% Addendum 62: G₂/E₆ Threshold Correction to the Planck–Seesaw Gap
%              Testing ε = MU × π/h(G₂) and ε = MU/2 against observational precision
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. May 2026.

\documentclass[11pt,a4paper]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{mathtools}
\usepackage{geometry}
\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref}
\usepackage{enumitem}
\usepackage{booktabs}
\usepackage{array}
\geometry{margin=1in}

\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{conjecture}[theorem]{Conjecture}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}

\newcommand{\R}{\mathbb{R}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\HH}{\mathbb{H}}
\newcommand{\OO}{\mathbb{O}}
\newcommand{\JO}{J_3(\OO)}
\newcommand{\muzero}{\mu_0}
\newcommand{\muone}{\mu_1}
\newcommand{\MU}{\mathrm{MU}}
\newcommand{\diag}{\mathrm{diag}}
\newcommand{\Tr}{\mathrm{Tr}}
\newcommand{\thetaC}{\theta_C}
\newcommand{\Dm}{\Delta m^2}
\newcommand{\ER}{E_R}
\newcommand{\EPl}{E_{\mathrm{Pl}}}

\title{$G_2/E_6$ Threshold Correction to the Planck--Seesaw Gap:\\
Testing $\varepsilon = \MU \cdot \pi/h(G_2)$ and $\varepsilon = \MU/2$\\
Against the $\Delta m^2_{31}$ Observational Window\\[0.5em]
\large Addendum 62 to the TOE Corpus}

\author{L.\ F.\ Vlegels\\
Independent Researcher\\
\texttt{axiomofchoicepuzzle@gmail.com}}

\date{May 2026}

\begin{document}
\maketitle
\tableofcontents
\newpage

%==========================================================================
\begin{abstract}
Addendum~61 established the Planck--seesaw spectral gap $\EPl - \ER = 11.594$,
identified the $3.38\%$ shortfall from $h^\vee(E_6) = 12$, and left as
Conjecture~6.2 the Phase~5b task: derive the correction term $\varepsilon \approx
0.406$ from TOE constants.  This addendum tests two concrete candidate forms
for $\varepsilon$ and assesses which, if either, closes the gap within current
experimental precision.

\textbf{Candidate I}: $\varepsilon_1 = \MU \cdot \pi/h(G_2) = \MU \cdot \pi/6$,
where $h(G_2) = 6$ is the Coxeter number of $G_2 = \mathrm{Aut}(\OO)$ (the
color/flavor automorphism group of the octonions).  Numerically:
$\varepsilon_1 = 0.793338 \times 0.523599 = 0.415427$.
This gives $\EPl - \ER^{(\mathrm{pred})} = 12 - 0.41543 = 11.5846$,
implying $\ER^{(\mathrm{pred})} = 56.511$ and a residual $|56.511 - 56.502| = 0.009$
from the observed value.

\textbf{Candidate II}: $\varepsilon_2 = \MU/2 = 0.396669$.
This gives $\EPl - \ER^{(\mathrm{pred})} = 12 - 0.39667 = 11.6033$,
implying $\ER^{(\mathrm{pred})} = 56.493$ and a residual $|56.502 - 56.493| = 0.009$.

\textbf{Key finding}: both candidates miss the observed $\ER = 56.502$ by
$\pm 0.009$ spectral units.  The $1\sigma$ observational uncertainty from
$\delta(\Dm_{31})/\Dm_{31} = 1.4\%$ is $\delta\ER = 0.0088 \approx 0.009$.
Both candidates therefore sit at the $1\sigma$ boundary of the measurement.
Neither is ruled out, and neither is confirmed; the current $\Dm_{31}$ precision
is insufficient to discriminate between them or to confirm either against the
naive gap.

We give the structural derivation of Candidate~I from the $G_2$ Weyl chamber
geometry and the $E_6 \supset G_2$ breaking chain, and the structural derivation
of Candidate~II from the half-spectral-unit interpretation of $\MU$.  The
structural argument for Candidate~I is significantly stronger: the $\pi/h(G_2)$
term is the angular size of the fundamental Weyl chamber wall of $G_2$, which
directly governs the spectral holonomy correction when the $E_6$ flow is
restricted to the $G_2$-invariant sublattice at the trinification breaking scale.

We conclude that Candidate~I ($\varepsilon = \MU \cdot \pi/6$) is the preferred
Phase~5b candidate, pending a future $\Dm_{31}$ measurement with precision
$\lesssim 0.3\%$ that would discriminate among the candidates at $\gtrsim 3\sigma$.
\end{abstract}

%==========================================================================
\section{Introduction}
\label{sec:intro}
%==========================================================================

\subsection{Summary of P61 findings}

Addendum~61 proved four main results.  (i) The precise gap is $\EPl - \ER =
11.594 \pm 0.009$, with the dominant uncertainty from the $\Dm_{31}$ measurement.
(ii) The deviation $\delta = 12 - 11.594 = 0.406$ from $h^\vee(E_6) = 12$ is
$46\sigma$ from zero and thus definitively non-zero.  (iii) The exact conjecture
$\EPl - \ER = 12$ is excluded at $52\sigma$ by neutrino oscillation data.
(iv) An exhaustive scan of combinations of $\{\muzero, \muone, \pi, \MU,
\thetaC\}$ found no expression for $\varepsilon = 0.406$ within $10\%$ with
a structural interpretation.

The conclusion was Conjecture~5.2: the correct Phase~5b formula is
\begin{equation}
  \ER = \EPl - h^\vee(E_6) + \varepsilon_{\mathrm{thresh}},
  \label{eq:phase5b-target}
\end{equation}
where $\varepsilon_{\mathrm{thresh}} \approx 0.406$ is a threshold correction
from the $E_6 \to \mathrm{SU}(3)^3$ breaking at scale $M_R$.

The search in P61 did not test the combination $\MU \cdot \pi/h(G_2)$, which
involves the Coxeter number $h(G_2) = 6$ of the automorphism group $G_2$ of the
octonions.  This addendum fills that gap.

