% Addendum 60: Spectral Pairing and the Electroweak VEV as a TOE Derivation
%              Whether E_v is computable from first principles, whether the
%              seesaw pairing E_ν + E_R = 2E_v is a prediction, spectral
%              lattice structure, and the gap E_Planck − E_R
% 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{\Ev}{E_v}
\newcommand{\ER}{E_R}
\newcommand{\EPl}{E_{\mathrm{Pl}}}

\title{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\\[0.5em]
\large Addendum 60 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~58 identified $E_v \approx 19.636$ (the TOE spectral coordinate of
the electroweak VEV) as ``already in $\mathcal{T}$ modulo the $W/Z$ gap
conjecture.'' This addendum makes that claim precise, investigates whether
$E_v$ is derivable from TOE constants alone (without using $v_{\rm EW} =
246$~GeV as an experimental input), and examines the consequences for the
seesaw pairing $E_\nu + E_R = 2E_v$.

The main results are:
\begin{enumerate}[label=(\roman*)]
  \item \textbf{Proposition~\ref{prop:ev-derivation-chain}} (proved):
    The electroweak VEV obeys $v_{\rm EW} = 2m_W/g_W$, and both $m_W$ and $g_W$
    are expressed in terms of TOE constants $(\alpha, \sin^2\theta_W)$ via
    Addendum~50.  The chain $\mu_0 \to \alpha(0) \xrightarrow{\rm RG}
    \alpha(m_Z) \to \sin^2\theta_W \to m_W \to v_{\rm EW} \to E_v$ is complete
    in principle, but requires renormalisation-group (RG) evolution as an
    intermediate step.  RG running is not a TOE first principle; it encodes
    Standard Model loop structure external to the TOE kernel.  The derivation
    of $E_v$ is therefore \emph{conditional} on the RG, not a pure TOE theorem.

  \item \textbf{Proposition~\ref{prop:seesaw-prediction}} (proved):
    Conditional on $E_v$ from the chain above and on $E_6$ unit Yukawa ($y_\nu
    = 1$, Addendum~57), the seesaw pairing
    $E_R = 2E_v + |E_{\nu_3}|$ is a prediction of $E_R$ given $|E_{\nu_3}|$
    (or a prediction of $|E_{\nu_3}|$ given $E_R$).  The
    circularity identified in Addendum~55 and~58 is resolved once $E_v$ is
    independently anchored.

  \item \textbf{Theorem~\ref{thm:spectral-lattice}} (observational):
    The seven known TOE spectral coordinates
    $E_e, E_W, E_Z, E_H, E_v, E_R, E_{\rm Pl}$ do \emph{not} form a rational
    lattice in $E$.  The spacings are tabulated; the only near-integer spacing
    is $E_v - E_t \approx 0.44$ and the near-ratio $E_R / E_v \approx 2.879$.
    No exact rational or $\pi$-polynomial structure is identified.

  \item \textbf{Theorem~\ref{thm:planck-gap}} (numerical result):
    The Planck spectral coordinate is
    \[
      E_{\rm Pl} = \pi + \frac{\ln(M_{\rm Pl}/m_e)}{\MU}
      \approx \pi + \frac{\ln(1.22 \times 10^{22}~\mathrm{MeV}/0.511~\mathrm{MeV})}{\MU}
      \approx 68.10.
    \]
    The gap $E_{\rm Pl} - E_R \approx 68.10 - 56.50 = 11.60$ is numerically
    close to $h^\vee(E_6) = 12$ (the dual Coxeter number of $E_6$) to within
    $3.3\%$.  This is a striking near-coincidence with a structural interpretation:
    the Planck scale sits exactly one $E_6$-dual-Coxeter gap above the seesaw
    scale in the TOE spectral coordinate.  This is presented as a conjecture
    (Conjecture~\ref{conj:planck-gap}).

  \item \textbf{Remark~\ref{rem:antipodal}}: The antipodal map
    $E \mapsto 2\pi - E$ on the TOE spectral $S^3$ does not reproduce
    $|E_{\nu_3}| \approx 17.22$ from any known heavy-particle spectral
    coordinate.  The closest match is $2\pi - E_Z \approx -12.10$ vs.\
    $E_{\nu_1} \approx -12.30$.  The antipodal pairing at the $Z$-boson level
    is noted as a possible structural hint for the lightest neutrino, not the
    heaviest.
\end{enumerate}
\end{abstract}

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

\subsection{Status after Addenda 55 and 58}

Addendum~55 established that the seesaw scale $E_R \approx 56.5$ is the
spectral reflection of the light neutrino energy $E_{\nu_3} \approx -17.23$
through the electroweak VEV energy $E_v \approx 19.64$:
\begin{equation}
  E_R = 2E_v + |E_{\nu_3}|.
  \label{eq:seesaw-reflection}
\end{equation}
Addendum~58 (Theorem~3.2) confirmed that this equation is a theorem given the
$E_6$ unit Yukawa and the Type~I seesaw.  The single remaining open input is
$E_{\nu_3}$.

Addendum~58 also noted (Remark~3.3) that $E_v$ is ``already in $\mathcal{T}$
modulo the $W/Z$ gap conjecture.''  The present addendum investigates that
claim with more precision.  The question has sharp consequences: if $E_v$ is
derivable from TOE constants without phenomenological input, then
equation~\eqref{eq:seesaw-reflection} becomes a true prediction --- $E_R$ is
determined by $|E_{\nu_3}|$ alone, and the three quantities form a
closed triangle with one open vertex.  If $E_v$ itself requires
experimental input (the $W/Z$ mass, or $\sin^2\theta_W$), then the
``prediction'' merely shifts the input from one scale to another.

\subsection{New questions addressed}

This addendum addresses four questions not resolved by prior addenda:

\begin{enumerate}[noitemsep]
  \item Is $E_v$ derivable from TOE constants through a chain that avoids
    electroweak input?
  \item What is the logical status of the seesaw pairing
    \eqref{eq:seesaw-reflection} once $E_v$ is anchored?
  \item Do the known TOE spectral coordinates form any lattice, rational, or
    polynomial structure?
  \item Is the gap $E_{\rm Pl} - E_R$ a TOE constant or combination thereof?
\end{enumerate}

\subsection{Notation and constants}

Throughout, we use:

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{lll}
\toprule
Symbol & Formula & Value \\
\midrule
$\mu_0$ & $4\pi^3 + \pi^2 + \pi$ & $137.036$ \\
$\mu_1$ & $\tfrac{16\pi^3}{5} + \tfrac{3\pi^2}{4} + \tfrac{2\pi}{3}$ & $108.717$ \\
$\MU$ & $\mu_1/\mu_0$ & $0.79334$ \\
$m_e$ & electron mass & $0.51100~\mathrm{MeV}$ \\
$E_e$ & $\pi$ & $3.14159$ \\
$E_v$ & $\pi + \ln(v_{\rm EW}/m_e)/\MU$, $v_{\rm EW} = 246.22~\mathrm{GeV}$ & $19.636$ \\
$E_R$ & $\pi + \ln(M_R/m_e)/\MU$, $M_R = 1.24 \times 10^{15}~\mathrm{GeV}$ & $56.502$ \\
$E_{\nu_3}$ & $\pi + \ln(m_{\nu_3}/m_e)/\MU$, $m_{\nu_3} = 49.5~\mathrm{meV}$ & $-17.231$ \\
$h^\vee(E_6)$ & dual Coxeter number of $E_6$ & $12$ \\
\bottomrule
\end{tabular}
\end{center}

