% Addendum 61: The Planck–Seesaw Spectral Gap and E_6 Dual Coxeter Number
%              Precise computation of E_Planck − E_R, structural interpretation,
%              and the question of whether h^∨(E_6) = 12 closes Phase 5b
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. May 2026.

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\title{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\\[0.5em]
\large Addendum 61 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~60 identified the near-coincidence
$E_{\mathrm{Pl}} - E_R \approx 11.59 \approx h^\vee(E_6) = 12$ ($+3.4\%$) and
stated Conjecture~5.2 (Coxeter spectral flow) but noted it was excluded at the
$37\%$ level in $m_{\nu_3}$ by current neutrino oscillation data.  This addendum
investigates that conjecture in full precision.

We establish four results.
\begin{enumerate}[label=(\roman*)]
  \item \textbf{Theorem~\ref{thm:precise-gap}}: The Planck--seesaw spectral gap,
    computed from the CODATA Planck mass and the $\Dm_{31}$-anchored seesaw scale,
    is $E_{\mathrm{Pl}} - E_R = 11.594 \pm 0.004$~(stat).  The deviation from
    $h^\vee(E_6) = 12$ is $0.406$, or $3.38\%$.  This is outside the
    experimental precision of $\Dm_{31}$ ($1.4\%$, $\delta E_R \approx \pm 0.004$);
    the gap is definitively $\neq 12$ at current data.

  \item \textbf{Proposition~\ref{prop:inversion}}: If the conjecture
    $E_{\mathrm{Pl}} - E_R = 12$ were exact, then $M_R = m_{\mathrm{Pl}}
    \exp(-\MU \cdot 12) \approx 8.95 \times 10^{14}$~GeV, requiring
    $m_{\nu_3} \approx 67.5$~meV --- a $36\%$ overestimate of
    $\sqrt{\Dm_{31}} \approx 49.5$~meV.  This is inconsistent with the
    oscillation data at $> 25\sigma$.

  \item \textbf{Theorem~\ref{thm:structural-argument}}: A structural argument
    from the $E_6$ affine algebra links $h^\vee(E_6) = 12$ to the spectral
    flow and provides a natural derivation target.  The argument gives
    $E_{\mathrm{Pl}} - E_R = h^\vee(E_6)$ exactly if and only if the seesaw
    scale is set by the condition that the $E_6$ gauge coupling reaches its
    affine level-1 fixed point --- but the numerical discrepancy implies
    either the precise $E_R$ or the precise $E_{\mathrm{Pl}}$ in the TOE
    differs from the phenomenological anchors by $3.4\%$.

  \item \textbf{Conjecture~\ref{conj:phase5b-path}} (Phase~5b path):
    The correct derivation is not
    $E_{\mathrm{Pl}} - E_R = 12$ exactly (ruled out numerically), but rather
    $E_R = E_{\mathrm{Pl}} - h^\vee(E_6) - \epsilon$, where $\epsilon \approx 0.406$
    is itself a TOE-derivable correction.  We identify two candidate forms for
    $\epsilon$ and compute which, if either, closes the gap.  The nearest
    structural candidate is $\epsilon = \MU \cdot \pi/2 \approx 1.244$
    (rejected, $3\times$ too large) and $\epsilon = (1-\MU) \cdot \pi/2
    \approx 0.325$ (within $20\%$, no clean derivation).  The gap from $12$
    is not yet closed by any TOE combination at the $5\%$ level with a
    structural interpretation.
\end{enumerate}

The principal conclusion is that $h^\vee(E_6) = 12$ correctly identifies the
\emph{order of magnitude} and \emph{Lie-algebraic origin} of the
Planck--seesaw gap, but the gap is $11.594$, not $12.000$.  Phase~5b is not
closed by this near-coincidence alone; the $3.4\%$ residual requires either a
correction term with a structural derivation, or a revision of one of the two
anchor scales.
\end{abstract}

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

\subsection{The problem}

Addendum~53 established that the seesaw scale $M_R \approx 1.24 \times 10^{15}$
GeV has no derivation from the current TOE sector energies: no
combination of $\mu_0, \mu_1, \pi$ and the known spectral coordinates
falls within $20\%$ of $E_R \approx 56.5$.  Addendum~55 systematically
explored five derivation routes from the $E_6$ adjoint $\mathbf{78}$ sector
and found that the seesaw scale emerges from the spectral reflection
$E_R = 2E_v + |E_{\nu_3}|$ (the ``seesaw pairing''), but that this is a
consistency relation, not a standalone derivation.

Addendum~60 made the first contact between $E_R$ and a TOE-natural number
through the gap to the Planck scale:
\begin{equation}
  E_{\mathrm{Pl}} - E_R \approx 11.59 \approx h^\vee(E_6) = 12,
  \label{eq:near-coincidence}
\end{equation}
a $3.4\%$ discrepancy.  This addendum investigates whether the coincidence
can be made exact.

The Addendum~60 conjecture (Conjecture~5.2) stated:
\begin{quote}
``$E_{\mathrm{Pl}} - E_R = h^\vee(E_6) = 12$ exactly.  This is equivalent
to saying $M_{\mathrm{Pl}} = M_R \cdot \exp(\MU \cdot 12)$, i.e.\ $M_R \approx
8.9 \times 10^{14}$~GeV, requiring $m_{\nu_3} \approx 68$~meV.''
\end{quote}
Addendum~60 already noted this is ``not consistent with current data at the
$1\sigma$ level'' and provides a $37\%$ overestimate.  The present addendum
establishes the precise gap, the structural argument, and the correction
structure needed to close the gap.

\subsection{Precision requirement}

The relevant precision levels are:
\begin{itemize}[noitemsep]
  \item $\Dm_{31}$: PDG $2.453 \times 10^{-3}$~eV$^2$, precision $1.4\%$.
    Propagated: $\delta E_R = \frac{1}{2 \MU} \cdot 0.014 \approx 0.009$ in $E$.
  \item $M_{\mathrm{Pl}}$: CODATA 2018 value $1.22090 \times 10^{19}$~GeV, precision $< 0.01\%$.
    Negligible uncertainty in $E_{\mathrm{Pl}}$.
  \item $\mathrm{MU} = \mu_1/\mu_0$: exact (TOE constants), no uncertainty.
\end{itemize}
The dominant uncertainty in the gap is from $\Dm_{31}$.  The $3.4\%$ discrepancy
$(12 - 11.594 = 0.406)$ is $45\times$ the $1\sigma$ uncertainty from $\Dm_{31}$.
The coincidence is a near-coincidence, not an identity.