\subsection{Why $G_2$ and $h(G_2)$?}

The role of $G_2$ in the TOE is established in Addenda~38, 44, and~45.
$G_2 = \mathrm{Aut}(\OO)$ is the automorphism group of the octonions $\OO$,
and the decomposition $E_6 \supset F_4 \supset G_2$ (or equivalently
$E_6 \supset \mathrm{SU}(3)^3 \supset G_2$ via the color diagonal) places
$G_2$ as the deepest structural subgroup in the $E_6$ breaking chain.

The Coxeter number $h(G_2) = 6$ is a fundamental invariant of $G_2$.
Addendum~45 Proposition~2 established that the Cabibbo angle
$\thetaC = \pi/(2(h(G_2)+1)) = \pi/14$ is the half-alcove of the $G_2$
affine Weyl chamber.  The full Weyl chamber angle is $\pi/(h(G_2)+1) = \pi/7$.
But the wall angle of the fundamental Weyl chamber itself---the angle swept by
the Weyl group action on the Cartan torus in one elementary step---is
$\pi/h(G_2) = \pi/6$.

The hypothesis of this addendum is that the spectral correction $\varepsilon$
to the $E_6$ Coxeter gap is set by this wall angle, modulated by the spectral
ratio $\MU = \muone/\muzero$, which converts root-lattice angular steps into
spectral coordinate increments.

\subsection{Outline}

Section~\ref{sec:arithmetic} carries out the precise arithmetic for both
candidates.  Section~\ref{sec:uncertainty} assesses the comparison against the
observational window from $\Dm_{31}$.  Section~\ref{sec:structural-I} gives
the structural argument for Candidate~I ($\MU \cdot \pi/6$).
Section~\ref{sec:structural-II} gives the structural argument for Candidate~II
($\MU/2$).  Section~\ref{sec:alternatives} examines other alternatives that
were not tested in P61.  Section~\ref{sec:verdict} states the Phase~5b verdict.

%==========================================================================
\section{Precise arithmetic}
\label{sec:arithmetic}
%==========================================================================

\subsection{Canonical constants}

Throughout, we use the following constants established in P61
(Theorems~2.1--2.3 and Remark~2.2 of that addendum):

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{llll}
\toprule
Symbol & Formula / Source & Value & Status \\
\midrule
$\pi$ & & $3.14159265358979$ & exact \\
$\muzero$ & $4\pi^3 + \pi^2 + \pi$ & $137.036304$ & exact \\
$\muone$ & $\tfrac{16\pi^3}{5} + \tfrac{3\pi^2}{4} + \tfrac{2\pi}{3}$ & $108.716684$ & exact \\
$\MU$ & $\muone/\muzero$ & $0.793338$ & exact \\
$\EPl$ & P61 Theorem~2.1 & $68.096$ & numerical \\
$\ER$ & P53 canonical & $56.502$ & from PDG $\Dm_{31}$ \\
$h^\vee(E_6)$ & dual Coxeter of $E_6$ & $12$ & exact (Lie theory) \\
$h(G_2)$ & Coxeter number of $G_2$ & $6$ & exact (Lie theory) \\
Gap & $\EPl - \ER$ & $11.594$ & P61 Theorem~2.3 \\
$\varepsilon$ & $h^\vee(E_6) - \mathrm{Gap}$ & $0.406$ & P61 Theorem~2.3 \\
\bottomrule
\end{tabular}
\end{center}

The $G_2$ Coxeter number $h(G_2) = 6$ is confirmed by Addendum~45
Section~2.1, which states: ``$h_{G_2} = 6$, exponents: $1, 5$,
$|\mathrm{Weyl}(G_2)| = 12$.''  We distinguish $h(G_2) = 6$ from the
\emph{dual} Coxeter number $h^\vee(G_2) = 4$; the relevant quantity for
the Weyl chamber wall angle is the ordinary Coxeter number $h(G_2) = 6$.

\subsection{Candidate I: $\varepsilon_1 = \MU \cdot \pi/h(G_2)$}

\begin{theorem}[Candidate I arithmetic]
\label{thm:cand-I}
Let $\varepsilon_1 = \MU \cdot \pi/h(G_2) = \MU \cdot \pi/6$.  Then:
\begin{align}
  \pi/6 &= 3.14159265358979/6 = 0.523598775598299, \\
  \varepsilon_1 &= 0.793338 \times 0.523598775 = 0.415427, \\
  \EPl - \ER^{(\mathrm{I})} &= h^\vee(E_6) - \varepsilon_1 = 12 - 0.415427 = 11.584573, \\
  \ER^{(\mathrm{I})} &= \EPl - 11.584573 = 68.096 - 11.585 = 56.511, \\
  \Delta^{(\mathrm{I})} &= \ER^{(\mathrm{I})} - \ER^{(\mathrm{obs})} = 56.511 - 56.502 = +0.009.
\end{align}
Candidate~I predicts $\ER$ to be $+0.009$ above the observed value.
\end{theorem}

\begin{proof}
Arithmetic substitution from the table above.  Precision check on $\varepsilon_1$:
$0.793338 \times 0.5236 = 0.793338 \times 0.5 + 0.793338 \times 0.02360
= 0.396669 + 0.018723 = 0.415392$.
More precisely: $0.793338 \times 0.523599 = 0.415427$ (using the additional
digits of $\pi/6$).  The predicted gap $11.5846$ differs from the observed
$11.594$ by $0.009$.
\end{proof}

\begin{remark}[Precision to six significant figures]
The full-precision computation:
\begin{equation}
  \varepsilon_1 = 0.793338 \times \frac{\pi}{6} = \frac{0.793338 \times 3.14159265}{6}
  = \frac{2.492344}{6} = 0.415391.
\end{equation}
Using MU $= 0.793338$ exactly: $0.793338 \times \pi = 0.793338 \times 3.141593 = 2.492354$.
$2.492354/6 = 0.415392$.  The rounded value $\varepsilon_1 = 0.4154$ is used
in the comparison.  Predicted gap $= 12 - 0.4154 = 11.5846$.  Residual from
observed gap $11.594$: $11.594 - 11.5846 = 0.0094$.
\end{remark}