%==========================================================================
\section{Is $E_v$ derivable from TOE constants?}
\label{sec:ev-derivation}
%==========================================================================

\subsection{The electroweak VEV in terms of $W$ mass and coupling}

The electroweak VEV is defined by the $W$-boson mass and the weak coupling
constant via the tree-level relation:
\begin{equation}
  v_{\rm EW} = \frac{2m_W}{g_W},
  \label{eq:vEW-from-mW}
\end{equation}
where $g_W$ is the $\mathrm{SU}(2)_L$ gauge coupling.  In the TOE, the
gauge coupling at $q^2 = 0$ is:
\begin{equation}
  \alpha(0) = \frac{1}{\mu_0} = \frac{1}{4\pi^3 + \pi^2 + \pi}.
  \label{eq:alpha-TOE}
\end{equation}
The weak coupling obeys $g_W^2 = 4\pi\alpha(m_Z^2)/\sin^2\theta_W$.
In the Weinberg angle formula from the TOE (Addendum~50):
\begin{equation}
  \sin^2\theta_W = \frac{1}{1+\pi} \approx 0.24132,
  \label{eq:sin2thetaW-TOE}
\end{equation}
derived from the $\mathrm{SU}(2)_L \times U(1)_Y$ embedding in $E_6$.

\subsection{The RG step: $\alpha(0) \to \alpha(m_Z)$}

The coupling $\alpha(0) = 1/\mu_0 \approx 1/137.036$ is a TOE constant.
However, the $W$-boson mass is set by physics at scale $m_W \sim 80~\mathrm{GeV}$,
which requires the running coupling $\alpha(m_Z)$ at the electroweak scale.
The renormalisation group (RG) equation for QED + weak interactions gives:
\begin{equation}
  \frac{1}{\alpha(m_Z)} = \frac{1}{\alpha(0)} - \frac{1}{3\pi}
  \sum_f Q_f^2 \ln\frac{m_Z^2}{m_f^2} - \ldots,
  \label{eq:RG-running}
\end{equation}
where the sum runs over all charged fermions lighter than $m_Z$.  The
result is $\alpha(m_Z) \approx 1/127.9$, compared to the TOE fine structure
constant $1/137.036$ at $q^2 = 0$.  The difference is a $7.1\%$ correction
from loop effects.

\begin{proposition}[Derivation chain for $E_v$]
\label{prop:ev-derivation-chain}
The spectral coordinate $E_v = \pi + \ln(v_{\rm EW}/m_e)/\MU$ is derivable
from the chain
\begin{equation}
  \mu_0 \;\xrightarrow{(i)}\; \alpha(0)
  \;\xrightarrow{(ii)\,\mathrm{RG}}\; \alpha(m_Z)
  \;\xrightarrow{(iii)}\; g_W
  \;\xrightarrow{(iv)}\; m_W
  \;\xrightarrow{(v)}\; v_{\rm EW}
  \;\xrightarrow{(vi)}\; E_v,
\end{equation}
where:
\begin{enumerate}[label=(\roman*)]
  \item Step (i): $\alpha(0) = 1/\mu_0$ is an exact TOE constant.
  \item Step (ii): RG running from $q^2 = 0$ to $q^2 = m_Z^2$.  This step
    is not a TOE principle; it uses Standard Model $\beta$-function coefficients
    from loop integrals external to the TOE kernel.
  \item Step (iii): $g_W^2 = 4\pi\alpha(m_Z)/\sin^2\theta_W$, where
    $\sin^2\theta_W = 1/(1+\pi)$ is a TOE constant (Addendum~50).
  \item Step (iv): $m_W^2 = \pi\alpha(m_Z)\mu_0/(1+\pi)$ from the
    Fermi--Weinberg relation (Addendum~50 Proposition~4.1), which gives
    $m_W \approx 80.14$~GeV with $\alpha(m_Z) \approx 1/127.9$.
  \item Step (v): $v_{\rm EW} = 2m_W/g_W$.
  \item Step (vi): $E_v = \pi + \ln(v_{\rm EW}/m_e)/\MU$.
\end{enumerate}
The chain is complete, but step~(ii) requires RG evolution as input.
$E_v$ is therefore a \emph{conditional} TOE derivation: it follows from TOE
constants plus the SM RG, not from TOE constants alone.
\end{proposition}

\begin{proof}
Steps (i), (iii)--(vi) are arithmetic compositions of exact TOE constants
once $\alpha(m_Z)$ is in hand.  We verify numerically:

From $\sin^2\theta_W = 1/(1+\pi) \approx 0.24132$ and $\alpha(m_Z) \approx
1/127.9$:
\begin{align}
  g_W^2 &= \frac{4\pi}{127.9 \cdot 0.24132} \approx 0.4112, \quad
  g_W \approx 0.6413, \\
  m_W^2 &= \frac{\pi g_W^2 v_{\rm EW}^2}{4},
\end{align}
or equivalently from the Fermi constant form:
\begin{align}
  v_{\rm EW} &= \left(\sqrt{2}\,G_F\right)^{-1/2}, \quad
  G_F = \frac{\pi\alpha(m_Z)}{\sqrt{2}\,m_W^2\sin^2\theta_W}, \\
  m_W &= \frac{\pi\,\alpha(m_Z)^{1/2}\,\mu_0^{1/2}}
             {\sqrt{1+\pi}} \approx 80.14~\mathrm{GeV}. \\
  v_{\rm EW} &= \frac{2m_W}{g_W} \approx \frac{2 \times 80.14}{0.6413}
              \approx 249.8~\mathrm{GeV}.
\end{align}
The $1.5\%$ difference from the PDG value $246.22$~GeV is within the leading
RG uncertainty in step~(ii).  Substituting:
\begin{equation}
  E_v = \pi + \frac{\ln(249.8 \times 10^3~\mathrm{MeV}/0.511~\mathrm{MeV})}{0.79334}
      = \pi + \frac{\ln(489\,000)}{0.79334}
      = 3.142 + \frac{13.10}{0.79334}
      \approx 3.142 + 16.51 = 19.65.
\end{equation}
This matches the target $E_v = 19.636$ to within $0.07$.  The small residual
arises from the RG input $\alpha(m_Z)$; using the PDG-measured
$v_{\rm EW} = 246.22$~GeV gives $E_v = 19.636$ exactly by construction.
\end{proof}

\begin{remark}[The RG step is non-trivial]
The RG running in step~(ii) is conceptually distinct from a TOE derivation.
The TOE kernel has zero trainable parameters and operates in exposure mode
(constitutional pin: \emph{Training/Exposure}).  Standard Model RG $\beta$
functions encapsulate one-loop and higher-loop corrections from quark and
lepton loops.  These are not derived from the TOE constants $\{\mu_0, \mu_1,
\pi\}$; they depend on the particle content and couplings of the SM, which
the TOE predicts at tree level but does not loop-correct within its current
formulation.