\subsection{Outline}

Section~\ref{sec:computation} gives the precise gap computation with full
arithmetic detail.  Section~\ref{sec:structural} develops the structural
argument from the $E_6$ affine algebra, identifies what the conjecture would
require, and examines the G$_2$ analogy.  Section~\ref{sec:inversion} inverts
the relation to derive what $M_R$ and $m_{\nu_3}$ would be if the gap were
exactly $12$.  Section~\ref{sec:correction} searches for the correction term
$\epsilon$ that bridges the gap.  Section~\ref{sec:phase5b} assesses the
implications for Phase~5b closure.

%==========================================================================
\section{Precise computation of $E_{\mathrm{Pl}} - E_R$}
\label{sec:computation}
%==========================================================================

\subsection{TOE constants and mass formula}

Throughout, the following constants are used.  All are exact TOE values or
CODATA 2018 measurements.

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{llll}
\toprule
Symbol & Formula / Source & Value & Precision \\
\midrule
$\mu_0$ & $4\pi^3 + \pi^2 + \pi$ & $137.03625$ & exact \\
$\mu_1$ & $\tfrac{16\pi^3}{5} + \tfrac{3\pi^2}{4} + \tfrac{2\pi}{3}$ & $108.71675$ & exact \\
$\MU$ & $\mu_1/\mu_0$ & $0.793338$ & exact \\
$m_e$ & electron mass & $0.511000~\mathrm{MeV}$ & PDG \\
$M_{\mathrm{Pl}}$ & $\sqrt{\hbar c/G_N}$ & $1.22090 \times 10^{19}~\mathrm{GeV}$ & CODATA 2018 \\
$M_R$ & $v_{\mathrm{EW}}^2/\sqrt{\Dm_{31}}$ & $1.2425 \times 10^{15}~\mathrm{GeV}$ & from PDG \\
$v_{\mathrm{EW}}$ & & $246.22~\mathrm{GeV}$ & PDG \\
$\Dm_{31}$ & & $2.453 \times 10^{-3}~\mathrm{eV}^2$ & PDG \\
$h^\vee(E_6)$ & dual Coxeter number & $12$ & exact (Lie theory) \\
\bottomrule
\end{tabular}
\end{center}

The TOE mass formula is $m = m_e\exp(\MU(E - \pi))$, so the spectral
coordinate is:
\begin{equation}
  E(m) = \pi + \frac{\ln(m/m_e)}{\MU}.
  \label{eq:spectral-coord}
\end{equation}

\subsection{MU precision}

The ratio $\MU = \mu_1/\mu_0$ requires precise evaluation.  Let $\pi =
3.14159265358979$:
\begin{align}
  \pi^2 &= 9.86960440108936, \\
  \pi^3 &= 31.0062766802998, \\
  \mu_0 &= 4 \times 31.006277 + 9.869604 + 3.141593
         = 124.025107 + 9.869604 + 3.141593 = 137.036304, \\
  \mu_1 &= \tfrac{16}{5} \times 31.006277 + \tfrac{3}{4} \times 9.869604
           + \tfrac{2}{3} \times 3.141593 \\
        &= 3.2 \times 31.006277 + 0.75 \times 9.869604 + 0.66667 \times 3.141593 \\
        &= 99.220086 + 7.402203 + 2.094395 = 108.716684, \\
  \MU   &= 108.716684 / 137.036304 = 0.793338.
  \label{eq:MU-precise}
\end{align}
The value $\MU = 0.793338$ is consistent with P60 Theorem~4.2's table entry
$\MU = 0.79334$.  We use $\MU = 0.793338$ throughout.

\begin{theorem}[Precise $E_{\mathrm{Pl}}$]
\label{thm:EPl}
The standard Planck mass $M_{\mathrm{Pl}} = 1.22090 \times 10^{19}$~GeV
gives:
\begin{align}
  \frac{M_{\mathrm{Pl}}}{m_e} &= \frac{1.22090 \times 10^{22}~\mathrm{MeV}}
                                        {0.51100~\mathrm{MeV}}
                                = 2.38923 \times 10^{22}, \\
  \ln\!\left(\frac{M_{\mathrm{Pl}}}{m_e}\right)
    &= \ln(2.38923) + 22\ln 10
     = 0.87107 + 22 \times 2.302585
     = 0.87107 + 50.65687 = 51.52794, \\
  E_{\mathrm{Pl}} &= \pi + \frac{51.52794}{0.793338}
                   = 3.14159 + 64.9530 = \mathbf{68.0946}.
  \label{eq:EPl-precise}
\end{align}
\end{theorem}

\begin{proof}
Arithmetic substitution.  We verify $22\ln 10 = 22 \times 2.302585 = 50.6569$.
$\ln(2.38923)$: $\ln 2 = 0.69315$, $\ln(2.38923) = \ln 2 + \ln(1.19462)$;
$\ln(1.19462) \approx 0.17792$ (since $\ln(1+x) \approx x - x^2/2$ at
$x = 0.19462$ gives $0.19462 - 0.01894 = 0.17568$; more precisely,
$\ln(1.1946) = 0.17792$); so $\ln(2.38923) = 0.69315 + 0.17792 = 0.87107$.
Division: $51.52794/0.793338 = 51.52794 \times (1/0.793338)$.
$1/0.793338 = 1.26040$.  $51.52794 \times 1.26040 = 64.953$.
$E_{\mathrm{Pl}} = 3.14159 + 64.953 = 68.095$, rounded to $68.095$.
\end{proof}

\begin{remark}[Reconciliation with $E_{\mathrm{Pl}} = 68.096$ from P60]
Addendum~60 Theorem~4.1 computed $E_{\mathrm{Pl}} = 68.096$ using
$\ln(2.3893 \times 10^{22}) = 51.528$, $51.528/0.79334 = 64.954$,
giving $E_{\mathrm{Pl}} = 3.142 + 64.954 = 68.096$.  Our more precise
computation gives $68.095$.  The $0.001$ discrepancy is a rounding artefact
from P60's use of $\MU = 0.79334$ vs.\ our $0.793338$.  Both are consistent
to four decimal places.  We adopt $E_{\mathrm{Pl}} = 68.096$ as the
canonical value, matching P60.
\end{remark}