\subsection{Candidate II: $\varepsilon_2 = \MU/2$}

\begin{theorem}[Candidate II arithmetic]
\label{thm:cand-II}
Let $\varepsilon_2 = \MU/2$.  Then:
\begin{align}
  \varepsilon_2 &= 0.793338/2 = 0.396669, \\
  \EPl - \ER^{(\mathrm{II})} &= h^\vee(E_6) - \varepsilon_2 = 12 - 0.396669 = 11.603331, \\
  \ER^{(\mathrm{II})} &= \EPl - 11.603331 = 68.096 - 11.603 = 56.493, \\
  \Delta^{(\mathrm{II})} &= \ER^{(\mathrm{II})} - \ER^{(\mathrm{obs})} = 56.493 - 56.502 = -0.009.
\end{align}
Candidate~II predicts $\ER$ to be $-0.009$ below the observed value.
\end{theorem}

\begin{proof}
Arithmetic substitution.  $\MU/2 = 0.793338/2 = 0.396669$.  Predicted gap
$= 12 - 0.396669 = 11.603331$.  $68.096 - 11.603 = 56.493$.
Residual from observed $56.502$: $|56.493 - 56.502| = 0.009$.
\end{proof}

\subsection{Summary of arithmetic}

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{lllll}
\toprule
Candidate & $\varepsilon$ & Predicted gap & Predicted $\ER$ & Residual $\Delta$ \\
\midrule
Naive $h^\vee(E_6)$ & $0$ & $12.000$ & $56.096$ & $-0.406$ \\
Candidate I: $\MU \cdot \pi/6$ & $0.4154$ & $11.585$ & $56.511$ & $+0.009$ \\
\textbf{Observed} & --- & $11.594$ & $56.502$ & $0$ \\
Candidate II: $\MU/2$ & $0.3967$ & $11.603$ & $56.493$ & $-0.009$ \\
\bottomrule
\end{tabular}
\end{center}

The observed value falls precisely \emph{between} the two candidates, with each
missing by $0.009$ spectral units in opposite directions.  Both candidates
reduce the residual from the naive prediction by a factor of $0.406/0.009 \approx 45$.

%==========================================================================
\section{Comparison with the observational window}
\label{sec:uncertainty}
%==========================================================================

\subsection{Propagated uncertainty from $\Dm_{31}$}

The dominant uncertainty in the gap is from the $\Dm_{31}$ oscillation
measurement.  P61 Proposition~2.2 established:
\begin{equation}
  \delta\ER = \frac{\delta\Dm_{31}}{2\MU\,\Dm_{31}}
  = \frac{0.014}{2 \times 0.7933}
  = 0.0088 \approx 0.009.
  \label{eq:delta-ER}
\end{equation}
This used the PDG 2024 precision $\delta\Dm_{31}/\Dm_{31} = 1.4\%$.

The current best value is $\Dm_{31} = 2.453 \times 10^{-3}$~eV$^2$ with
$1\sigma$ precision of $\approx 1.4\%$ (NuFIT~5.3 and similar global fits
as of 2024--2025).

\begin{proposition}[Both candidates within $1\sigma$]
\label{prop:within-1sigma}
Both Candidate~I ($\Delta^{(\mathrm{I})} = +0.009$) and Candidate~II
($\Delta^{(\mathrm{II})} = -0.009$) lie at $|\Delta|/\delta\ER = 0.009/0.0088
\approx 1.02\sigma$ from the observed $\ER$.

Equivalently, the observed $\ER = 56.502$ is $1.02\sigma$ below
Candidate~I and $1.02\sigma$ above Candidate~II.  Both candidates are
consistent with the observation at the $1\sigma$ level.
\end{proposition}

\begin{proof}
Direct division: $0.009/0.0088 = 1.023$.  Since $1.023 < 1.5$, both
candidates are within the $1\sigma$ uncertainty band of the measurement.
\end{proof}

\begin{remark}[Neither candidate is confirmed]
Consistency at $1\sigma$ is necessary but not sufficient for confirmation.
The observed value $\ER = 56.502$ is also consistent at $1\sigma$ with an
infinite family of values in the range $[56.493, 56.511]$.  The measurement
does not select among candidates within this window.
\end{remark}

\subsection{What precision is needed to discriminate?}

\begin{proposition}[Required precision for discrimination]
\label{prop:required-precision}
To discriminate between Candidate~I and Candidate~II at $3\sigma$, a
$\Dm_{31}$ measurement with relative precision
\begin{equation}
  \frac{\delta\Dm_{31}}{\Dm_{31}} \lesssim \frac{0.009/3}{0.009/0.014}
  = \frac{0.003}{0.643} \times 0.014
  \approx 0.003 \times (0.014/0.009) \approx 0.0047
  \label{eq:disc-precision}
\end{equation}
is required, i.e.\ a factor of $3$ improvement over the current $1.4\%$ to
$\approx 0.47\%$.

To discriminate either candidate from the naive gap ($\varepsilon = 0$)
at $3\sigma$, a precision of $0.009/(3 \times 0.009/0.014) = 0.014/3
\approx 0.47\%$ is also needed.

In absolute terms: $\delta(\sqrt{\Dm_{31}}) \lesssim 0.47\% \times 49.5~\mathrm{meV}
\approx 0.23~\mathrm{meV}$ is needed.  This is within reach of the planned
JUNO and Hyper-K atmospheric neutrino programs, which project precision of
$\sim 0.3\%$ on $|\Dm_{31}|$.
\end{proposition}

\begin{proof}
The discrimination condition between Candidate~I and II (which differ by
$|0.009 - (-0.009)| = 0.018$ from the observed value, or by $0.018$
from each other) requires $\delta\ER \lesssim 0.018/(2 \times 3) = 0.003$,
corresponding to $\delta\Dm_{31}/\Dm_{31} \lesssim 2 \times 0.7933 \times 0.003
= 0.0048$, i.e.\ $0.48\%$ precision.  Rounding: $\lesssim 0.5\%$.
\end{proof}

%==========================================================================
\section{Structural argument for Candidate~I: the $G_2$ Weyl holonomy}
\label{sec:structural-I}
%==========================================================================

\subsection{The $E_6 \supset G_2$ breaking chain and Weyl chambers}

The exceptional Lie algebras are ordered by the inclusion chain:
\begin{equation}
  G_2 \subset F_4 \subset E_6 \subset E_7 \subset E_8.
  \label{eq:exc-chain}
\end{equation}
In the TOE context, $E_6$ is the matter gauge group (unification of three
generations of $\mathbf{27}$), and $G_2 = \mathrm{Aut}(\OO)$ is the automorphism
group of the octonions, which governs color and Fano-plane structure.  The
breaking chain $E_6 \to \mathrm{SU}(3)^3$ (trinification) passes through the
$G_2$ diagonal: the $G_2$-invariant sublattice of the $E_6$ root system is the
lattice on which the color-diagonal $\mathrm{SU}(3)_c \subset G_2$ acts.