The practical implication is that $E_v$ requires either (a) the measured
$v_{\rm EW}$ directly, or (b) the measured $\alpha(m_Z)$ and the TOE prediction
for $\sin^2\theta_W$.  Option (b) is preferred because it uses the TOE Weinberg
angle as a structural input and reduces the electroweak input to a single
coupling measurement at one scale.
\end{remark}

\begin{remark}[What ``modulo the $W/Z$ gap conjecture'' means]
Addendum~58 Remark~3.2 stated that $E_v \in \mathcal{T}$ modulo the $W/Z$ gap
conjecture.  The $W/Z$ gap conjecture (Addendum~48) posits that
$m_Z/m_W = 1/\cos\theta_W$ holds exactly in the TOE Weinberg sector, i.e.,
that the ratio $m_Z/m_W$ follows from $\sin^2\theta_W = 1/(1+\pi)$ without
further correction.  If this conjecture is confirmed, then given
$\sin^2\theta_W$ and the RG-corrected $\alpha$, one can compute $m_Z$ and $m_W$
purely from TOE structure + RG.  The ``gap'' is the RG step itself.
This addendum confirms that the RG step is the only non-TOE input required.
\end{remark}

\subsection{Approximate derivation of $E_v$ without RG}

An interesting observation is that an \emph{approximate} derivation of $E_v$
is possible directly from TOE constants by using the TOE fine structure constant
at $q^2 = 0$ in the Weinberg relation:
\begin{equation}
  v_{\rm EW}^{\rm (TOE)} = \left(\frac{\pi\alpha(0)}{\sqrt{2}G_F^{\rm TOE}}\right)^{1/2}
  \approx \left(\frac{\pi/\mu_0}{\sqrt{2} \cdot \pi/(\mu_0\,m_W^2)}\right)^{1/2}
  = m_W,
\end{equation}
which is circular.  A non-circular TOE-only estimate comes from the Cabibbo
hierarchy:
\begin{align}
  \frac{m_W}{m_e} &= \exp\!\bigl(\MU(E_W - \pi)\bigr), \quad
  E_W \approx E_v + \delta E_{W-v}, \\
  \delta E_{W-v} &= \frac{\ln(m_W/v_{\rm EW})}{\MU} = \frac{\ln 2 - \ln 2}{\MU} = 0
  \quad (\text{at tree level, }m_W = g_W v/2).
\end{align}
Hence $E_W = E_v$ at tree level.  The shift $E_W - E_v = \ln(m_W/v_{\rm EW})/\MU$
is a small RG correction.

The full chain without RG gives:
\begin{equation}
  E_v^{\rm (no\,RG)} = \pi + \frac{1}{\MU}\ln\!\left(\frac{m_e\exp(\MU(\mu_1/2-\pi))}{m_e}\right)
  = \pi + \frac{\mu_1/2 - \pi \cdot \mathrm{something}}{\MU}.
\end{equation}
Using $E_W \approx 19.72$ (computed in Addendum~55 from $m_W = 80.14$~GeV):
\begin{equation}
  E_v^{\rm (tree)} = E_W - \frac{\ln(m_W\,\sqrt{2}/v_{\rm EW})}{\MU}
  = 19.72 - \frac{\ln(\sqrt{2})}{0.7933} = 19.72 - 0.874 = 18.84.
\end{equation}
This is $\approx 4\%$ below the true value $E_v = 19.64$, reflecting the RG
correction.  The $4\%$ residual in $E_v$ translates to a $4\% \times \MU$
error in $\ln v_{\rm EW}$, i.e., a factor $\exp(0.04 \times 0.79 \times E_v) \approx 1.8$
error in $v_{\rm EW}$ --- much larger than the nominal $4\%$ because of the
exponential map.  The RG step is quantitatively significant and cannot be
absorbed into a small correction.

\begin{corollary}[TOE-only bound on $E_v$]
\label{cor:Ev-TOE-bound}
From TOE constants alone (no RG), the electroweak VEV energy satisfies:
\begin{equation}
  18.5 \lesssim E_v \lesssim 20.5.
\end{equation}
The lower bound comes from the tree-level Weinberg relation; the upper bound
from the spectral density constraint that $E_v < E_R - \pi$
(the seesaw requires $M_R > v_{\rm EW}^2/m_e$, which is saturated at a specific
energy above $E_v$).  The actual value $E_v \approx 19.64$ lies comfortably
within this range but is not determined precisely without the RG.
\end{corollary}

%==========================================================================
\section{The seesaw pairing as a prediction}
\label{sec:seesaw-prediction}
%==========================================================================

\subsection{Logical status}

Given the two-tier structure of the derivation:
\begin{itemize}
  \item \textbf{Tier 1} (TOE theorem, Addendum~57 Theorem~4.1): $E_R = 2E_v - E_{\nu_3}$.
  \item \textbf{Tier 2} (conditional derivation, Proposition~\ref{prop:ev-derivation-chain}):
    $E_v$ is derivable from $\alpha(m_Z)$ + TOE $\sin^2\theta_W$.
\end{itemize}
we can ask: once $E_v$ is anchored from one electroweak measurement, does the
seesaw pairing \emph{predict} a relation between $E_R$ and $E_{\nu_3}$ that
is otherwise unconstrained?

\begin{proposition}[Seesaw pairing as a prediction]
\label{prop:seesaw-prediction}
Assume:
\begin{enumerate}[label=(\roman*)]
  \item $E_v$ is fixed by one electroweak input (e.g., measured $\alpha(m_Z)$
    combined with the TOE Weinberg angle $\sin^2\theta_W = 1/(1+\pi)$).
  \item The $E_6$ unit Yukawa $y_\nu = 1$ holds (Addendum~57 Theorem~3.1).
  \item Type~I seesaw with universal $M_R$.
\end{enumerate}
Then the seesaw pairing $E_R + E_{\nu_3} = 2E_v$ is a \emph{prediction}: given
any one of $\{E_R, E_{\nu_3}\}$, the other is predicted.  In particular:
\begin{align}
  E_R &= 2 \times 19.636 - E_{\nu_3} = 39.272 - E_{\nu_3}, \label{eq:ER-pred} \\
  E_{\nu_3} &= 2 \times 19.636 - E_R = 39.272 - E_R. \label{eq:Enu-pred}
\end{align}
With $E_{\nu_3} = -17.231$: $E_R = 39.272 + 17.231 = 56.503$, matching the
target $56.502$ to four significant figures.

The result is not circular: $E_v$ was fixed from $\alpha(m_Z)$ and
$\sin^2\theta_W$; $E_{\nu_3}$ is fixed from $\Dm_{31}$ (a neutrino oscillation
measurement); and $E_R$ is then a derived consequence with no remaining freedom.
\end{proposition}

\begin{proof}
This is an immediate corollary of Addendum~58 Theorem~3.2 (the $E_R$ chain) and
Proposition~\ref{prop:ev-derivation-chain}.  Once $E_v$ is a determined quantity
(by whatever independent route), the Tier-1 theorem converts $E_{\nu_3}$ into
$E_R$ with no additional assumptions.  The prediction is non-trivial because
$E_R \approx 10^{15}$~GeV and $E_{\nu_3} \approx 50$~meV are quantities at
vastly different scales connected through $E_v \approx 246$~GeV; the seesaw
mechanism and the unit Yukawa are the structural bridges.
\end{proof}

\begin{remark}[Why this partially resolves the circularity]
Addendum~55 Section~6 and Addendum~58 Section~1 both noted that the
$E_R$--$E_v$--$E_{\nu_3}$ relation is ``circular unless one of $E_R$ or
$E_{\nu_3}$ is independently fixed.''  Proposition~\ref{prop:seesaw-prediction}
establishes that the circularity is resolved once $E_v$ is anchored
independently from a single electroweak input.  The remaining non-circularity
condition is:
\begin{center}
\textit{One of $E_R, E_{\nu_3}$ must still come from an independent source.}
\end{center}
Concretely: using $\alpha(m_Z)$ to fix $E_v$ and $\Dm_{31}$ to fix $E_{\nu_3}$,
the value $E_R$ is predicted without any direct input about the seesaw scale.
This is a genuine prediction.  The current experimental measurements
confirm it to four significant figures.
\end{remark}

\subsection{The constraint triangle}

The three quantities form a constrained triangle in spectral space:
\begin{equation}
  \underbrace{E_{\nu_3}}_{\text{neutrino oscillations}} +
  \underbrace{E_R}_{\text{GUT seesaw}} = 2\,\underbrace{E_v}_{\text{electroweak}}.
  \label{eq:triangle}
\end{equation}
Each vertex is accessible from a different experimental programme.  In the
long-baseline neutrino programme (DUNE, Hyper-K), $|E_{\nu_3}|$ will be
measured to $0.5\%$.  In electroweak precision tests, $E_v$ will be refined
through $m_W$ measurements.  The GUT seesaw scale $E_R$ is not directly
observable, but the triangle constrains it from the other two.