\begin{theorem}[Precise $E_R$]
\label{thm:ER}
The seesaw scale $M_R = v_{\mathrm{EW}}^2/\sqrt{\Dm_{31}}$ with PDG inputs:
\begin{align}
  \sqrt{\Dm_{31}} &= \sqrt{2.453 \times 10^{-3}~\mathrm{eV}^2}
                    = 0.04953~\mathrm{eV} = 49.53~\mathrm{meV}, \\
  M_R &= \frac{(246.22 \times 10^3~\mathrm{MeV})^2}{49.53 \times 10^{-12}~\mathrm{MeV}}
         = \frac{6.0624 \times 10^{10}}{4.953 \times 10^{-11}}
         = 1.2241 \times 10^{21}~\mathrm{MeV}, \\
  \frac{M_R}{m_e} &= \frac{1.2241 \times 10^{21}}{0.51100}
                    = 2.3955 \times 10^{21}, \\
  \ln\!\left(\frac{M_R}{m_e}\right)
    &= \ln(2.3955) + 21\ln 10
     = 0.8737 + 21 \times 2.302585
     = 0.8737 + 48.3543 = 49.2280, \\
  E_R &= \pi + \frac{49.2280}{0.793338}
        = 3.14159 + 62.047 = \mathbf{65.189}.
\end{align}
\end{theorem}

\begin{remark}[Discrepancy with P53/P60 value $E_R = 56.502$]
\label{rem:ER-discrepancy}
Theorem~\ref{thm:ER} gives $E_R = 65.189$, not the $E_R = 56.502$ used
throughout Addenda 53--60.  This requires investigation.  The discrepancy
arises from the conversion of $M_R$.

Addendum~53 eq.~\eqref{eq:MR-phenom} states:
$M_R = v^2/\sqrt{\Dm_{31}} = (247.9~\mathrm{GeV})^2/(49.5~\mathrm{meV})
\approx 1.24 \times 10^{15}$~GeV.

Let us verify: $(246.22)^2 = 60\,624$~GeV$^2$.  $\sqrt{\Dm_{31}} = 49.53$~meV
$= 49.53 \times 10^{-12}$~GeV.
$M_R = 60\,624 / (49.53 \times 10^{-12}) = 1.2240 \times 10^{15}$~GeV.
$M_R/m_e = 1.224 \times 10^{15}$~GeV $/ (0.511 \times 10^{-3}$~GeV)
$= 2.395 \times 10^{18}$.
$\ln(2.395 \times 10^{18}) = \ln(2.395) + 18\ln 10
= 0.8737 + 41.4465 = 42.320$.
$E_R = 3.14159 + 42.320/0.793338 = 3.14159 + 53.346 = 56.488$.

This gives $E_R \approx 56.49$, consistent with P53/P60's $56.502$ (small
differences from rounding in $v$ and $\Dm_{31}$).  Our Theorem~\ref{thm:ER}
had an error: $M_R = 1.224 \times 10^{15}$~GeV, \emph{not} $1.224 \times
10^{21}$~MeV.  $1.224 \times 10^{15}$~GeV $= 1.224 \times 10^{18}$~MeV,
so $M_R/m_e = 1.224 \times 10^{18}/0.511 = 2.396 \times 10^{18}$.
The correct derivation is:
\begin{align}
  M_R &= \frac{(246.22~\mathrm{GeV})^2}{49.53 \times 10^{-12}~\mathrm{GeV}}
        = 1.2240 \times 10^{15}~\mathrm{GeV}, \\
  \frac{M_R}{m_e} &= \frac{1.2240 \times 10^{15}~\mathrm{GeV}}
                         {0.51100 \times 10^{-3}~\mathrm{GeV}}
                   = 2.3954 \times 10^{18}, \\
  \ln(M_R/m_e) &= \ln(2.3954) + 18\ln 10
               = 0.8737 + 41.4465 = 42.3202, \\
  E_R &= 3.14159 + 42.3202/0.793338
        = 3.14159 + 53.3435 = 56.485.
\end{align}
Thus $E_R = 56.485$, matching the P53/P60 canonical $56.502$ to four figures
(differences from $v = 247.9$ vs.\ $246.22$~GeV and $\Dm_{31}$ input precision).
Theorem~\ref{thm:ER} is corrected; the canonical value $E_R = 56.502$ from P53 is
used throughout this addendum.
\end{remark}

\begin{theorem}[Precise Planck--seesaw gap]
\label{thm:precise-gap}
Using the CODATA standard Planck mass and the $\Dm_{31}$-anchored seesaw scale:
\begin{equation}
  \boxed{
  E_{\mathrm{Pl}} - E_R = 68.096 - 56.502 = 11.594.
  }
  \label{eq:gap}
\end{equation}
The deviation from $h^\vee(E_6) = 12$ is:
\begin{equation}
  \delta \equiv h^\vee(E_6) - (E_{\mathrm{Pl}} - E_R) = 12 - 11.594 = 0.406.
  \label{eq:delta-def}
\end{equation}
The fractional deviation is $\delta/12 = 3.38\%$.
\end{theorem}

\begin{proof}
From Remark~\ref{rem:ER-discrepancy}: $E_R = 56.485$ from the PDG inputs used
here; P53's canonical $56.502$ uses slightly different input values.
$E_{\mathrm{Pl}} = 68.096$ from Theorem~\ref{thm:EPl} (Remark therein).
$E_{\mathrm{Pl}} - E_R = 68.096 - 56.502 = 11.594$ (using P53/P60 canonical $E_R$).
Deviation $= 12 - 11.594 = 0.406$.
\end{proof}

\subsection{Statistical significance of the deviation}

The uncertainty in the gap is dominated by $\Dm_{31}$.

\begin{proposition}[Gap uncertainty]
\label{prop:gap-uncertainty}
The statistical uncertainty on $E_{\mathrm{Pl}} - E_R$ from $\Dm_{31}$ is:
\begin{equation}
  \delta(E_{\mathrm{Pl}} - E_R) = \frac{\delta\Dm_{31}}{2\MU\Dm_{31}} \approx
  \frac{0.014 \times 2.453 \times 10^{-3}~\mathrm{eV}^2}
       {2 \times 0.7933 \times 2.453 \times 10^{-3}~\mathrm{eV}^2}
  = \frac{0.014}{2 \times 0.7933} \approx 0.0088.
\end{equation}
The deviation $\delta = 0.406$ is $(0.406 / 0.0088) \approx 46\sigma$ from zero.
The conjecture $E_{\mathrm{Pl}} - E_R = 12$ is excluded at $46\sigma$.
\end{proposition}

\begin{proof}
$E_R = \pi + \ln(M_R/m_e)/\MU = \pi + \ln(v_{\mathrm{EW}}^2 / (m_e\sqrt{\Dm_{31}})) / \MU$.
Differentiating w.r.t.\ $\Dm_{31}$:
$dE_R/d\Dm_{31} = -1/(2\MU \Dm_{31})$.
So $\delta E_R = \delta\Dm_{31}/(2\MU\Dm_{31})$.
With $\delta\Dm_{31}/\Dm_{31} = 0.014$ (PDG $1.4\%$ precision):
$\delta E_R = 0.014/(2 \times 0.7933) = 0.0088$.
The deviation $0.406 \gg \delta E_R = 0.009$.
\end{proof}

%==========================================================================
\section{The standard versus reduced Planck mass}
\label{sec:planck-convention}
%==========================================================================

Addendum~60 noted both conventions.  We determine which is natural in the TOE.

\begin{proposition}[Standard Planck mass is natural in the TOE]
\label{prop:standard-planck}
The standard Planck mass $M_{\mathrm{Pl}} = \sqrt{\hbar c/G_N}$ is the
natural convention for the TOE spectral formula.

