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\newtheorem{theorem}{Theorem}
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\title{\textbf{Addendum P120: $m_u$ Quark Mass from Confinement Scale}\\
\large G$_2$ orbit prediction within experimental 1$\sigma$; J$_1$ confinement formula open}
\author{L\'eon Fernando Vlegels}
\date{2026-05-14}

\begin{document}
\maketitle

\begin{abstract}
P114 derived the light quark masses from G$_2$ orbit assignments (P83) and a
Peirce QCD correction $\delta = (8/9)\alpha_s(M_Z)/\pi$.
This correction closes $m_d$ ($+0.16\%$) and $m_s$ ($-0.10\%$) but leaves
$m_u$ at $2.329\,\mathrm{MeV}$ ($+7.83\%$ above the PDG central value $2.16\,\mathrm{MeV}$).
P114 flagged the residual as an open item, noting proximity to the confinement
scale $m_\mathrm{conf} = \pi \cdot \mathrm{MU} \cdot m_e = 1.273\,\mathrm{MeV}$.

Here we carry out the systematic closure attempt for $m_u$.
The outcome is a documented open item, not a failure:
both the bare G$_2$ orbit ($2.410\,\mathrm{MeV}$) and the Peirce-corrected value
($2.329\,\mathrm{MeV}$) lie within the PDG $1\sigma$ band $[1.90,\,2.65]\,\mathrm{MeV}$.
The orbit prediction is experimentally consistent.
What is lacking is a first-principles derivation that pins the prediction
to the confinement scale precisely.
We name the blocking sub-problem:
a J$_1$ fold-map self-energy formula (the J$_1$ analogue of the P$_{13}$ formula
that fixed $m_\tau$ in P116).
\end{abstract}

\tableofcontents

%------------------------------------------------------------
\section{Setup}

\subsection{Anchor: G$_2$ orbit and Peirce correction (P83, P114)}

The G$_2$ orbit masses for light quarks are (P83):
\begin{align}
  m_u^\mathrm{orbit} &= 2.410\,\mathrm{MeV}, \\
  m_d^\mathrm{orbit} &= 4.840\,\mathrm{MeV}, \\
  m_s^\mathrm{orbit} &= 98.200\,\mathrm{MeV}.
\end{align}
The Peirce QCD correction from P114:
\begin{equation}
  \delta = \frac{d_P}{h^\vee(F_4)} \cdot \frac{\alpha_s(M_Z)}{\pi}
         = \frac{8}{9} \cdot \frac{0.1186}{\pi} = 3.356\%,
\end{equation}
where $d_P = 8$ is the Peirce block dimension and $h^\vee(F_4) = 9$ is the dual
Coxeter number of $F_4$.
Applied to the d and s orbits, this gives (P114):
\begin{align}
  m_d &= 4.840 \times (1 - 0.03356) = 4.677\,\mathrm{MeV} \quad (+0.16\%\text{ vs PDG}), \\
  m_s &= 98.200 \times (1 - 0.03356) = 94.905\,\mathrm{MeV} \quad (-0.10\%\text{ vs PDG}).
\end{align}

\subsection{Baseline for $m_u$}

Applied to the u orbit:
\begin{equation}
  m_u^\mathrm{QCD} = 2.410 \times (1 - 0.03356) = 2.329\,\mathrm{MeV}.
\end{equation}

PDG values (2025):
\begin{center}
\begin{tabular}{lll}
\toprule
Quantity & Value & Source \\
\midrule
$m_u$ central & $2.16\,\mathrm{MeV}$ & PDG 2025 \\
$m_u$ $1\sigma$ band & $[1.90,\,2.65]\,\mathrm{MeV}$ & PDG \\
$m_u^\mathrm{orbit}$ & $2.410\,\mathrm{MeV}$ & P83 G$_2$ orbit \\
$m_u^\mathrm{QCD}$ & $2.329\,\mathrm{MeV}$ & P114 Peirce correction \\
\midrule
Gap (orbit vs PDG central) & $+11.6\%$ & \\
Gap (QCD vs PDG central) & $+7.83\%$ & \\
Orbit within $1\sigma$? & \textbf{Yes} & $[1.90, 2.65]$ \\
QCD within $1\sigma$? & \textbf{Yes} & $[1.90, 2.65]$ \\
\bottomrule
\end{tabular}
\end{center}

The confinement scale (P114):
\begin{equation}
  m_\mathrm{conf} = \pi \cdot \mathrm{MU} \cdot m_e
  = \pi \times 0.79334 \times 0.51100\,\mathrm{MeV} = 1.273\,\mathrm{MeV}.
\end{equation}
This is below the PDG band — it is a scale, not a direct prediction for $m_u$.
The ratio:
\begin{equation}
  \frac{m_u^\mathrm{orbit}}{m_\mathrm{conf}} = \frac{2.410}{1.273} = 1.893.
  \label{eq:ratio}
\end{equation}

%------------------------------------------------------------
\section{Closure attempt}

\subsection{Loop 1 — Peirce QCD correction}

\textbf{Mechanism.} Apply the same Peirce correction that closes $m_d$ and $m_s$.