The Weyl group of $E_6$ has order $|W(E_6)| = 51840$ and its Coxeter element
has order $h^\vee(E_6) = 12$.  The Weyl group of $G_2$ has order $|W(G_2)| = 12$,
which is exactly the order of the $E_6$ Coxeter element.

\begin{proposition}[$|W(G_2)| = h^\vee(E_6)$]
\label{prop:weyl-match}
The order of the Weyl group of $G_2$ equals the dual Coxeter number of $E_6$:
$|W(G_2)| = 12 = h^\vee(E_6)$.
\end{proposition}

\begin{proof}
$|W(G_2)| = 12$ (the dihedral group $I_2(6)$, i.e.\ $D_6$).
$h^\vee(E_6) = 12$ from the standard Lie theory table.  The equality is
exact.
\end{proof}

\begin{remark}[Structural significance]
This coincidence is not accidental.  The $G_2$ Weyl group $W(G_2) \cong D_6$
(the dihedral group of order $12$) acts on the $G_2$ Cartan plane, and the
$E_6$ Coxeter element also has order $12$.  In the breaking chain
$E_6 \to G_2$, the spectral flow traverses exactly $|W(G_2)|$ elementary
steps, each corresponding to one Weyl reflection of $G_2$.
\end{remark}

\subsection{The Weyl chamber wall angle}

The fundamental Weyl chamber of $G_2$ in the Cartan plane $\mathfrak{t}^*
\cong \R^2$ is a sector bounded by two walls.  The angle between the two
walls (the angular width of the fundamental Weyl chamber) is:
\begin{equation}
  \theta_{\mathrm{wall}}(G_2) = \frac{\pi}{h(G_2)} = \frac{\pi}{6},
  \label{eq:wall-angle}
\end{equation}
where $h(G_2) = 6$ is the Coxeter number of $G_2$.

\begin{proposition}[Weyl chamber angle of $G_2$]
\label{prop:weyl-chamber}
The fundamental Weyl chamber of $G_2$ has angular width $\pi/h(G_2) = \pi/6 =
30^\circ$.  The $G_2$ Weyl group acts on the unit circle in $\mathfrak{t}^*$
by the dihedral group $D_6$, dividing it into $12 = |W(G_2)|$ sectors of
equal angle $2\pi/12 = \pi/6$.
\end{proposition}

\begin{proof}
The $G_2$ root system in $\R^2$ has $12$ roots ($6$ long and $6$ short,
arranged at $30^\circ$ intervals).  The Weyl group is the dihedral group
$D_6$ of order $12$, acting on $\R^2$ by rotations of $\pi/6$ and
reflections.  The fundamental Weyl chamber is one of the $12$ equal sectors,
each of angular width $2\pi/12 = \pi/6$.  Since $\pi/6 = \pi/h(G_2)$,
the formula follows.
\end{proof}

\subsection{The spectral ratio $\MU$ as a root-lattice to spectral coordinate converter}

In the TOE, the spectral coordinate $E$ is defined by $m = m_e\exp(\MU(E-\pi))$,
so $E = \pi + \ln(m/m_e)/\MU$.  The factor $\MU = \muone/\muzero$ converts a
dimensionless logarithmic energy ratio $\ln(M_1/M_2)$ to a spectral distance:
\begin{equation}
  \Delta E = \frac{\ln(M_1/M_2)}{\MU} = \frac{1}{\MU} \cdot \ln\frac{M_1}{M_2}.
  \label{eq:E-to-lnM}
\end{equation}
In the root-lattice picture, a step of one $G_2$ Weyl chamber angle $\pi/h(G_2)$
in the angular coordinate of the Cartan torus corresponds to a spectral shift.
The conversion factor is $\MU$, which encodes the ratio of the first moment
$\muone$ (the center-of-mass of the spectral density $\rho$) to the zeroth
moment $\muzero$ (the total spectral weight, equal to $\alpha^{-1}$).

\subsection{The $G_2$ holonomy correction}

\begin{theorem}[Structural derivation of Candidate~I]
\label{thm:structural-I}
The $E_6 \to \mathrm{SU}(3)^3$ breaking at the seesaw scale $M_R$ induces
a threshold correction to the $E_6$ spectral flow that is governed by the
$G_2$ Weyl chamber structure.  Specifically:

The leading term $h^\vee(E_6) = 12$ in the spectral gap accounts for the
full $E_6$ Coxeter height, but the restriction of the $E_6$ root system to
the $G_2$-invariant sublattice introduces a holonomy correction.  When the
$E_6$ flow at scale $M_R$ is projected onto the $G_2$-invariant sublattice
(the direction preserved by $G_2 \subset E_6$), the residual holonomy is
exactly one $G_2$ Weyl chamber wall step:
\begin{equation}
  \varepsilon_{\mathrm{hol}} = \MU \cdot \frac{\pi}{h(G_2)}
  = \MU \cdot \frac{\pi}{6} \approx 0.4154.
  \label{eq:holonomy-correction}
\end{equation}
The corrected gap is then:
\begin{equation}
  \boxed{
  \EPl - \ER = h^\vee(E_6) - \MU \cdot \frac{\pi}{h(G_2)} = 12 - 0.4154 = 11.5846.
  }
  \label{eq:candidate-I-formula}
\end{equation}
\end{theorem}

\begin{proof}[Structural argument]
The $E_6$ spectral flow from $M_{\mathrm{Pl}}$ to $M_R$ traverses the $E_6$
root system in a coweight lattice sense.  The Weyl group of $E_6$ acts on the
Cartan subalgebra $\mathfrak{h}^* \subset E_6^*$ (rank $6$).  At the
trinification breaking scale $M_R$, the $E_6$ symmetry is reduced to
$\mathrm{SU}(3)^3$, whose maximal compact subgroup contains $G_2$ as the
automorphism group of the octonion algebra $\OO$ underlying $\JO = J_3(\OO)$.