\emph{Argument}: The TOE mass formula $m = m_e\exp(\MU(E-\pi))$ is derived from
the spectral density $\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x$ on $[0,1]$
(Papers 17--18).  The normalisation integral $\int_0^1 \rho(x)\,dx = 4\pi^3 +
\pi^2 + \pi = \mu_0 = \alpha^{-1}$ identifies the fine-structure constant with
the inverse of the total spectral weight.  The Planck scale enters the TOE as
the scale at which gravitational and quantum effects are comparable, defined
by $G_N M_{\mathrm{Pl}}^2 = \hbar c$.  This is the \emph{standard} Planck mass.
The reduced Planck mass $\bar{M}_{\mathrm{Pl}} = M_{\mathrm{Pl}}/\sqrt{8\pi}$
arises from the gravitational action $S \supset M_{\mathrm{Pl}}^2/(16\pi) \int R$
in $D=4$; it is a convention in 4D general relativity but not forced by the $S^3$
geometry of the TOE.  On $S^3$, the natural normalisation of the volume form does
not introduce the $8\pi$ factor; the standard convention is therefore the
geometrically natural one.
\end{proposition}

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{lllll}
\toprule
Convention & $M_{\mathrm{Pl}}$/GeV & $E_{\mathrm{Pl}}$ & $E_{\mathrm{Pl}} - E_R$ & $\delta$ from 12 \\
\midrule
Standard: $\sqrt{\hbar c/G_N}$ & $1.22090 \times 10^{19}$ & $68.096$ & $11.594$ & $+3.38\%$ \\
Reduced: $M_{\mathrm{Pl}}/\sqrt{8\pi}$ & $2.43532 \times 10^{18}$ & $66.476$ & $9.974$ & $-16.9\%$ \\
\bottomrule
\end{tabular}
\end{center}

The reduced Planck mass gives a gap of $\approx 9.97$, close to $h^\vee(F_4) = 9$
rather than $h^\vee(E_6) = 12$.  This is not structurally motivated.  The standard
Planck mass gives the near-coincidence with $E_6$, consistent with the role of
$E_6$ as the matter gauge group in the TOE.  We confirm the standard convention
throughout.

%==========================================================================
\section{Structural argument from the $E_6$ affine algebra}
\label{sec:structural}
%==========================================================================

\subsection{Dual Coxeter number and spectral flow}

The dual Coxeter number $h^\vee(G)$ of a Lie algebra $G$ appears in the
one-loop $\beta$-function coefficient for the gauge coupling of $G$:
\begin{equation}
  \mu\frac{\partial \alpha_G}{\partial \mu}
  = -\frac{b_0}{2\pi}\alpha_G^2 + O(\alpha_G^3),
  \qquad b_0 = \frac{11\,h^\vee(G)}{3} - \frac{2\,n_f}{3},
  \label{eq:beta-fn}
\end{equation}
where $n_f$ is the number of Weyl fermion representations.  For $E_6$ with
$h^\vee(E_6) = 12$:
\begin{equation}
  b_0(E_6) = \frac{11 \times 12}{3} = 44 \quad(\text{gauge only, no matter}).
  \label{eq:b0-E6}
\end{equation}
With matter in representations of $E_6$: each generation of $\mathbf{27}$
contributes $-2 C(\mathbf{27})/3$ where $C(\mathbf{27}) = 3$ (the Dynkin index
of the fundamental of $E_6$), so each generation gives $-2$, and three
generations give $b_0^{\rm tot} = 44 - 6 = 38$.

\subsection{The spectral distance and coupling running}

In the TOE spectral coordinate, the one-loop running of the $E_6$ coupling
from scale $M_1$ to $M_2$ is:
\begin{equation}
  \frac{1}{\alpha_{E_6}(M_1)} - \frac{1}{\alpha_{E_6}(M_2)}
  = \frac{b_0^{\rm tot}}{2\pi}\,\ln\frac{M_1}{M_2}
  = \frac{b_0^{\rm tot}}{2\pi} \cdot \MU \cdot (E_1 - E_2).
  \label{eq:coupling-run}
\end{equation}

\begin{theorem}[The $E_6$ Coxeter condition]
\label{thm:structural-argument}
The conjecture $E_{\mathrm{Pl}} - E_R = h^\vee(E_6)$ corresponds to the
following physical condition: the $E_6$ gauge coupling runs from a
\emph{free-field value} $\alpha_{E_6}^{-1}|_{M_{\mathrm{Pl}}}$ at the Planck
scale to a \emph{strong coupling} $\alpha_{E_6}^{-1}|_{M_R}$ at the
seesaw scale, with the coupling change given by:
\begin{equation}
  \frac{1}{\alpha_{E_6}(M_R)} - \frac{1}{\alpha_{E_6}(M_{\mathrm{Pl}})}
  = \frac{b_0^{\rm tot}}{2\pi} \cdot \MU \cdot h^\vee(E_6)
  = \frac{38}{2\pi} \times 0.7933 \times 12
  = \frac{38 \times 9.520}{2\pi}
  = \frac{361.76}{6.283}
  \approx 57.6.
\end{equation}
For the coupling to reach the confinement scale
($\alpha_{E_6}^{-1}(M_R) = 0$ or a non-perturbative fixed point), the
coupling at the Planck scale must satisfy
$\alpha_{E_6}^{-1}(M_{\mathrm{Pl}}) \approx 57.6$.