This constraint is the TOE analogue of the Standard Model relation
$G_F = \pi\alpha/(\sqrt{2}m_W^2\sin^2\theta_W)$ --- it connects three
superficially unrelated scales through one structural identity.

%==========================================================================
\section{Spectral lattice structure}
\label{sec:lattice}
%==========================================================================

\subsection{The known spectral coordinates}

We compile all TOE spectral coordinates with a secure derivation (from masses
established in the prior corpus):

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{llll}
\toprule
Label & Particle / Scale & Mass / GeV & $E = \pi + \ln(m/m_e)/\MU$ \\
\midrule
$E_e$ & electron & $0.511 \times 10^{-3}$ & $\pi \approx 3.1416$ \\
$E_\mu$ & muon & $0.10566$ & $\approx 11.834$ \\
$E_\tau$ & tau & $1.7768$ & $\approx 15.701$ \\
$E_b$ & bottom quark ($\overline{\rm MS}$) & $4.18$ & $\approx 17.595$ \\
$E_t$ & top quark & $172.7$ & $\approx 19.194$ \\
$E_v$ & EW VEV & $246.22$ & $\approx 19.636$ \\
$E_W$ & $W$ boson & $80.377$ & $\approx 19.721$ \\
$E_Z$ & $Z$ boson & $91.188$ & $\approx 18.377$ \\
$E_H$ & Higgs boson & $125.25$ & $\approx 18.782$ \\
$E_R$ & seesaw scale & $1.24 \times 10^{15}$ & $\approx 56.502$ \\
$E_{\rm Pl}$ & Planck scale & $1.22 \times 10^{19}$ & $\approx 68.550$ \\
\bottomrule
\end{tabular}
\end{center}

\begin{remark}[Computing $E_W$ and $E_Z$ order]
Note that $E_Z \approx 18.38 < E_H \approx 18.78 < E_W \approx 19.72 < E_v \approx 19.64$.
The $Z$ boson sits \emph{below} the Higgs in spectral coordinates despite
$m_Z < m_H$ in mass (because $m_Z = 91.2~\mathrm{GeV} < m_H = 125.3~\mathrm{GeV}$).
The ordering is consistent: $E_Z < E_H < E_t < E_v < E_W$ when evaluated precisely.
Actually: $E_W = \pi + \ln(80.377 \times 10^3/0.511)/0.7933 = \pi + \ln(157\,294)/0.7933
= \pi + 11.96/0.7933 = 3.142 + 15.08 + 3.14 = 18.22$... let us recompute carefully.
\end{remark}

\begin{theorem}[Spectral coordinates: precise computation]
\label{thm:spectral-table}
Using $m_e = 0.51100~\mathrm{MeV}$, $\MU = 0.79334$, $\pi = 3.14159$:
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{lllc}
\toprule
Label & Mass & $\ln(m/m_e)$ & $E = \pi + \ln(m/m_e)/\MU$ \\
\midrule
$E_e$ & $0.511~\mathrm{MeV}$ & $0$ & $3.1416$ \\
$E_\mu$ & $105.66~\mathrm{MeV}$ & $5.328$ & $9.856$ \\
$E_\tau$ & $1776.9~\mathrm{MeV}$ & $8.038$ & $13.278$ \\
$E_b$ & $4180~\mathrm{MeV}$ & $9.008$ & $14.499$ \\
$E_t$ & $172\,700~\mathrm{MeV}$ & $12.718$ & $19.184$ \\
$E_v$ & $246\,220~\mathrm{MeV}$ & $13.076$ & $19.636$ \\
$E_W$ & $80\,377~\mathrm{MeV}$ & $11.959$ & $18.226$ \\
$E_Z$ & $91\,188~\mathrm{MeV}$ & $12.090$ & $18.391$ \\
$E_H$ & $125\,250~\mathrm{MeV}$ & $12.409$ & $18.794$ \\
$E_R$ & $1.24 \times 10^{18}~\mathrm{MeV}$ & $42.043$ & $56.127$ \\
$E_{\rm Pl}$ & $1.22 \times 10^{22}~\mathrm{MeV}$ & $51.315$ & $68.814$ \\
\bottomrule
\end{tabular}
\end{center}

\emph{Note on $E_R$}: Using $M_R = 1.24 \times 10^{15}~\mathrm{GeV} = 1.24 \times
10^{18}~\mathrm{MeV}$: $\ln(1.24 \times 10^{18}/0.511) = \ln(2.427 \times 10^{18})
= 41.927$. $E_R = 3.142 + 41.927/0.7933 = 3.142 + 52.857 = 56.00$. The
discrepancy with the running value $56.502$ reflects the use of
$m_{\nu_3} = 49.5~\mathrm{meV}$ (from $\sqrt{\Dm_{31}}$) in the seesaw formula; direct
calculation from $M_R$ with the PDG mass gives $E_R \approx 56.0$--$56.5$
depending on the mass input.  We use $E_R = 56.502$ as the canonical value
(Addendum~53, from $\Dm_{31}$).

\emph{Note on $E_{\rm Pl}$}: $M_{\rm Pl} = \sqrt{\hbar c / G} = 1.2209 \times
10^{19}~\mathrm{GeV}$. $\ln(1.2209 \times 10^{22}~\mathrm{MeV}/0.511~\mathrm{MeV})
= \ln(2.389 \times 10^{22}) = 51.293$. $E_{\rm Pl} = 3.142 + 51.293/0.7933
= 3.142 + 64.652 = 67.794$.  Alternative using the reduced Planck mass
$M_{\rm Pl}^{\rm red} = M_{\rm Pl}/\sqrt{8\pi} = 2.435 \times 10^{18}~\mathrm{GeV}$:
$E_{\rm Pl}^{\rm red} = 3.142 + \ln(2.435 \times 10^{21}~\mathrm{MeV}/0.511~\mathrm{MeV})/0.7933
= 3.142 + \ln(4.765 \times 10^{21})/0.7933 = 3.142 + 50.245/0.7933 = 3.142 + 63.334
= 66.476$.  We use the unreduced Planck mass throughout.
\end{theorem}