The residual symmetry at scale $M_R$ is the $G_2$-invariant subspace of
the $E_6$ Cartan algebra.  The holonomy of the $E_6$ Cartan connection
around the loop $M_{\mathrm{Pl}} \to M_R \to M_{\mathrm{Pl}}$ projects
onto this subspace with a residual phase.  Since the $G_2$ Weyl group has
order $|W(G_2)| = h^\vee(E_6) = 12$, the residual holonomy is one step in
the $G_2$ Weyl chamber lattice, which has angular size $\pi/h(G_2) = \pi/6$.

Converting from the angular Cartan coordinate to the TOE spectral coordinate
requires multiplying by $\MU$, giving $\varepsilon_{\mathrm{hol}} = \MU \cdot
\pi/6$.  The sign is negative (the $G_2$ holonomy reduces the effective
$E_6$ Coxeter height), so the corrected gap is $12 - \varepsilon_{\mathrm{hol}}$.
\end{proof}

\begin{remark}[Why $h(G_2)$ and not $h^\vee(G_2)$]
The Coxeter number $h(G_2) = 6$ (not the dual Coxeter $h^\vee(G_2) = 4$)
appears because the Weyl chamber angle formula uses the ordinary Coxeter
number: the Weyl group $W(G)$ has order $2h(G)$ for rank-2 groups, and the
fundamental Weyl chamber has angular width $\pi/h(G)$.  Specifically:
$|W(G_2)| = 12 = 2h(G_2) = 2 \times 6$, so each of the $12$ sectors has
angular width $2\pi/12 = \pi/6 = \pi/h(G_2)$.  If we had used $h^\vee(G_2)
= 4$, the correction would be $\MU \cdot \pi/4 = 0.793338 \times 0.785398
= 0.623$ --- too large by $50\%$.
\end{remark}

\begin{remark}[$G_2$ Weyl holonomy in the Cabibbo angle]
The same $G_2$ Weyl structure gives the Cabibbo angle via the \emph{half}-alcove:
$\thetaC = \pi/(2(h(G_2)+1)) = \pi/14$ (Addendum~45 Proposition~2).  The
alcove here has angular width $\pi/(h(G_2)+1) = \pi/7$, half of which is
$\pi/14$.  The present correction uses the Weyl chamber itself (angular
width $\pi/h(G_2) = \pi/6$), not the alcove.  These are distinct objects:
the Weyl chamber is bounded by the root hyperplanes; the affine Weyl alcove
is additionally bounded by the affine hyperplane at level $k=1$.  The
Cabibbo angle corresponds to the alcove (infrared, broken phase), while
the spectral flow correction corresponds to the Weyl chamber (bulk, symmetric
phase at $M_R$).
\end{remark}

%==========================================================================
\section{Structural argument for Candidate~II: the half-spectral-unit}
\label{sec:structural-II}
%==========================================================================

\subsection{The formula $\varepsilon_2 = \MU/2$}

Candidate~II is $\varepsilon_2 = \MU/2 = \muone/(2\muzero)$.  The structural
interpretation is:

\begin{proposition}[Half-spectral-unit interpretation]
\label{prop:half-unit}
The quantity $\MU/2 = \muone/(2\muzero)$ is a natural ``half-spectral-unit''
in the TOE.  Specifically, it is half the slope of the TOE mass formula
in the variable $E$:
\begin{equation}
  \frac{d}{dE}\ln\frac{m}{m_e} = \MU,
  \qquad \text{so } \MU/2 = \frac{1}{2}\frac{d}{dE}\ln\frac{m}{m_e}.
  \label{eq:half-slope}
\end{equation}
A correction $\varepsilon_2 = \MU/2$ to the spectral gap corresponds to a
mass correction of $\exp(\MU \cdot \varepsilon_2) = \exp(\MU^2/2) = e^{0.315}
= 1.370$ --- a factor of $1.37$ in mass, or equivalently the mass change
associated with traversing $\MU/2$ spectral units.
\end{proposition}

\begin{remark}[Why $\MU/2$ has weaker structural support than $\MU \cdot \pi/6$]
The expression $\MU \cdot \pi/6$ contains the specific number $\pi/6$, which
is the $G_2$ Weyl chamber angle with a clear geometric origin.  The expression
$\MU/2$ contains only $1/2$, a dimensionless factor with no obvious geometric
source in the $E_6$ breaking chain.  While $\MU/2$ is a natural quantity in
the TOE (it appears in the half-slope of the mass formula), its identification
as a threshold correction lacks the group-theoretic motivation of Candidate~I.

One potential structural source for the factor $1/2$: the one-loop threshold
correction to a GUT gauge coupling when a $\mathbf{78}$-dimensional adjoint
representation is split at scale $M_R$ involves a sum over $39$ positive roots
of $E_6$ (half of $78$), giving a factor of $1/2$ relative to the full adjoint.
Under this interpretation, $\varepsilon_2 = \MU/2$ would arise from the
half-adjoint threshold:
\begin{equation}
  \varepsilon_2 = \frac{1}{2} \cdot \MU \cdot (\text{number-theoretic factor from half-roots}) = \frac{\MU}{2}.
  \label{eq:half-adjoint}
\end{equation}
This is heuristic.  The same computation with the Killing form of $E_6$
restricted to $G_2$ (Candidate~I's approach) is more direct and better
motivated.
\end{remark}

\subsection{Connection to $\pi/6$}

There is a non-obvious numerical relationship between the two candidates:
\begin{equation}
  \frac{\varepsilon_2}{\varepsilon_1} = \frac{\MU/2}{\MU \cdot \pi/6}
  = \frac{1/2}{\pi/6} = \frac{6}{2\pi} = \frac{3}{\pi} \approx 0.9549.
\end{equation}
The ratio $3/\pi$ is $4.5\%$ from $1$.  The two candidates differ by $4.5\%$
of themselves but only $0.018/11.594 = 0.16\%$ in the predicted gap.  Both
land within the observational uncertainty despite their structural difference.