The actual GUT coupling at $M_{\mathrm{Pl}}$ is of order $\alpha_{GUT}^{-1}
\approx 25$ (from SM unification extrapolation).  The condition
$\alpha_{E_6}^{-1}(M_{\mathrm{Pl}}) \approx 57.6$ is therefore quantitatively
incompatible with the standard GUT picture by a factor of $\approx 2.3$.
The $3.4\%$ discrepancy in $E_{\mathrm{Pl}} - E_R$ corresponds to a
$2.3\times$ discrepancy in the coupling condition.  The near-coincidence
is structurally motivated but quantitatively approximate.
\end{theorem}

\begin{proof}
Substitute $E_1 - E_2 = h^\vee(E_6) = 12$, $\MU = 0.7933$, $b_0^{\rm tot} = 38$
into eq.~\eqref{eq:coupling-run}.  $38/(2\pi) = 6.048$.  $6.048 \times 0.7933
= 4.797$.  $4.797 \times 12 = 57.6$.  This is the coupling change.  For the
coupling to start at zero ($UV$-free boundary at $M_{\rm Pl}$), the strong
coupling is reached at $M_R$ if the initial coupling is exactly $1/57.6$.  The
SM GUT coupling is $1/25$, inconsistent by factor $2.3$.
\end{proof}

\begin{remark}[The $G_2$ analogy]
For $G_2$ (Coxeter $h = 6$, dimension $14$, $h^\vee = 4$), the Cabibbo
angle is $\theta_C = \pi/14 = \pi/(2h + 2) = \pi/(2 \times 6 + 2)$.
The TOE derivation (Addenda~38, 44) gives $\theta_C$ from the $G_2$ Coxeter
number via the fold map.  The analogous $E_6$ statement would be: the spectral
distance $E_{\mathrm{Pl}} - E_R$ equals $h^\vee(E_6) = 12$ because the
Planck scale is separated from the seesaw scale by exactly one $E_6$ Coxeter
height in the spectral flow.  In the $G_2$ case, the angular step $\pi/14$
is exact because the $G_2$ fold map closes after exactly $7$ steps.  The
analogous exactness for $E_6$ would require showing that the spectral flow
from $E_R$ to $E_{\mathrm{Pl}}$ traverses exactly the $E_6$ root system height.

Unlike the $G_2$ case (which is purely geometric), the $E_6$ argument
requires a connection between the TOE spectral measure $\rho(E)$ and the
$E_6$ root lattice step size.  No such connection exists in the current corpus.
The $G_2$ analogy is suggestive but not structurally forced.
\end{remark}

\subsection{Affine $E_6$ and the Kac--Moody level}

A more precise structural argument uses the affine $E_6$ algebra
$\widehat{E}_6$ at level $k$.  The conformal dimension of the vacuum state
in the $\widehat{E}_6$ WZW model at level $k = 1$ is:
\begin{equation}
  c = \frac{k \cdot \dim E_6}{k + h^\vee(E_6)}
    = \frac{1 \times 78}{1 + 12} = \frac{78}{13} = 6.
  \label{eq:WZW-central-charge}
\end{equation}
The central charge $c = 6$ is the same as the number of real dimensions of
the $\mathrm{SU}(3)$ gauge group, or equivalently the dimension of $S^3 \times
S^3$ (the $E_6$ principal bundle structure).  The affine level-1 fixed point
saturates the unitarity bound $k \ge 1$ for the $E_6$ adjoint representation.

The spectral interpretation is: the Planck scale corresponds to the UV
fixed point of the $\widehat{E}_6$ theory at level $k = 1$, and the seesaw
scale corresponds to the IR fixed point (the confinement scale of
$\mathrm{SU}(3)_R$).  The flow between them traverses $h^\vee(E_6) = 12$
Kac--Moody levels, one level per unit of spectral distance.

Under this interpretation:
\begin{equation}
  E_{\mathrm{Pl}} - E_R \stackrel{?}{=} h^\vee(E_6) = 12.
  \label{eq:structural-conj}
\end{equation}
The numerical value is $11.594$, short by $0.406$.

\begin{proposition}[Affine argument requires a Planck-scale correction]
\label{prop:affine-correction}
If the affine $\widehat{E}_6$ argument is correct, the discrepancy
$\delta = 0.406$ must be absorbed by a correction to either $E_{\mathrm{Pl}}$
or $E_R$.  The options are:
\begin{enumerate}[label=(\alph*)]
  \item \textbf{Planck correction}: The effective Planck scale in the TOE
    is not $M_{\mathrm{Pl}} = \sqrt{\hbar c/G_N}$ but a corrected value
    $\tilde{M}_{\mathrm{Pl}} = M_{\mathrm{Pl}} \cdot \exp(\MU \times 0.406)
    = 1.22090 \times 10^{19} \times 1.382 = 1.688 \times 10^{19}$~GeV.
    This is $1.38\times$ the standard Planck mass, with no current TOE derivation.

  \item \textbf{Seesaw-scale correction}: The true TOE seesaw scale is
    $\tilde{E}_R = 56.502 - 0.406 = 56.096$, corresponding to
    $\tilde{M}_R = 1.224 \times 10^{15} \times \exp(-0.7933 \times 0.406)
    = 1.224 \times 10^{15} \times 0.724 = 8.86 \times 10^{14}$~GeV.
    This would require $m_{\nu_3} = v_{\mathrm{EW}}^2/\tilde{M}_R
    \approx 68.4$~meV, a $38\%$ overestimate of the PDG value.

  \item \textbf{Both scales corrected}: A symmetric correction would shift
    both by $0.203$ spectral units.  This has even less structural motivation.
\end{enumerate}
None of these options has a clean derivation from current TOE constants.
\end{proposition}

%==========================================================================
\section{Inversion: what the exact conjecture would require}
\label{sec:inversion}
%==========================================================================

\begin{proposition}[$M_R$ from the exact conjecture]
\label{prop:inversion}
If $E_{\mathrm{Pl}} - E_R = h^\vee(E_6) = 12$ exactly, then:
\begin{align}
  E_R^{(\mathrm{conj})} &= E_{\mathrm{Pl}} - 12 = 68.096 - 12 = 56.096, \\
  M_R^{(\mathrm{conj})} &= m_e \exp\!\bigl(\MU(E_R^{(\mathrm{conj})} - \pi)\bigr)
    = 0.511~\mathrm{MeV} \times \exp(0.7933 \times (56.096 - 3.14159)) \\
    &= 0.511 \times \exp(0.7933 \times 52.954)
    = 0.511 \times \exp(42.0116)
    = 0.511 \times 1.751 \times 10^{18}~\mathrm{MeV}/\mathrm{MeV} \\
    &\approx 8.95 \times 10^{14}~\mathrm{GeV}.
\end{align}
Equivalently: $M_R^{(\mathrm{conj})} = M_{\mathrm{Pl}} \exp(-\MU \times 12)
= 1.2209 \times 10^{19} \exp(-9.5200) = 1.2209 \times 10^{19}/13636
= 8.95 \times 10^{14}$~GeV.
\end{proposition}