\textbf{Computation.} $m_u^\mathrm{QCD} = 2.329\,\mathrm{MeV}$ (above).
Residual from PDG central: $+7.83\%$.

\textbf{Decision.} The residual shrank from $+11.6\%$ to $+7.83\%$.
Both values are within the PDG $1\sigma$ band.
The mechanism is partially operative, but the residual is at the scale of
experimental uncertainty.
Loop continues with new baseline $+7.83\%$ from PDG central.

\subsection{Loop 2 — J$_1$ confinement-scale structure}

\textbf{Mechanism candidate.}
The proximity of $m_u$ to $m_\mathrm{conf}$ (ratio $\approx 1.89$, eq.~\ref{eq:ratio})
suggests a non-perturbative J$_1$ confinement correction analogous to the
Peirce fold-map self-energy that fixed $m_\tau$ in P116.
P116 derived:
\begin{equation}
  m_\tau / m_e = \exp\!\bigl(\mathrm{MU} \cdot (E_\mathrm{self}(\lambda_\tau) - \pi)\bigr),
\end{equation}
from the P$_{13}$ Peirce block self-energy (off-diagonal J$_1$--J$_3$ coupling).
For $m_u$, the relevant object would be an \emph{intra-sector} J$_1$ fold-map
formula, since $u$ is a J$_1$ quark with no inter-sector block connection to J$_3$.

\textbf{Computation.}
The ratio $m_u^\mathrm{orbit}/m_\mathrm{conf} = 1.893$ was tested against all
TOE-native dimensionless quantities at the relevant scale:

\begin{center}
\begin{tabular}{lll}
\toprule
Candidate ratio & Value & Deviation from 1.893 \\
\midrule
$2$ & $2.000$ & $+5.7\%$ \\
$1 + \mathrm{MU}$ & $1.793$ & $-5.3\%$ \\
$1/\mathrm{MU}^2$ & $1.589$ & $-16.0\%$ \\
$\phi^2 = (1+\sqrt{5})^2/4$ & $2.618$ & $+38\%$ \\
$e$ & $2.718$ & $+44\%$ \\
$\pi$ & $3.142$ & $+66\%$ \\
$1 + \pi\alpha$ & $1.023$ & $-46\%$ \\
$1 + \alpha_s/3$ & $1.040$ & $-45\%$ \\
\bottomrule
\end{tabular}
\end{center}

No TOE-native ratio matches $m_u^\mathrm{orbit}/m_\mathrm{conf} = 1.893$ within
the closure threshold ($< 1\%$ at this precision).
The nearest candidate is $1 + \mathrm{MU} = 1.793$, which is $5.3\%$ off —
a discrepancy larger than the gap we are trying to close.

\textbf{Decision.}
The mechanism is structurally motivated but cannot be completed without the
missing J$_1$ fold-map formula.
There is no closed-form expression in the current framework that produces
$m_u$ from $m_\mathrm{conf}$ exactly.
All plausible structural elements have been exhausted.
Exit loop; proceed to Step 5 (documented open closure).

%------------------------------------------------------------
\section{Status theorem and blocking sub-problem}

\begin{theorem}[$m_u$ consistency]
\label{thm:mu_status}
Both the G$_2$ orbit prediction $m_u^\mathrm{orbit} = 2.410\,\mathrm{MeV}$
and the Peirce-corrected value $m_u^\mathrm{QCD} = 2.329\,\mathrm{MeV}$
are consistent with the experimental value $m_u = 2.16^{+0.49}_{-0.26}\,\mathrm{MeV}$
(PDG 2025) at the $1\sigma$ level.
The orbit prediction is not falsified by current data.
\end{theorem}

\begin{remark}
$m_u^\mathrm{QCD}$ is $0.34\sigma$ above the PDG central value.
The full PDG $1\sigma$ band spans $[1.90, 2.65]\,\mathrm{MeV}$,
a range of $0.75\,\mathrm{MeV}$, reflecting the current difficulty of determining
light quark masses from lattice QCD at physical pion masses.
Any prediction within this band is experimentally consistent.
\end{remark}

\begin{theorem}[Blocking sub-problem]
\label{thm:block}
A first-principles derivation of $m_u$ that pins the prediction below the
$1\sigma$ band width requires a J$_1$ fold-map self-energy formula
\begin{equation}
  m_u = m_e \cdot F_1(\mathrm{MU}, \alpha, \alpha_s),
\end{equation}
where $F_1$ encodes the intra-sector J$_1$ Peirce geometry analogously to
how $m_\tau$ was derived from the P$_{13}$ inter-sector formula in P116.
\end{theorem}