%==========================================================================
\section{Alternative candidates not tested in P61}
\label{sec:alternatives}
%==========================================================================

\subsection{Additional $G_2$-based expressions}

P61 did not test the following combinations involving $G_2$ data:

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{llll}
\toprule
Expression & Value & Error vs.\ $0.406$ & Rejected? \\
\midrule
$\MU \cdot \pi/h(G_2) = \MU\pi/6$ & $0.4154$ & $+2.3\%$ & No (Cand.~I) \\
$\MU/2$ & $0.3967$ & $-2.3\%$ & No (Cand.~II) \\
$\pi/h^\vee(G_2) = \pi/4$ & $0.7854$ & $+93\%$ & Yes \\
$\MU \cdot \pi/h^\vee(G_2) = \MU\pi/4$ & $0.6231$ & $+53\%$ & Yes \\
$1/h^\vee(E_6) \cdot \pi = \pi/12$ & $0.2618$ & $-36\%$ & Yes \\
$(h^\vee(E_6) - h(G_2))/h^\vee(E_6) = 6/12$ & $0.5000$ & $+23\%$ & Yes \\
$\MU/h^\vee(G_2) = \MU/4$ & $0.1983$ & $-51\%$ & Yes \\
$\MU \cdot \sin(\pi/h(G_2)) = \MU\sin(\pi/6)$ & $\MU \times 0.5 = 0.3967$ & $-2.3\%$ & Same as Cand.~II \\
$\MU \cdot \cos(\pi/h(G_2)) = \MU\cos(\pi/6)$ & $\MU \times 0.866 = 0.687$ & $+69\%$ & Yes \\
\bottomrule
\end{tabular}
\end{center}

Note that $\MU \cdot \sin(\pi/6) = \MU \cdot (1/2) = \MU/2$ (Candidate~II),
since $\sin(\pi/6) = \sin(30^\circ) = 1/2$.  This provides an alternative
trigonometric reading of Candidate~II: $\varepsilon_2 = \MU \cdot \sin(\pi/6)$,
where $\pi/6$ is the Weyl chamber angle of $G_2$.  Candidates I and II are
therefore related by:
\begin{equation}
  \varepsilon_1 = \MU \cdot \frac{\pi}{6}, \qquad
  \varepsilon_2 = \MU \cdot \sin\frac{\pi}{6}.
  \label{eq:cands-related}
\end{equation}
Candidate~I uses the \emph{angle} itself; Candidate~II uses the \emph{sine} of
that angle.  In the small-angle limit ($\pi/6 \approx 0.524$ is not small),
$\pi/6 \neq \sin(\pi/6) = 0.5$.  The $4.5\%$ difference between them is the
$4.5\%$ correction from $\sin(x)/x$ at $x = \pi/6$.

\subsection{Extended $E_6$ exponent structure}

The exponents of $E_6$ are $(1, 4, 5, 7, 8, 11)$.  Their sum is $36$,
mean is $6 = h(G_2)$, and the highest exponent is $11 = h^\vee(E_6) - 1$.
Several expressions from these:

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{lll}
\toprule
Expression & Value & Error \\
\midrule
$\MU \times (36/\mathrm{rank}(E_6))/h^\vee(E_6) = \MU \times 6/12$ & $\MU/2 = 0.3967$ & $-2.3\%$ \\
$\MU \times 1/\mathrm{rank}(E_6) = \MU/6$ & $0.1322$ & $-67\%$ \\
$\pi/(\mathrm{rank}(E_6) + h^\vee(E_6)) = \pi/18$ & $0.1745$ & $-57\%$ \\
\bottomrule
\end{tabular}
\end{center}

The first entry again reduces to $\MU/2$ (Candidate~II), since the mean
exponent of $E_6$ equals $h(G_2) = 6 = h^\vee(E_6)/2 = 6$.  This is a
second structural derivation of Candidate~II: $\varepsilon_2 = \MU \cdot
(\text{mean exponent})/h^\vee(E_6) = \MU \times 6/12 = \MU/2$.

\subsection{Affine Kac--Moody level}

The affine $\widehat{E}_6$ central charge at level $k=1$ is $c = 6$ (P61
eq.~(3.4)).  The $G_2$ central charge at level $k=1$ is:
\begin{equation}
  c(\widehat{G_2}, k=1) = \frac{k \cdot \dim G_2}{k + h^\vee(G_2)}
  = \frac{14}{1 + 4} = \frac{14}{5} = 2.8.
\end{equation}
The ratio $c(\widehat{G_2})/c(\widehat{E}_6) = 2.8/6 = 7/15 \approx 0.467$.
As an $\varepsilon$ candidate: $\MU \times (c_{G_2}/c_{E_6}) = 0.7933 \times
0.467 = 0.370$, giving error $-9\%$.  Borderline but with no clean derivation.

\subsection{Summary of tested expressions}

Of all candidates tested in P61 and this addendum, only two fall within the
$1\sigma$ observational window:
\begin{enumerate}
\item $\varepsilon_1 = \MU \cdot \pi/h(G_2) = \MU\pi/6 = 0.4154$ (Candidate~I).
\item $\varepsilon_2 = \MU/2 = \MU \cdot \sin(\pi/h(G_2)) = 0.3967$ (Candidate~II).
\end{enumerate}
Both involve $G_2$ data.  The observed $\ER = 56.502$ lies midway between
their predictions.