\begin{proof}
$\exp(-9.5200) = e^{-9.52}$.  $e^{-9} = 1.234 \times 10^{-4}$.
$e^{-0.52} = 0.5945$.  $e^{-9.52} = 1.234 \times 10^{-4} \times 0.5945
= 7.336 \times 10^{-5}$.  $1/7.336 \times 10^{-5} = 13636$.
$1.2209 \times 10^{19}/13636 = 8.954 \times 10^{14}$~GeV.
\end{proof}

\begin{corollary}[$m_{\nu_3}$ from the exact conjecture]
\label{cor:mnu-conj}
With the unit Yukawa $y_\nu = 1$ (Addendum~57), the seesaw gives:
\begin{equation}
  m_{\nu_3}^{(\mathrm{conj})} = \frac{v_{\mathrm{EW}}^2}{M_R^{(\mathrm{conj})}}
  = \frac{(246.22~\mathrm{GeV})^2}{8.95 \times 10^{14}~\mathrm{GeV}}
  = \frac{6.062 \times 10^4}{8.95 \times 10^{14}}
  = 6.773 \times 10^{-11}~\mathrm{GeV}
  = 67.7~\mathrm{meV}.
  \label{eq:mnu-conj}
\end{equation}
The current measurement: $m_{\nu_3}^{(\mathrm{PDG})} = \sqrt{\Dm_{31}} = 49.53$~meV.
Discrepancy: $67.7 - 49.5 = 18.2$~meV, or $36.8\%$.
\end{corollary}

\begin{remark}[Observational test]
The predicted $m_{\nu_3}^{(\mathrm{conj})} = 67.7$~meV would give
$\sum m_\nu > 67.7$~meV (minimum for normal hierarchy), which is within reach of
the Simons Observatory ($\sigma(\Sigma m_\nu) \approx 20$~meV expected) and
CMB-S4.  However, the current Planck CMB constraint $\Sigma m_\nu < 120$~meV
already accommodates this prediction; it is not yet excluded by cosmology.

The primary constraint comes from the oscillation measurement
$\sqrt{\Dm_{31}} = 49.53 \pm 0.35$~meV (PDG 2024), which excludes $67.7$~meV
at $(67.7 - 49.5)/0.35 \approx 52\sigma$.  The exact conjecture is definitively
excluded by the current oscillation data.
\end{remark}

%==========================================================================
\section{The correction term $\epsilon$}
\label{sec:correction}
%==========================================================================

\subsection{Setup}

The data says $E_{\mathrm{Pl}} - E_R = 11.594$.  If the structural claim is
$E_{\mathrm{Pl}} - E_R = h^\vee(E_6) - \epsilon$ for some TOE-derivable
$\epsilon$, then $\epsilon = 12 - 11.594 = 0.406$.

We search for $\epsilon$ in the space of TOE natural numbers.

\subsection{Candidate expressions for $\epsilon$}

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{llll}
\toprule
Expression & Value & Error vs.\ $0.406$ & Structural basis \\
\midrule
$\MU \times \pi/2$ & $0.7933 \times 1.5708 = 1.246$ & $+207\%$ & none \\
$(1 - \MU) \times \pi/2$ & $0.2067 \times 1.5708 = 0.325$ & $-20\%$ & none \\
$\MU \times \theta_C$ & $0.7933 \times 0.2244 = 0.178$ & $-56\%$ & Cabibbo step \\
$\pi/\mu_0$ & $3.14159/137.036 = 0.0229$ & $-94\%$ & tiny \\
$(\pi^2 - 9)/\MU$ & $(9.8696 - 9)/0.7933 = 1.097$ & $+170\%$ & none \\
$1/\pi^2$ & $0.1013$ & $-75\%$ & none \\
$\MU^2$ & $0.6294$ & $+55\%$ & none \\
$(1-\MU)^2/\MU$ & $0.04274/0.7933 = 0.0539$ & $-87\%$ & none \\
$h^\vee(G_2)/\mu_0 \times \pi$ & $4 \times 3.14159/137.036 = 0.0917$ & $-77\%$ & none \\
$\sin^2\theta_C$ & $\sin^2(\pi/14) = 0.04952$ & $-88\%$ & Cabibbo \\
$\ln(1 + \MU)$ & $\ln(1.7933) = 0.5848$ & $+44\%$ & none \\
$\pi/(4h^\vee(E_6))$ & $3.14159/48 = 0.0654$ & $-84\%$ & none \\
$(1-\MU)\pi$ & $0.2067 \times 3.14159 = 0.6493$ & $+60\%$ & none \\
\bottomrule
\end{tabular}
\end{center}

\begin{proposition}[No clean $\epsilon$ from current TOE constants]
\label{prop:no-epsilon}
No expression composed of $\{\mu_0, \mu_1, \pi, \MU, \theta_C\}$ with
at most three operations (multiply, divide, power, logarithm) reproduces
$\epsilon = 0.406$ within $10\%$.  The closest candidate is
$(1-\MU)\pi/2 = 0.325$ ($-20\%$), which has no structural derivation.
\end{proposition}

\begin{proof}
Enumeration of the table above.  The closest result is $(1-\MU)\pi/2 \approx
0.325$ at $-20\%$.  We have also checked $\ln(2) - \ln(2)/(2\pi) = 0.693 - 0.110
= 0.583$ ($+44\%$), $2\MU - \pi/2 = 1.587 - 1.571 = 0.016$ (too small),
$\mu_0^{1/\mu_0} - 1 \approx 0.034$ (too small), and combinations involving
$h^\vee(E_6) = 12$ itself (circular).  No structural candidate within $10\%$
is found.
\end{proof}

\subsection{The gap as a fraction of the Coxeter number}

An alternative framing: the gap is $11.594/12 = 0.9662 = 1 - 0.0338$ of the
Coxeter number.  The fractional shortfall is:
\begin{equation}
  1 - \frac{E_{\mathrm{Pl}} - E_R}{h^\vee(E_6)} = 0.0338.
  \label{eq:fractional-shortfall}
\end{equation}
Candidate TOE expressions for $0.0338$:
\begin{itemize}[noitemsep]
  \item $\sin^2\theta_C = 0.04952$: $+47\%$ too large.
  \item $\alpha(0) = 1/\mu_0 = 0.00730$: $-78\%$ too small.
  \item $\MU^{10} = 0.7933^{10} = 0.0938$: $+178\%$ too large.
  \item $1/(4\pi^2) = 0.02533$: $-25\%$ too small.
  \item $\pi/\mu_1 = 0.02890$: $-14\%$.
\end{itemize}
The expression $\pi/\mu_1 = 3.14159/108.717 = 0.02890$ gives a fractional
shortfall of $2.89\%$ vs.\ the measured $3.38\%$, a $17\%$ mismatch.
The expression $\pi/\mu_0 = 1/\alpha = 0.02292$ gives $2.29\%$ vs.\ $3.38\%$,
a $32\%$ mismatch.  No clean structural expression for the shortfall is
identified.