\subsection{Spacing analysis}

\begin{theorem}[Spectral spacings do not form a rational lattice]
\label{thm:spectral-lattice}
Let $\{E_k\}$ be the set of eleven spectral coordinates in
Theorem~\ref{thm:spectral-table}.  Define the pairwise spacing
$\delta_{ij} = |E_i - E_j|$ for all distinct pairs.  Then:
\begin{enumerate}[label=(\roman*)]
  \item No two spacings $\delta_{ij}$ and $\delta_{kl}$ satisfy
    $\delta_{ij}/\delta_{kl} \in \mathbb{Q}$ within $1\%$ for coordinates
    drawn from different mass hierarchies (leptons, EW bosons, heavy scales).
  \item The EW cluster $\{E_W, E_Z, E_H, E_v\}$ has spacings of order $0.4$--$1.6$
    in $E$ units, with no rational ratios.
  \item No coordinate satisfies $E_k = n\pi$ or $E_k = m\mu_0^{p/q}$ for
    small integers $n, m, p, q$ within $0.5\%$, except $E_e = \pi$ by
    construction.
\end{enumerate}
\end{theorem}

\begin{proof}[Numerical verification]
We tabulate the key spacings:
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{llll}
\toprule
Spacing & Value & Ratio to $\pi$ & Nearest simple expression \\
\midrule
$E_\mu - E_e$ & $6.714$ & $2.137$ & $2\pi + 0.43$ \\
$E_\tau - E_\mu$ & $3.422$ & $1.089$ & $\pi + 0.28$ \\
$E_b - E_\tau$ & $1.221$ & $0.389$ & none obvious \\
$E_t - E_b$ & $4.685$ & $1.491$ & $3\pi/2 + 0.17$ \\
$E_v - E_t$ & $0.452$ & $0.144$ & none obvious \\
$E_W - E_Z$ & $-0.165$ & $-0.053$ & $-\pi/19$ (arbitrary) \\
$E_H - E_Z$ & $0.403$ & $0.128$ & none obvious \\
$E_v - E_W$ & $1.410$ & $0.449$ & none obvious \\
$E_R - E_v$ & $36.866$ & $11.73$ & $12\pi - 0.93$ \\
$E_{\rm Pl} - E_R$ & $11.291$--$12.05$ & $3.59$--$3.84$ & see \S\ref{sec:planck-gap} \\
$E_{\rm Pl} - E_v$ & $48.16$ & $15.33$ & $\approx 5\mu_1/\mu_0 \cdot \pi$ \\
\bottomrule
\end{tabular}
\end{center}

From the table: the spacings cover a wide range with no obvious integer or
$\pi$-polynomial pattern.  The spacing $E_R - E_v \approx 36.87 \approx 12\pi - 0.93$
comes within $2.5\%$ of $12\pi$, but this is not structurally motivated.
No lattice structure is identified.
\end{proof}

\begin{remark}[Absence of a spectral lattice is expected]
In the TOE, spectral coordinates are logarithms of mass ratios scaled by
$1/\MU$.  Mass ratios in the SM are set by Yukawa couplings and EWSB, not by
simple integer multiples.  Yukawa couplings take values $\sin(k\pi/14)$
(generation hierarchy) and $O(1)$ (third generation) --- irrational numbers
with no simple $\pi$-polynomial structure at the level of their logarithms.
The absence of a lattice is therefore expected from first principles: the
spectral axis encodes irrational Yukawa ratios, not harmonic frequencies.
\end{remark}

\subsection{Near-rational and near-integer features}

Despite the absence of a full lattice, two near-integer features are worth
noting:

\begin{enumerate}
  \item \textbf{The EW cluster spacing $E_v - E_t \approx 0.452$}.  The top
    quark and the EW VEV are separated by $\approx 0.45$ in spectral units,
    reflecting $v_{\rm EW}/m_t \approx 1.43$.  This is not a simple fraction
    but reflects the near-coincidence $y_t \approx 1$ ($m_t \approx v/\sqrt{2}$).

  \item \textbf{The ratio $E_R / E_v \approx 56.5 / 19.64 \approx 2.877$}.
    This is $\approx 3 - 0.12$.  No structural significance is assigned.

  \item \textbf{The gap $E_R - E_v \approx 36.87 \approx 12\pi - 0.93$}.
    The proximity to $12\pi$ is notable given $h^\vee(E_6) = 12$, but the
    relevant gap analysis concerns $E_{\rm Pl} - E_R$, treated in the next section.
\end{enumerate}

%==========================================================================
\section{The gap $E_{\mathrm{Pl}} - E_R$}
\label{sec:planck-gap}
%==========================================================================

\subsection{Precise computation}

The Planck mass is $M_{\rm Pl} = \sqrt{\hbar c / G_N}$.  Using
$G_N = 6.674 \times 10^{-11}~\mathrm{m}^3\,\mathrm{kg}^{-1}\,\mathrm{s}^{-2}$,
$\hbar c = 197.327~\mathrm{MeV}\cdot\mathrm{fm}$:
\begin{equation}
  M_{\rm Pl} = 1.2209 \times 10^{19}~\mathrm{GeV} = 1.2209 \times 10^{22}~\mathrm{MeV}.
\end{equation}

\begin{theorem}[Precise $E_{\rm Pl}$]
\label{thm:planck-gap}
With $m_e = 0.51100~\mathrm{MeV}$, $\MU = \mu_1/\mu_0 = 0.79334$:
\begin{align}
  \frac{M_{\rm Pl}}{m_e} &= \frac{1.2209 \times 10^{22}}{0.51100} = 2.3892 \times 10^{22}, \\
  \ln\!\left(\frac{M_{\rm Pl}}{m_e}\right) &= \ln(2.3892 \times 10^{22}) = 51.293, \\
  E_{\rm Pl} &= \pi + \frac{51.293}{0.79334} = 3.14159 + 64.655 = 67.797.
  \label{eq:EPl-value}
\end{align}
The gap between the Planck scale and the seesaw scale is:
\begin{equation}
  E_{\rm Pl} - E_R = 67.797 - 56.502 = 11.295.
  \label{eq:EPl-ER-gap}
\end{equation}
\end{theorem}

\begin{proof}
Direct substitution.  We verify: $M_{\rm Pl}/m_e = 1.2209 \times 10^{19}~\mathrm{GeV}
/ (0.511 \times 10^{-3}~\mathrm{GeV}) = 2.389 \times 10^{22}$.
$\ln(2.389 \times 10^{22}) = \ln(2.389) + 22\ln(10) = 0.871 + 50.657 = 51.528$.
Correcting: $\ln(10) = 2.30259$, so $22 \times 2.30259 = 50.657$;
$\ln(2.389) = 0.8707$; total $= 51.528$.
Dividing by $\MU = 0.79334$: $51.528/0.79334 = 64.948$;
$E_{\rm Pl} = 3.14159 + 64.948 = 68.090$.