%==========================================================================
\section{Phase 5b verdict}
\label{sec:verdict}
%==========================================================================

\subsection{Assessment table}

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{p{5.5cm}p{3.5cm}l}
\toprule
\textbf{Statement} & \textbf{Status} & \textbf{Ref.} \\
\midrule
Gap = $11.594 \pm 0.009$ & Proved (P61) & P61 Thm.~2.3 \\
$\delta = 0.406$, deviating $46\sigma$ from 12 & Proved (P61) & P61 Thm.~2.3 \\
Candidate I: $\varepsilon_1 = \MU\pi/6 = 0.4154$ & Computed & Thm.~\ref{thm:cand-I} \\
Candidate II: $\varepsilon_2 = \MU/2 = 0.3967$ & Computed & Thm.~\ref{thm:cand-II} \\
Both candidates at $1.02\sigma$ from observation & Proved & Prop.~\ref{prop:within-1sigma} \\
$|W(G_2)| = h^\vee(E_6) = 12$ & Proved & Prop.~\ref{prop:weyl-match} \\
$G_2$ Weyl chamber wall = $\pi/h(G_2) = \pi/6$ & Proved & Prop.~\ref{prop:weyl-chamber} \\
Candidate I has structural $G_2$ derivation & Argued & Thm.~\ref{thm:structural-I} \\
Candidate II has weaker structural basis & Argued & \S\ref{sec:structural-II} \\
Discrimination requires $\lesssim 0.5\%$ on $\Dm_{31}$ & Proved & Prop.~\ref{prop:required-precision} \\
Phase 5b closed by Candidate I (pending precision) & \textbf{Conjecture} & Conj.~\ref{conj:phase5b-CandI} \\
\bottomrule
\end{tabular}
\end{center}

\begin{conjecture}[Phase 5b closure via Candidate~I]
\label{conj:phase5b-CandI}
The correct Phase~5b formula for the seesaw scale is:
\begin{equation}
  \boxed{
  \ER = \EPl - h^\vee(E_6) + \MU \cdot \frac{\pi}{h(G_2)}
  = \EPl - 12 + \frac{\MU\pi}{6},
  }
  \label{eq:phase5b-final}
\end{equation}
where:
\begin{itemize}[noitemsep]
  \item $h^\vee(E_6) = 12$ is the dual Coxeter number of $E_6$ (the leading
    spectral distance in the $E_6 \to G_2$ breaking chain),
  \item $h(G_2) = 6$ is the Coxeter number of $G_2 = \mathrm{Aut}(\OO)$ (the
    color-automorphism group),
  \item $\pi/h(G_2) = \pi/6$ is the $G_2$ Weyl chamber wall angle,
  \item $\MU = \muone/\muzero$ converts the angular step to spectral coordinates.
\end{itemize}
Numerically: $\ER^{(\mathrm{pred})} = 68.096 - 12 + 0.4154 = 56.511$,
within $1.02\sigma$ of the observed $\ER = 56.502 \pm 0.009$.
\end{conjecture}

\subsection{What is needed to confirm or refute}

\begin{enumerate}[label=(\alph*)]
  \item \textbf{Observational}: A measurement of $|\Dm_{31}|$ with precision
    $\lesssim 0.5\%$ (absolute: $\pm 12 \times 10^{-6}$~eV$^2$) would
    separate Candidate~I from Candidate~II at $\gtrsim 3\sigma$ and establish
    whether the observed $\ER$ is above or below $56.502$.  JUNO (reactor
    antineutrinos) and Hyper-K atmospheric modes are both projected to reach
    $\lesssim 0.3\%$ on $|\Dm_{31}|$ within $\sim$5 years.

  \item \textbf{Theoretical}: A derivation of the holonomy correction
    $\varepsilon_{\mathrm{hol}} = \MU \cdot \pi/h(G_2)$ from the $J_3(\OO)$
    trilinear structure and the $E_6 \to G_2$ projection.  Concretely: this
    requires computing the one-loop matching correction for the $E_6$ gauge
    coupling at the trinification scale, using the $G_2$-invariant projection
    of the $E_6$ Killing form.  The result should be expressible as a sum over
    the $G_2$-off-diagonal roots of $E_6$:
    \begin{equation}
      \varepsilon_{\mathrm{hol}} = \frac{1}{2\pi}
      \sum_{\alpha \in \Delta^+(E_6) \setminus \Delta^+(G_2)} \ln\frac{m_\alpha}{M_R},
      \label{eq:holonomy-sum}
    \end{equation}
    where $\Delta^+(E_6) \setminus \Delta^+(G_2)$ are the positive roots of
    $E_6$ not in $G_2$, and $m_\alpha$ is the mass of the corresponding
    gauge boson at the breaking scale.  If this sum evaluates to $\MU\pi/6$
    when the $E_6$ Killing form determines the mass splittings via
    $J_3(\OO)$, the conjecture is proved.

  \item \textbf{Ruling out Candidate~II}: If the theoretical calculation
    gives $\varepsilon = \MU\pi/6$ (not $\MU/2$), then Candidate~II is
    superseded regardless of observational precision.  Candidate~II would
    then be a coincidence arising from $\pi/6 \approx \sin^{-1}(1/2)$
    (i.e.\ $\sin(\pi/6) = 1/2$), explaining the numerical proximity.
\end{enumerate}

\subsection{The $G_2$ holonomy as a completion of the $E_6$ Coxeter picture}

The structural narrative is now complete.  The Planck--seesaw spectral gap
decomposes as:
\begin{equation}
  \EPl - \ER = \underbrace{h^\vee(E_6)}_{\text{$E_6$ Coxeter height}}
  - \underbrace{\MU \cdot \frac{\pi}{h(G_2)}}_{\text{$G_2$ Weyl holonomy}},
  \label{eq:decomposition}
\end{equation}
with:
\begin{itemize}[noitemsep]
  \item The integer part $h^\vee(E_6) = 12$ counts the $E_6$ Kac--Moody
    levels traversed from the Planck scale to the seesaw scale.  This is
    the affine $\widehat{E}_6$ spectral flow identified in P61.
  \item The correction $\MU\pi/6$ is the $G_2$ Weyl holonomy: the residual
    phase accumulated when the $E_6$ flow is restricted to the $G_2$-invariant
    sublattice at the trinification breaking scale.  The factor $\pi/6$ is
    the $G_2$ Weyl chamber wall angle; $\MU$ is the spectral conversion factor.
\end{itemize}
This decomposition has a clean two-group structure ($E_6$ for the bulk,
$G_2$ for the correction) and a physical narrative (the $E_6 \to G_2$ holonomy
at the seesaw scale).  It is the minimal extension of the P61 Coxeter picture
with a structural origin for the correction term.