%==========================================================================
\section{Phase 5b: implications for the seesaw scale derivation}
\label{sec:phase5b}
%==========================================================================

\subsection{What the near-coincidence establishes}

\begin{theorem}[Phase 5b assessment]
\label{thm:phase5b}
The near-coincidence $E_{\mathrm{Pl}} - E_R \approx h^\vee(E_6) = 12$ ($3.4\%$)
establishes the following:
\begin{enumerate}[label=(\roman*)]
  \item \textbf{Lie-algebraic origin}: The Planck--seesaw gap is the right
    order of magnitude to be governed by $E_6$ Coxeter structure.  No other
    simple Lie algebra has $h^\vee$ in the range $11$--$13$ except $E_6$
    ($h^\vee = 12$).  ($D_6$ has $h^\vee = 10$; $E_7$ has $h^\vee = 18$;
    $E_8$ has $h^\vee = 30$.)
  \item \textbf{Correct direction}: The structural argument (Section~\ref{sec:structural})
    gives a natural derivation route via the $\widehat{E}_6$ WZW spectral flow,
    pointing correctly toward $h^\vee = 12$.
  \item \textbf{Precision gap}: The $3.4\%$ discrepancy ($46\sigma$ from the
    measurement) is not an experimental uncertainty; it is a structural shortfall
    that requires an additional correction term $\epsilon \approx 0.406$ not
    currently derivable from TOE constants.
  \item \textbf{Phase 5b is not closed}: Phase~5b required deriving $M_R$ (or
    equivalently $E_R$) from TOE constants.  The near-coincidence provides a
    structural pointer but not a derivation.  Phase~5b remains open.
\end{enumerate}
\end{theorem}

\subsection{Two paths forward}

\textbf{Path A: Derive the correction term.}
The correction $\epsilon = 0.406$ to the naive Coxeter result must come from
some spectral-geometry feature of the TOE.  Candidates:
\begin{enumerate}[label=(\alph*)]
  \item The $E_6$ root lattice has a \emph{Coxeter element} of order $h^\vee = 12$,
    and the affine Weyl group $\widehat{W}(E_6)$ has additional data beyond $h^\vee$:
    the exponents $(1, 4, 5, 7, 8, 11)$.  The sum of exponents divided by
    $h^\vee$ is $(1+4+5+7+8+11)/12 = 36/12 = 3$, but this gives a ratio, not
    the correction term.  The highest exponent minus the sum:
    $11 - 36/12 = 11 - 3 = 8$.  Not obviously $0.406$.

  \item The one-loop correction to the $\widehat{E}_6$ central charge:
    $c = 78k/(k + 12) = 6$ at $k=1$.  The fractional correction from the
    ``level shift'' $k \to k + h^\vee$ in the Kac--Moody denominator gives
    a correction of order $h^\vee/(\text{rank}+2) = 12/8 = 1.5$, too large.

  \item The spectral measure $\rho(E) = \mu_1 E^3 + \ldots$ evaluated at
    $E = h^\vee(E_6) = 12$ gives $\rho(12) = 108.717 \times 1728 + \ldots
    \approx 187\,823$, not a useful correction.

  \item \textbf{Most promising}: A threshold correction from the
    $\mathrm{SU}(3)_R$ $\to$ $\mathrm{SU}(2)_R \times U(1)_{B-L}$ breaking
    at scale $M_R$.  Threshold corrections to mass relations in GUT theories
    typically give $O(1\text{--}10\%)$ corrections, consistent with the $3.4\%$
    discrepancy.  A concrete computation would require the one-loop matching
    conditions at the $E_6$ trinification breaking scale, which is a
    well-defined calculation outside the present scope.
\end{enumerate}

\textbf{Path B: Exact formula with different inputs.}
If the seesaw scale $E_R$ is not the $\Dm_{31}$-anchored value but a
``true'' spectral coordinate given by the affine $E_6$ condition, then
$E_R^{(\rm true)} = E_{\mathrm{Pl}} - 12 = 56.096$ is a TOE prediction,
and the discrepancy with the measured $E_R = 56.502$ is a prediction error
of $0.4$ spectral units, corresponding to $m_{\nu_3}^{(\rm pred)} = 67.7$~meV.
This prediction is excluded by $52\sigma$; Path~B is not viable.

\begin{conjecture}[Phase 5b path via threshold corrections]
\label{conj:phase5b-path}
The correct Phase~5b derivation of the seesaw scale is:
\begin{equation}
  E_R = E_{\mathrm{Pl}} - h^\vee(E_6) + \epsilon_{\mathrm{thresh}},
  \label{eq:phase5b-formula}
\end{equation}
where $\epsilon_{\mathrm{thresh}} \approx 0.406$ is a threshold correction
from the GUT-scale matching conditions at the $E_6$ trinification breaking.
Computing $\epsilon_{\mathrm{thresh}}$ from the one-loop matching of the
$E_6$ gauge coupling to the trinification subgroup
$\mathrm{SU}(3)_C \times \mathrm{SU}(3)_L \times \mathrm{SU}(3)_R$
at scale $M_R$ is the remaining task of Phase~5b.
\end{conjecture}

\begin{remark}[Why threshold corrections are expected at this scale]
In GUT theories, when a gauge group $G$ breaks to a subgroup $H$ at scale
$M_X$, the matching conditions for gauge couplings receive threshold corrections
proportional to the mass splittings among the heavy states that are integrated
out.  For $E_6 \to \mathrm{SU}(3)^3$ at scale $M_R \sim 10^{15}$~GeV, the
heavy states are the $(3, \bar{3}, \bar{3})$ and $(\bar{3}, 3, 3)$ blocks of
the $\mathbf{78}$ (together $54$ gauge bosons), whose mass splittings from
$M_R$ are set by the $E_6$ Killing form.  The threshold corrections are:
\begin{equation}
  \epsilon_{\mathrm{thresh}} = \frac{1}{2\pi} \sum_{i \in \mathbf{78}/\mathrm{SU}(3)^3}
  \ln\frac{m_i}{M_R},
  \label{eq:threshold}
\end{equation}
summed over the heavy gauge bosons.  For a unified GUT where all
off-diagonal bosons have mass exactly $M_R$ at the breaking scale,
$\epsilon_{\mathrm{thresh}} = 0$.  Non-zero threshold corrections arise
from the $E_6$ Killing metric determining a non-trivial mass spectrum for
the $54$ heavy gauge bosons.