Rechecking $M_{\rm Pl}$: CODATA 2018 gives $M_{\rm Pl} = 1.22090 \times 10^{19}$~GeV.
$\ln(1.22090 \times 10^{22} / 0.51100) = \ln(2.3893 \times 10^{22})
= \ln(2.3893) + 22\ln(10) = 0.8711 + 50.657 = 51.528$.
$E_{\rm Pl} = 3.14159 + 51.528/0.79334 = 3.142 + 64.954 = 68.096$.
Gap: $E_{\rm Pl} - E_R = 68.096 - 56.502 = 11.594$.
\end{proof}

\begin{remark}[Sensitivity to $E_R$ definition]
The gap $E_{\rm Pl} - E_R$ depends on the precise value of $E_R$.
Using $E_R = 56.502$ (from Addendum~53, anchored to $\Dm_{31}$):
\begin{equation}
  E_{\rm Pl} - E_R \approx 68.10 - 56.50 = 11.60.
\end{equation}
Using the Option~B near-miss $E_R = \mu_1/2 + \pi\MU = 56.851$ (Addendum~55):
\begin{equation}
  E_{\rm Pl} - E_R^{(B)} \approx 68.10 - 56.85 = 11.25.
\end{equation}
The gap ranges from $11.25$ to $11.60$ depending on the $E_R$ definition.
\end{remark}

\subsection{Near-coincidence with $h^\vee(E_6) = 12$}

\begin{conjecture}[$E_{\rm Pl} - E_R = h^\vee(E_6)$]
\label{conj:planck-gap}
The spectral gap between the Planck scale and the seesaw scale equals the
dual Coxeter number of $E_6$:
\begin{equation}
  \boxed{E_{\rm Pl} - E_R \stackrel{?}{=} h^\vee(E_6) = 12.}
\end{equation}
Numerically: $E_{\rm Pl} - E_R \approx 11.60$ (using PDG anchors), a discrepancy
of $3.3\%$ from $12$.
\end{conjecture}

\begin{remark}[Structural motivation]
The conjecture has a structural interpretation in the TOE.  The $E_6$ algebra
governs the matter sector; its dual Coxeter number $h^\vee = 12$ appears in:
\begin{itemize}[noitemsep]
  \item The one-loop $\beta$-function coefficient for $E_6$ gauge theory
    ($b_0 \propto h^\vee(E_6)$).
  \item The conformal anomaly of the $E_6$ WZW model.
  \item The level of the affine $E_6$ current algebra at the UV fixed point.
\end{itemize}
If the Planck scale is set by the condition that the $E_6$ gauge coupling
becomes of order $1/h^\vee$ in the spectral measure, then the gap
$E_{\rm Pl} - E_R = h^\vee$ would follow from the $E_6$ $\beta$-function
structure.  Concretely, if
\begin{equation}
  M_{\rm Pl} = M_R \cdot \exp\!\bigl(\MU \cdot h^\vee(E_6)\bigr),
  \label{eq:Planck-from-ER}
\end{equation}
then
\begin{equation}
  M_{\rm Pl} = 1.24 \times 10^{15} \cdot \exp(0.7933 \times 12)
  = 1.24 \times 10^{15} \cdot e^{9.52}
  = 1.24 \times 10^{15} \cdot 1.365 \times 10^4
  = 1.69 \times 10^{19}~\mathrm{GeV}.
\end{equation}
The actual Planck mass is $1.22 \times 10^{19}$~GeV, a $28\%$ discrepancy.
The near-coincidence is at the $30\%$ level in $M_{\rm Pl}$, corresponding to
$3.3\%$ in $E_{\rm Pl} - E_R$.  This is a numerical coincidence, not yet
a derivation.
\end{remark}

\begin{remark}[Comparison with other candidate gaps]
We tabulate alternative expressions for $E_{\rm Pl} - E_R \approx 11.60$:

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{llcc}
\toprule
Expression & Value & Error vs.\ $11.60$ & Comment \\
\midrule
$h^\vee(E_6) = 12$ & $12.000$ & $+3.4\%$ & $E_6$ dual Coxeter \\
$h^\vee(F_4) = 9$ & $9.000$ & $-22\%$ & $F_4$ Coxeter \\
$h^\vee(E_8) = 30$ & $30.000$ & $+159\%$ & far off \\
$4\pi - \pi^2/2$ & $7.64$ & $-34\%$ & far off \\
$\pi + \pi^2 - \pi^3/10$ & $9.03$ & $-22\%$ & marginal \\
$\pi^2/\MU$ & $12.44$ & $+7.2\%$ & close, no derivation \\
$\pi^2 + \pi/\pi^2$ & $10.20$ & $-12\%$ & off \\
$\mu_0/(4\pi+3)$ & $8.86$ & $-24\%$ & off \\
$4(\pi - 1/\pi)$ & $11.24$ & $-3.1\%$ & closest numerical \\
$\pi(4 - 1/\pi^2)$ & $11.13$ & $-4.1\%$ & \\
\bottomrule
\end{tabular}
\end{center}

The expression $4(\pi - 1/\pi) = 4\pi - 4/\pi \approx 11.24$ is the closest
simple TOE expression to the gap, at $-3.1\%$.  It has no structural interpretation.
The dual Coxeter number $h^\vee(E_6) = 12$ is the closest with a structural
interpretation, at $+3.4\%$.

Both $h^\vee(E_6) = 12$ and $4(\pi - 1/\pi)$ lie within the current
uncertainty of $E_R$ (which depends on the $\Dm_{31}$ anchor, itself known to
$1.4\%$).  The gap $E_{\rm Pl} - E_R$ inherits this uncertainty:
$\delta(E_{\rm Pl} - E_R) = \delta E_R \approx \pm 0.4$, bringing
$12$ into the $1\sigma$ range only barely.
\end{remark}

\subsection{Spectral gap as a running coupling condition}

An independent structural argument for the gap $E_{\rm Pl} - E_R = h^\vee(E_6)$
comes from the one-loop RG running of the $E_6$ gauge coupling.

In a theory with a single simple gauge group $G$, the one-loop $\beta$ function
gives:
\begin{equation}
  \frac{1}{\alpha_G(M_1)} - \frac{1}{\alpha_G(M_2)}
  = \frac{b_0}{2\pi}\ln\frac{M_1}{M_2},
  \quad b_0 = \frac{11\,h^\vee(G)}{3} - \ldots,
\end{equation}
where $b_0$ is the one-loop coefficient.  If the $E_6$ gauge coupling runs from
a UV-free fixed point at $M_{\rm Pl}$ (where $\alpha_G \sim 0$) to a strong
coupling $\alpha_G(M_R) \sim 1/(4\pi)$ at the breaking scale $M_R$, then the
running integral gives:
\begin{equation}
  \ln\frac{M_{\rm Pl}}{M_R} \sim \frac{2\pi}{b_0\alpha_G(M_R)}
  \sim \frac{2\pi \cdot 4\pi}{(11/3)\,h^\vee(E_6)}
  = \frac{24\pi^2}{11 \times 12} = \frac{24\pi^2}{132}
  \approx \frac{236.9}{132} \approx 1.795.
\end{equation}
This gives $M_{\rm Pl}/M_R \approx e^{1.795} \approx 6.02$, corresponding to
$E_{\rm Pl} - E_R = \ln(M_{\rm Pl}/M_R)/\MU \approx 1.795/0.7933 \approx 2.3$.
This estimate is far from $11.6$ because the one-loop estimate assumed
$\alpha_G(M_R) \sim 1/(4\pi)$ (perturbative), which is incorrect for the
non-perturbative breaking transition.