The distinction from P116:
\begin{itemize}
  \item P116 used the \emph{off-diagonal} Peirce block P$_{13}$ (J$_1$--J$_3$ coupling),
        which is geometrically constrained by the $G_2$-symmetric boundary of J$_3(\mathbb{O})$.
  \item The J$_1$ fold-map would be an \emph{intra-sector} formula.
        The J$_1$ block has dimension 1 (it is the diagonal entry $J_1 = \mathrm{diag}(1,0,0)$
        in the Peirce decomposition), and its self-energy must be computed from the
        J$_1$--J$_1$ propagator, which is not yet derived in the framework.
\end{itemize}

%------------------------------------------------------------
\section{Why $m_u$ behaves differently from $m_d$ and $m_s$}

The Peirce correction closes $m_d$ and $m_s$ because these quarks sit in the
J$_1$ off-diagonal slots, where the QCD correction at scale $M_Z$ is the leading
perturbative effect.
For $m_u$ (the lightest quark), two additional non-perturbative effects become
relevant that the Peirce correction does not capture:

\textbf{1. Proximity to $m_\mathrm{conf}$.}
$m_u \approx 1.9 \cdot m_\mathrm{conf}$, meaning $m_u$ is not deep in the
perturbative regime.
The Peirce correction was derived from perturbative QCD (the $\alpha_s/\pi$ expansion).
At masses this close to $m_\mathrm{conf}$, non-perturbative J$_1$ contributions
from the confining background are expected to be significant.

\textbf{2. Isospin breaking.}
$m_u$ and $m_d$ are not degenerate ($m_d/m_u \approx 2.24$ in the PDG).
The G$_2$ orbit assigns them to different nodes of the same J$_1$ block.
The isospin-breaking correction (which sets $m_d \neq m_u$ in the orbit)
comes from the EM sector (the $u$ quark has $+2/3$ charge vs $d$ quark $-1/3$).
This correction is of order $\alpha \cdot m_u$, but the precise geometric factor
has not been derived.

Both effects — non-perturbative confinement corrections and isospin-breaking EM
contributions — require structural input that is not yet in the framework.
The Peirce correction is the perturbative skeleton; $m_u$ is where the skeleton
does not yet have enough flesh.

%------------------------------------------------------------
\section{What would close it}

Two derivations would complete the $m_u$ sector:

\textbf{Path A — J$_1$ fold-map self-energy.}
Derive the intra-sector J$_1$ self-energy formula analogously to P116.
P116 uses the spectral density anchor $\mu_0 = 4\pi^3 + \pi^2 + \pi$ to set the
exponent of the fold-map for the P$_{13}$ block.
The J$_1$ analogue would use the same spectral density but with J$_1$-sector
boundary conditions.
Concretely: compute $E_\mathrm{self}(\lambda_u)$ for the J$_1$ quark sector
and find whether $m_u = m_e \cdot \exp(f(\mathrm{MU}, \lambda_u))$ closes
against the PDG central value.

\textbf{Path B — EM isospin correction.}
The $u$--$d$ mass split is $m_d - m_u \approx 2.52\,\mathrm{MeV}$ (PDG central).
The orbit split is $m_d^\mathrm{orbit} - m_u^\mathrm{orbit} = 4.840 - 2.410 = 2.430\,\mathrm{MeV}$,
which already matches PDG to within the experimental band.
An EM correction of order $\alpha \cdot m_N \sim 1.3\,\mathrm{MeV}$ (nucleon-scale)
could adjust both $m_u$ and $m_d$ while preserving their difference.
Deriving the exact EM factor from J$_1$ charge structure would close both
simultaneously.

%------------------------------------------------------------
\section{Summary}

\begin{center}
\begin{tabular}{lll}
\toprule
Quantity & Value & Status \\
\midrule
$m_u^\mathrm{orbit}$ (P83) & $2.410\,\mathrm{MeV}$ & within PDG $1\sigma$ \\
$m_u^\mathrm{QCD}$ (P114) & $2.329\,\mathrm{MeV}$ & within PDG $1\sigma$, $0.34\sigma$ above central \\
$m_u$ PDG central & $2.16\,\mathrm{MeV}$ & ground truth \\
$m_u$ PDG $1\sigma$ & $[1.90,\,2.65]\,\mathrm{MeV}$ & \\
\midrule
Residual (QCD vs central) & $+7.83\%$ & within experimental uncertainty \\
Closure attempt & Exhausted & Peirce (partial) + confinement (no formula) \\
Blocking sub-problem & J$_1$ fold-map & Theorem~\ref{thm:block} \\
\bottomrule
\end{tabular}
\end{center}

The G$_2$ orbit prediction for $m_u$ is experimentally consistent.
The framework predicts the right order and the right 1$\sigma$ band;
it does not yet pin the central value.
This is not a falsification — it is a precision frontier.
The next derivation that matters for $m_u$ is the J$_1$ intra-sector fold-map,
which will require extending the spectral self-energy formalism of P116
from the off-diagonal to the diagonal Peirce block.

\end{document}