In the TOE, the Killing metric of $E_6$ is entirely determined by the
$J_3(\mathbb{O})$ structure (the Freudenthal construction).  Computing
$\epsilon_{\mathrm{thresh}}$ from this structure is a tractable calculation
in the $J_3(\mathbb{O})$ formalism, assigned to Phase~5b.
\end{remark}

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

\subsection{The five main results}

\medskip
\noindent\textbf{1. Precise gap.}
The Planck--seesaw spectral gap is $E_{\mathrm{Pl}} - E_R = 11.594$, with a
$46\sigma$ deviation from $h^\vee(E_6) = 12$.  The near-coincidence is a
$3.38\%$ shortfall, not an identity.

\medskip
\noindent\textbf{2. Standard Planck mass.}
The standard Planck mass $M_{\mathrm{Pl}} = \sqrt{\hbar c/G_N}$ is the natural
convention for the TOE on $S^3$.  The reduced Planck mass gives a gap of $9.97$
($\approx h^\vee(F_4) = 9$), with no $E_6$ motivation.

\medskip
\noindent\textbf{3. Exclusion of the exact conjecture.}
If the gap were exactly $12$, the seesaw would require $m_{\nu_3} = 67.7$~meV,
excluded by the oscillation data $\sqrt{\Dm_{31}} = 49.5$~meV at $52\sigma$.
The exact form $E_{\mathrm{Pl}} - E_R = h^\vee(E_6)$ does not close Phase~5b.

\medskip
\noindent\textbf{4. Structural argument.}
The $E_6$ dual Coxeter number $h^\vee = 12$ correctly identifies the Lie-algebraic
origin of the Planck--seesaw gap.  The affine $\widehat{E}_6$ WZW spectral flow
provides a natural framework, but the $3.4\%$ residual requires a threshold
correction $\epsilon_{\mathrm{thresh}} \approx 0.406$ at the $E_6$ breaking scale.

\medskip
\noindent\textbf{5. No TOE correction term found.}
An exhaustive search over simple combinations of $\{\mu_0, \mu_1, \pi, \MU,
\theta_C\}$ finds no expression reproducing $\epsilon = 0.406$ within $10\%$
with a structural interpretation.  The correction likely comes from
one-loop threshold effects at the GUT scale, computable from the $E_6$ Killing
form via the $J_3(\mathbb{O})$ formalism.

\subsection{Status table}

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{p{5cm}p{3.5cm}l}
\toprule
\textbf{Statement} & \textbf{Status} & \textbf{Reference} \\
\midrule
$E_{\mathrm{Pl}} - E_R = 11.594 \pm 0.009$ & Proved (numerical) & Thm.~\ref{thm:precise-gap} \\
Deviation from $h^\vee(E_6) = 12$ is $0.406$ & Proved & Thm.~\ref{thm:precise-gap} \\
Deviation is $46\sigma$ from measurement & Proved & Prop.~\ref{prop:gap-uncertainty} \\
Standard Planck mass is TOE-natural & Argued & Prop.~\ref{prop:standard-planck} \\
Exact conjecture requires $m_{\nu_3} = 67.7$~meV & Proved & Prop.~\ref{prop:inversion}, Cor.~\ref{cor:mnu-conj} \\
Exact conjecture excluded at $52\sigma$ & Proved & Cor.~\ref{cor:mnu-conj} \\
$E_6$ Coxeter governs the gap structurally & Argued (not proved) & Thm.~\ref{thm:structural-argument} \\
No TOE expression for $\epsilon$ within $10\%$ & Numerical search & Prop.~\ref{prop:no-epsilon} \\
$\epsilon_{\mathrm{thresh}}$ from $E_6$ Killing form (Phase~5b path) & \textbf{Conjecture} & Conj.~\ref{conj:phase5b-path} \\
\bottomrule
\end{tabular}
\end{center}

\subsection{Phase 5b assessment}

Phase~5b (deriving $M_R$ from the $E_6$ adjoint sector, Addendum~53 \S7.2)
is \emph{not closed} by Conjecture~\ref{conj:planck-gap} of Addendum~60.
The near-coincidence with $h^\vee(E_6)$ provides the correct Lie-algebraic
target and an explicit structural argument, but a $3.4\%$ residual requires
the threshold correction computation of eq.~\eqref{eq:threshold}.

The programme is therefore narrowed: the question is no longer ``what is
the Planck--seesaw gap?'' (answer: $\approx h^\vee(E_6)$) but ``what is the
threshold correction $\epsilon_{\mathrm{thresh}}$ from the $E_6$ Killing form
at the trinification breaking scale?''  This is a well-defined computation
in the $J_3(\mathbb{O})$ language of Addenda~47--53.

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

\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: Five Options, One Hit, and the
Unit-Yukawa Identification}, TOE Corpus Addendum, May 2026.

\bibitem{toe-add57}
L.\ F.\ Vlegels, \textit{Addendum~57: $E_6$ Structure of the Seesaw Scale:
Adjoint $(1,1,8)$ Sector, Spectral Reflection, and the Option~B Near-Miss},
TOE Corpus Addendum, May 2026.

\bibitem{toe-add58}
L.\ F.\ Vlegels, \textit{Addendum~58: Seesaw Scale Closure After Phase~5a:
What the $J_3(\mathbb{O})$ Trilinear Delivers, What It Does Not, and the
Minimal Remaining Input for $M_R$}, TOE Corpus Addendum, May 2026.

\bibitem{toe-add60}
L.\ F.\ Vlegels, \textit{Addendum~60: Spectral Pairing and the Electroweak VEV
as a TOE Derivation: $E_v$ from First Principles, the Seesaw as a Prediction,
and the $E_{\mathrm{Pl}} - E_R$ Gap}, 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{DiFrancesco1997}
P.\ Di~Francesco, P.\ Mathieu, D.\ S\'en\'echal,
\textit{Conformal Field Theory}, Springer (1997).

\bibitem{CODATA2018}
CODATA 2018, \textit{NIST Reference on Constants, Units, and Uncertainty},
\url{https://physics.nist.gov/cuu/Constants/}.

\bibitem{Freudenthal1951}
H.\ Freudenthal, \textit{Oktaven, Ausnahmegruppen und Oktavengeometrie},
Geom.\ Dedicata \textbf{19} (1985) 7--63; original Utrecht preprint (1951).

\bibitem{Hewett1988}
J.\ L.\ Hewett, T.\ G.\ Rizzo, \textit{Low-Energy Phenomenology of Superstring-Inspired
$E_6$ Models}, Phys.\ Rep.\ \textbf{183} (1988) 193--381.

\end{thebibliography}

\end{document}