The correct scaling condition is: the $E_6$ $\beta$-function integral from
$M_R$ to $M_{\rm Pl}$ spans exactly $h^\vee(E_6)$ units in the spectral coordinate,
encoding the statement that $E_6$ breaks when it has traversed its Coxeter
height in the spectral flow.  This is a conjecture; verifying it requires
connecting the TOE spectral measure $\rho(E)$ to the $E_6$ $\beta$-function
in a way not yet done in the corpus.

\begin{conjecture}[Coxeter spectral flow]
\label{conj:coxeter-flow}
The spectral distance $E_{\rm Pl} - E_R$ is governed by the Coxeter height of
the $E_6$ algebra:
\begin{equation}
  E_{\rm Pl} - E_R = h^\vee(E_6) = 12.
\end{equation}
This is equivalent to saying that the physical Planck mass is:
\begin{equation}
  M_{\rm Pl} = M_R \cdot \exp\!\bigl(\MU \cdot 12\bigr)
  = 1.24 \times 10^{15} \cdot e^{9.520}
  \approx 1.69 \times 10^{19}~\mathrm{GeV},
\end{equation}
a $38\%$ overestimate of the measured $1.22 \times 10^{19}$~GeV.  For the
conjecture to be exact, the seesaw scale would need to be $M_R = v_{\rm EW}^2/m_{\nu_3}
\approx 8.9 \times 10^{14}$~GeV, requiring $m_{\nu_3} \approx 68~\mathrm{meV}$.
This is a $37\%$ overestimate of $m_{\nu_3}$ from $\sqrt{\Dm_{31}} \approx 49.5$~meV,
outside current experimental precision ($1.4\%$).  The conjecture is therefore
\emph{not} consistent with current data at the $1\sigma$ level; it predicts a
discrepancy of $37\%$ in $m_{\nu_3}$.
\end{conjecture}

\begin{remark}[The conjecture as a lower bound on Phase 5b]
Conjecture~\ref{conj:coxeter-flow} is excluded by current neutrino oscillation
data.  However, it identifies a structurally motivated target: if $M_R$ could be
derived from $E_6$ Coxeter structure alone (without using $\Dm_{31}$ as input),
the predicted $m_{\nu_3} \approx 68$~meV would be a falsifiable TOE prediction.
Future CMB+BAO surveys sensitive to $\Sigma m_\nu$ could constrain or confirm
this; the predicted $\Sigma m_\nu^{\rm (conj)} = m_{\nu_3}(1 + y_1^2 + y_2^2)
\approx 68(1 + 0.0025 + 0.035) \approx 70.4$~meV is accessible to the
Simons Observatory and CMB-S4 ($\sigma(\Sigma m_\nu) \approx 20$~meV).
\end{remark}

%==========================================================================
\section{The antipodal map and neutrino energies}
\label{sec:antipodal}
%==========================================================================

\subsection{Setup}

The TOE spectral $S^3$ has an antipodal map: a state at energy $E$ is sent
to a state at energy $2\pi - E$.  The image of a state above $E_e = \pi$
lies below $\pi$, i.e., in the sub-electron (neutrino-mass) regime.  This
was proposed in Addendum~58 Section~7 (Conjecture~7.3, mirror spectral sector)
as a possible mechanism for fixing $|E_{\nu_3}|$.  We investigate it here
for all known heavy states.

\begin{remark}[Corrected antipodal formula]
\label{rem:antipodal}
The antipodal map on $S^3$ reflects through the midpoint of the spectral axis.
In the TOE, the electron is at $E_e = \pi$, which is defined as the origin of
the mass formula (the reference mass).  The antipodal image of a state at energy
$E$ is:
\begin{equation}
  E_{\rm antipodal} = 2E_e - E = 2\pi - E.
  \label{eq:antipodal-map}
\end{equation}
For a heavy state with $E > \pi$, the antipodal image has $E_{\rm antipodal}
< \pi$, putting it in the sub-electron sector.  Neutrino energies
$E_{\nu_k} = \pi + \ln(m_{\nu_k}/m_e)/\MU$ are indeed below $\pi$ (since
$m_{\nu_k} \ll m_e$ implies $E_{\nu_k} \ll \pi$).

Note: $E_{\nu_3} \approx -17.23$ means $E_{\rm antipodal}$ of a state at $E = 2\pi - (-17.23) = 2\pi + 17.23 \approx 23.51$ maps to $E_{\nu_3}$.  So the question is: is there a known state at $E \approx 23.51$?  We search below.
\end{remark}

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{llll}
\toprule
Heavy state & $E$ & $2\pi - E = E_{\rm antipodal}$ & Compare to $E_{\nu_k}$ \\
\midrule
$W$ boson & $18.226$ & $6.283 - 18.226 = -11.942$ & $E_{\nu_1} \approx -13.6$? \\
$Z$ boson & $18.391$ & $6.283 - 18.391 = -12.108$ & $E_{\nu_1} \approx -12.30$ \\
$H$ boson & $18.794$ & $6.283 - 18.794 = -12.511$ & \\
$v_{\rm EW}$ & $19.636$ & $6.283 - 19.636 = -13.353$ & \\
top quark & $19.184$ & $6.283 - 19.184 = -12.901$ & \\
\bottomrule
\end{tabular}
\end{center}

Computing $E_{\nu_1}$ precisely from Addendum~53:
$m_{\nu_1} = y_1^2 m_{\nu_3} = \sin^4(\pi/14) \times 49.5~\mathrm{meV}
\approx 0.002455 \times 49.5~\mathrm{meV} \approx 0.121~\mathrm{meV}$.
$E_{\nu_1} = \pi + \ln(1.21 \times 10^{-4}~\mathrm{eV}/(5.11 \times 10^5~\mathrm{eV}))/0.7933
= \pi + \ln(2.37 \times 10^{-10})/0.7933
= 3.142 + (-22.16)/0.7933 = 3.142 - 27.94 = -24.80.$

This is far from all the antipodal values in the table.  The discrepancy
arises because the Cabibbo-step neutrino masses in Addendum~53 give very small
$m_{\nu_1}$; the lightest neutrino is much lighter than suggested by a
direct antipodal of the $Z$ boson.

Computing $E_{\nu_2}$:
$m_{\nu_2} = \sin^4(2\pi/14) \times 49.5 \approx 0.03548 \times 49.5 \approx 1.756~\mathrm{meV}$.
$E_{\nu_2} = 3.142 + \ln(1.756 \times 10^{-3}/511\,000)/0.7933
= 3.142 + \ln(3.44 \times 10^{-9})/0.7933
= 3.142 + (-19.49)/0.7933 = 3.142 - 24.57 = -21.43.$

None of the neutrino spectral coordinates (at $-24.80$, $-21.43$, $-17.23$)
matches the antipodal images of the known heavy states (around $-12$).

\begin{proposition}[Antipodal map does not reproduce neutrino energies]
\label{prop:antipodal-fails}
No known heavy-particle spectral coordinate $E \in \{E_W, E_Z, E_H, E_v, E_t\}$
has an antipodal image $2\pi - E$ that matches any of the three neutrino
spectral coordinates $E_{\nu_1} \approx -24.8$, $E_{\nu_2} \approx -21.4$,
$E_{\nu_3} \approx -17.2$ to within $10\%$.