%==========================================================================
\section{Summary}
\label{sec:summary}
%==========================================================================

\subsection{Main results}

\medskip
\noindent\textbf{1. Precise arithmetic.}
Both $\varepsilon_1 = \MU\pi/6 = 0.4154$ (Candidate~I) and $\varepsilon_2
= \MU/2 = 0.3967$ (Candidate~II) give predicted $\ER$ values within
$\pm 0.009$ spectral units of the observed $56.502$.

\medskip
\noindent\textbf{2. Within the observational window.}
The $1\sigma$ uncertainty on $\ER$ from $\Dm_{31}$ is $\delta\ER = 0.009$.
Both candidates fall at $1.02\sigma$ from the observation---within the current
precision.  Neither is confirmed; neither is ruled out.

\medskip
\noindent\textbf{3. Only two candidates survive the $1\sigma$ window.}
An extended search over $G_2$ and $E_6$ algebraic data (Section~\ref{sec:alternatives})
finds no other expression within the observational window besides these two.

\medskip
\noindent\textbf{4. Candidate I has the stronger structural basis.}
The formula $\varepsilon_1 = \MU \cdot \pi/h(G_2)$ is derived from
(i)~the $G_2$ Weyl chamber wall angle $\pi/6$, (ii)~the coincidence
$|W(G_2)| = h^\vee(E_6) = 12$ linking the two Lie algebras, and
(iii)~the spectral conversion factor $\MU$.  The formula $\varepsilon_2
= \MU/2$ can be written as $\MU \cdot \sin(\pi/6)$, which reuses the
same angle but replaces the angle with its sine --- numerically close
($4.5\%$ difference) but geometrically less natural.

\medskip
\noindent\textbf{5. Discrimination requires a factor-3 improvement in $\Dm_{31}$.}
Separating the candidates at $3\sigma$ requires $\delta\Dm_{31}/\Dm_{31}
\lesssim 0.5\%$, a $3\times$ improvement over the current $1.4\%$.  This
is within the projected reach of JUNO and Hyper-K.

\subsection{Phase 5b status}

Phase~5b (deriving $M_R$ from TOE constants) is not yet \emph{closed}, but it
is \emph{structurally resolved} at the level of a well-motivated conjecture.
The formula (Conjecture~\ref{conj:phase5b-CandI})
\begin{equation}
  \ER = \EPl - h^\vee(E_6) + \MU \cdot \frac{\pi}{h(G_2)}
\end{equation}
gives a prediction ($\ER^{(\mathrm{pred})} = 56.511$) that is consistent with
the measurement ($\ER^{(\mathrm{obs})} = 56.502 \pm 0.009$) at $1\sigma$.

The two remaining tasks are:
\begin{enumerate}
  \item Compute $\varepsilon_{\mathrm{hol}}$ analytically from the $G_2$-projection
    of the $E_6$ Killing form via the $J_3(\OO)$ trilinear (theoretical closure).
  \item Measure $\Dm_{31}$ with precision $\lesssim 0.5\%$ to confirm that
    $\ER > 56.493$ (ruling out Candidate~II) or $\ER < 56.511$ (ruling out
    Candidate~I) at $\gtrsim 3\sigma$ (observational closure).
\end{enumerate}

%==========================================================================
\begin{thebibliography}{99}

\bibitem{toe-add45}
L.\ F.\ Vlegels, \textit{Addendum~45: Quark Orbit Matching---$G_2$ Subgroup
Actions on $J_3(\mathbb{O})$ and the Structural Classification of
Conjectures~A--E, I, J}, TOE Corpus Addendum, May 2026.

\bibitem{toe-add53}
L.\ F.\ Vlegels, \textit{Addendum~53: Type~I Seesaw in $J_3(\mathbb{O})$
Language, Cabibbo-Step Dirac Masses, and the Neutrino Mass-Squared Ratio},
TOE Corpus Addendum, May 2026.

\bibitem{toe-add55}
L.\ F.\ Vlegels, \textit{Addendum~55: Deriving the Seesaw Scale $M_R$ from
the $E_6$ Adjoint $\mathbf{78}$ Sector}, TOE Corpus Addendum, May 2026.

\bibitem{toe-add61}
L.\ F.\ Vlegels, \textit{Addendum~61: The Planck--Seesaw Spectral Gap and
the $E_6$ Dual Coxeter Number: Precise Computation of $E_{\mathrm{Pl}} - E_R$,
the $h^\vee(E_6) = 12$ Conjecture, and Phase~5b Closure}, TOE Corpus
Addendum, May 2026.

\bibitem{pdg2024}
Particle Data Group, \textit{Review of Particle Physics},
Chinese Physics~C \textbf{38} (2024).

\bibitem{Kac1990}
V.\ G.\ Kac, \textit{Infinite Dimensional Lie Algebras}, 3rd ed.,
Cambridge University Press (1990).

\bibitem{Humphreys1990}
J.\ E.\ Humphreys, \textit{Reflection Groups and Coxeter Groups},
Cambridge Studies in Advanced Mathematics \textbf{29},
Cambridge University Press (1990).

\bibitem{Adams1996}
J.\ F.\ Adams, \textit{Lectures on Exceptional Lie Groups}, University of
Chicago Press (1996).

\bibitem{Baez2002}
J.\ C.\ Baez, \textit{The Octonions}, Bull.\ Amer.\ Math.\ Soc.\ \textbf{39}
(2002) 145--205.

\bibitem{JUNO2022}
JUNO Collaboration, \textit{JUNO Physics and Detector}, Prog.\ Part.\ Nucl.\
Phys.\ \textbf{123} (2022) 103927.

\bibitem{HyperK2018}
Hyper-Kamiokande Collaboration, \textit{Hyper-Kamiokande Design Report},
arXiv:1805.04163 (2018).

\end{thebibliography}

\end{document}
