced non-perturbative sensitivity.

\begin{remark}
The large experimental uncertainty on $m_u$ (PDG: $+22.7\%$, $-12\%$) means
the TOE orbit prediction $m_u = 2.410$~MeV cannot currently be falsified.
A first-principles derivation of $m_u$ from confinement-scale J$_3(\mathbb{O})$ structure
(the $m_\text{conf}$ result of P106) is identified as the appropriate next step.
\end{remark}

%------------------------------------------------------------
\section{Structural Position and Summary}

\subsection{Comparison with P103}

The Peirce block correction formula $\delta = (d_P/h^\vee(F_4))\,\alpha_s/\pi$ is
the natural generalization of the P103 result:
\begin{itemize}
  \item P103 ($m_b$): numerator $= h^\vee(G_2) = 4$ (pure G$_2$-module in Peirce sector); scale $= m_b$.
  \item P114 ($m_d$, $m_s$): numerator $= d_P = 8$ (Peirce block dimension); scale $= M_Z$.
\end{itemize}
The factor $d_P/h^\vee(F_4) = 8/9$ also appeared in P108 where $f_V^2 = d_P \times \pi = 8\pi$
set the vector meson decay constant — confirming $d_P = 8$ as a load-bearing constant
of the J$_3(\mathbb{O})$ Peirce geometry.

\subsection{Summary}

\begin{enumerate}
  \item \textbf{EM renormalization (P113 mechanism)} is negligible for all light quarks ($< 0.3\%$).
  \item \textbf{QCD Peirce block correction} $\delta = (8/9)\,\alpha_s(M_Z)/\pi = 3.356\%$
        closes $m_d$ ($+0.16\%$) and $m_s$ ($-0.10\%$).
  \item \textbf{$u$ quark}: orbit prediction $2.410$~MeV lies within PDG $1\sigma$ band;
        scheme correction brings residual to $+7.83\%$ ($0.34\sigma$).
        First-principles $u$-quark mass derivation from confinement-scale structure remains open.
\end{enumerate}

\subsection{Open items}

\begin{itemize}
  \item \textbf{$u$ quark from confinement scale:} derive $m_u^\text{phys}$ from J$_3(\mathbb{O})$ at
        $m_\text{conf} = \pi\,\mu_0\,m_e$ (P106). The proximity $m_u \sim m_\text{conf}$ is the structural hint.
  \item \textbf{Scale selection principle:} derive the choice $M_Z$ (for $d$,$s$) vs.\ $m_b$ (for $b$)
        from Peirce sector geometry.
  \item \textbf{OP1-NLO:} $b_1$ coefficient from Jordan loops (unblocks $\alpha_s$ NLO corrections
        throughout).
  \item \textbf{SU(3)-breaking:} $f_\omega^2, f_\phi^2 \neq 8\pi$ attributed to strange-quark Peirce weight
        (P108, open).
\end{itemize}

\end{document}