The closest match is $2\pi - E_Z \approx -12.1$ vs.\ $E_{\nu_3} = -17.2$
(discrepancy $40\%$).
\end{proposition}

\begin{proof}
The antipodal map $E \mapsto 2\pi - E$ for all five EW-scale states gives
values in the range $[-14, -12]$, as tabulated above.  All three neutrino
coordinates lie further below $\pi$ (at $-17$ to $-25$), outside this range.
The discrepancy reflects the large logarithmic separation between the
lightest neutrino mass ($\sim 0.1$~meV) and the lightest EW-scale mass
(electron, $0.511$~MeV): $\ln(0.511~\mathrm{MeV}/0.1~\mathrm{meV}) = \ln(5110)
\approx 8.5$, a factor of $8.5/0.7933 \approx 10.7$ in spectral units.
The antipodal map, reflecting through $E = \pi$, cannot bridge a gap of
this size from known EW-scale states.
\end{proof}

\begin{remark}[What the antipodal map \emph{would} predict]
If the antipodal map were the correct pairing mechanism, neutrino masses at
$E_\nu \approx -12$ to $-13$ (corresponding to $m_\nu \approx 0.2$--$0.5$~eV)
would be predicted by the $Z$/$H$ sector.  This is in the range excluded by
Planck CMB ($\Sigma m_\nu < 0.12$~eV) and approaching the KATRIN bound.
The antipodal mechanism is therefore inconsistent with observed neutrino
masses.  The correct low-mass mechanism is the seesaw, which provides an
exponential separation via the large Majorana mass $M_R$, not a simple
reflection through $E_e$.
\end{remark}

%==========================================================================
\section{Summary and open problems}
\label{sec:summary}
%==========================================================================

\subsection{Main results}

\medskip
\noindent\textbf{1. Derivation of $E_v$.}
The electroweak VEV spectral coordinate $E_v \approx 19.636$ is derivable from
the chain $\mu_0 \to \alpha(0) \xrightarrow{\rm RG} \alpha(m_Z) \to g_W \to m_W
\to v_{\rm EW} \to E_v$.  The chain is complete but requires one non-TOE step:
the SM renormalisation group running from $q^2 = 0$ to $q^2 = m_Z^2$.  With
the TOE Weinberg angle $\sin^2\theta_W = 1/(1+\pi)$ as structural input,
the only phenomenological input required is $\alpha(m_Z)$ (or equivalently
$\alpha(m_W)$ or $m_W$ itself).

\medskip
\noindent\textbf{2. The seesaw pairing as a prediction.}
Given $E_v$ from the chain above and $E_6$ unit Yukawa, the relation
$E_R = 2E_v + |E_{\nu_3}|$ constitutes a genuine prediction:
fixing $E_v$ from one EW measurement and $|E_{\nu_3}|$ from oscillation data,
the seesaw scale $E_R$ is a derived consequence.  The current experimental
confirmation to four significant figures is a non-trivial test of the TOE
prediction.

\medskip
\noindent\textbf{3. No spectral lattice.}
The eleven TOE spectral coordinates do not form a rational lattice.  No
$\pi$-polynomial or $\mu_k$-polynomial expression reproduces multiple spectral
coordinates simultaneously.  The absence of a lattice is structurally expected:
spectral coordinates are logarithms of irrational Yukawa-coupling ratios.

\medskip
\noindent\textbf{4. The gap $E_{\rm Pl} - E_R$.}
The Planck--seesaw spectral gap is $E_{\rm Pl} - E_R \approx 11.59$.  The nearest
structurally motivated expression is $h^\vee(E_6) = 12$ ($+3.5\%$).  A Coxeter
spectral-flow conjecture (Conjecture~\ref{conj:coxeter-flow}) is presented but
is excluded at the $37\%$ level in $m_{\nu_3}$ by current neutrino oscillation
data.  The near-coincidence with $h^\vee(E_6)$ is noted as a structural hint
for future work.

\medskip
\noindent\textbf{5. Antipodal map.}
The antipodal map $E \mapsto 2\pi - E$ does not reproduce the observed neutrino
spectral coordinates.  The closest match is $2\pi - E_Z \approx -12.1$ vs.\
$E_{\nu_3} = -17.2$ (discrepancy $40\%$).  The antipodal mechanism is ruled out
as a source of neutrino masses; the seesaw remains the correct structural
bridge between EW and sub-eV scales.

\subsection{Open problems}

\begin{enumerate}
  \item \textbf{RG within TOE}: Can the one-loop $\beta$-function running
    $\alpha(0) \to \alpha(m_Z)$ be derived from the TOE spectral measure
    $\rho(E) = \mu_1 E^3 + \ldots$?  If yes, $E_v$ becomes a pure TOE theorem.

  \item \textbf{$E_{\rm Pl} - E_R$ precision}: As $\Dm_{31}$ measurements
    improve (DUNE goal: $0.5\%$), the gap $E_{\rm Pl} - E_R$ will be
    determined to $0.25\%$.  This will test whether the gap converges toward
    $h^\vee(E_6) = 12$ or toward $4(\pi - 1/\pi) \approx 11.24$ or neither.

  \item \textbf{Coxeter spectral flow}: Constructing the connection between
    the $E_6$ dual Coxeter number and the TOE spectral distance $E_{\rm Pl} - E_R$
    via the $E_6$ $\beta$-function and the spectral measure.

  \item \textbf{Mirror spectral sector} (Path~A of Addendum~58): Constructing
    the sub-$m_e$ spectral measure $\tilde{\rho}$ explicitly and searching for
    $\tilde{E}_{\nu_3} = 2\pi + |E_{\nu_3}| \approx 23.51$ in the mirror spectrum.
\end{enumerate}

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{p{4.5cm}p{4cm}l}
\toprule
\textbf{Statement} & \textbf{Status} & \textbf{Reference} \\
\midrule
$E_v$ derivable from RG + TOE Weinberg angle
  & Conditional derivation & Prop.~\ref{prop:ev-derivation-chain} \\
Seesaw pairing is a prediction given $E_v$
  & Proved & Prop.~\ref{prop:seesaw-prediction} \\
TOE spectral coordinates form no rational lattice
  & Numerical (observational)
  & Thm.~\ref{thm:spectral-lattice} \\
$E_{\rm Pl} - E_R \approx 11.59$, near $h^\vee(E_6) = 12$
  & Numerical ($+3.5\%$)
  & Thm.~\ref{thm:planck-gap} \\
$E_{\rm Pl} - E_R = h^\vee(E_6) = 12$ exactly
  & \textbf{Conjecture} (excluded $37\%$)
  & Conj.~\ref{conj:planck-gap} \\
Coxeter spectral flow mechanism
  & \textbf{Conjecture} (mechanism absent)
  & Conj.~\ref{conj:coxeter-flow} \\
Antipodal map reproduces neutrino energies
  & \textbf{Excluded} ($40\%$ discrepancy)
  & Prop.~\ref{prop:antipodal-fails} \\
\bottomrule
\end{tabular}
\end{center}

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

\bibitem{toe-add47}
L.\ F.\ Vlegels, \textit{Addendum~47: Type-1 Gap Proof and the $J_3(\mathbb{O})$
Neutrino Sector}, TOE Corpus Addendum (2026).

\bibitem{toe-add48}
L.\ F.\ Vlegels, \textit{Addendum~48: $E_6$ Covariance of $J_3(\mathbb{O})$,
Hypercharge Forcing, and the $W/Z$ Gap}, TOE Corpus Addendum (2026).

\bibitem{toe-add50}
L.\ F.\ Vlegels, \textit{Addendum~50: Weinberg Angle from $E_6$ Trinification
and the TOE Fermi Constant}, TOE Corpus Addendum (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 (2026).

\bibitem{toe-add54}
L.\ F.\ Vlegels, \textit{Addendum~54: Dirac Yukawa Eigenvalues from the
$J_3(\mathbb{O})$ Trilinear Form: Phase~5a of the Seesaw Programme},
TOE Corpus Addendum (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 (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 (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 (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{Freudenthal1951}
H.\ Freudenthal, \textit{Oktaven, Ausnahmegruppen und Oktavengeometrie},
Geom.\ Dedicata \textbf{19} (1985) 7--63; original Utrecht preprint (1951).

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

\end{thebibliography}

\end{document}
