% Addendum 70: Phase 5b Closure — The Midpoint Conjecture and the
%   Wolfenstein O(θ_C) Threshold Correction
%   Computing the midpoint formula E_R = E_Planck − h^∨(E₆) + MU(\pi+3)/12,
%   its structural (non-)derivation from G₂/E₆ geometry, and whether the
%   O(θ_C) Wolfenstein corrections from the PMNS sector resolve the
%   Phase 5b degeneracy.
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. May 2026.

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\title{Phase~5b Closure: the Midpoint Conjecture and\\
the Wolfenstein $O(\thetaC)$ Threshold Correction\\[0.5em]
\large Addendum 70 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~62 (P62) left Phase~5b with two equally viable candidates for the
threshold correction $\varepsilon$ to the Planck--seesaw spectral gap
$\EPl - \ER = h^\vee(E_6) - \varepsilon$:
\begin{itemize}[noitemsep]
  \item Candidate~I: $\varepsilon_1 = \MU\pi/6$ (the $G_2$ Weyl chamber angle,
    predicting $\ER^{(\mathrm{I})} = 56.511$, residual $+0.009$).
  \item Candidate~II: $\varepsilon_2 = \MU/2$ (the mean $E_6$ exponent formula,
    predicting $\ER^{(\mathrm{II})} = 56.493$, residual $-0.009$).
\end{itemize}
Both sit at $1.02\sigma$ from the observed $\ER = 56.502$ in opposite directions.
The present addendum addresses two questions: (1)~is the midpoint of the two
candidates itself a structural formula? and (2)~do the $O(\thetaC)$ Wolfenstein
corrections computed in the PMNS sector (Addenda 65--68) provide the additional
$\delta \approx 0.009$ needed to break this degeneracy?

\textbf{Main results.}
\begin{enumerate}[label=(\roman*)]
\item \textbf{Proposition~\ref{prop:midpoint-numerical}} (proved): the arithmetic
  midpoint of the two Phase~5b predictions is
  \[
    \tfrac{1}{2}\bigl(\ER^{(\mathrm{I})} + \ER^{(\mathrm{II})}\bigr)
    = E_{\mathrm{Pl}} - 12 + \MU\cdot\frac{\pi+3}{12} = 56.502,
  \]
  coinciding with the observed $\ER$ to three decimal places.

\item \textbf{Proposition~\ref{prop:midpoint-algebra}} (proved): $(\pi+3)/12$ is
  the arithmetic mean of $\pi/6$ (Candidate~I) and $1/2$ (Candidate~II), and
  equals $(\pi + N_{\rm gen})/h^\vee(E_6)$ with $N_{\rm gen} = 3$.  Thus the
  midpoint formula has a natural bookkeeping identity in terms of $G_2/E_6$
  data and the number of generations, but this identity does not constitute
  a structural derivation.

\item \textbf{Theorem~\ref{thm:wolfenstein-too-large}} (proved): the leading
  $O(\thetaC^2)$ Wolfenstein correction to $\ER$ from the atmospheric PMNS
  angle $\theta_{23}$ has magnitude
  \[
    \delta\ER^{(\mathrm{W},2)} = \frac{\sin^2\thetaC}{2\MU} \approx 0.0312,
  \]
  which is $3.3\times$ larger than the required $0.009$.  It therefore overshoots
  the splitting and does not resolve the degeneracy at this order.

\item \textbf{Proposition~\ref{prop:wolfenstein-cubic}} (proved): the
  $O(\thetaC^3)$ Wolfenstein term $\MU\cdot\sin^3\thetaC \approx 0.00875$ and
  the $O(\thetaC^3)$ angular term $\MU\thetaC^3 \approx 0.00897$ bracket
  the observed residual $0.009$ from below and above respectively, at the
  $1$--$3\%$ level.  This proximity is numerically suggestive but does not
  constitute a structural derivation.

\item \textbf{Conjecture~\ref{conj:midpoint}} (conjecture, not theorem): the
  correct Phase~5b formula is the midpoint formula
  \[
    \ER = \EPl - h^\vee(E_6) + \MU\cdot\frac{\pi + N_{\rm gen}}{h^\vee(E_6)},
  \]
  which is the \emph{average} of the two geometric threshold corrections from
  Candidate~I (the $G_2$ Weyl chamber angle $\pi/6$) and Candidate~II (the
  mean $E_6$ exponent $1/2$).  The status is conjecture because the observed
  $\ER = 56.502$ depends on $\Dm_{31}$ at $1.4\%$ precision, and the midpoint
  coincidence is only verified numerically, not from a structural averaging
  principle.

\item \textbf{Conclusion}: the Phase~5b degeneracy is \emph{not} resolved by
  the Wolfenstein O($\thetaC$) PMNS corrections at any order in a clean
  structural way.  The midpoint formula is a striking numerical identity that
  warrants further investigation, but its derivation from first principles
  in the $G_2 \subset E_6$ geometry remains open.
\end{enumerate}
\end{abstract}

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

\subsection{The Phase~5b situation after P62}

Addendum~62 established that the Planck--seesaw spectral gap
$\EPl - \ER = 11.594 \pm 0.009$ deviates from the naive $h^\vee(E_6) = 12$
by $\varepsilon = 0.406$.  Two candidates were shown to land within the $1\sigma$
observational window:
\begin{equation}
  \ER^{(\mathrm{I})} = 56.511 \quad \text{(Candidate~I: $\varepsilon_1 = \MU\pi/6$, residual $+0.009$)},
\end{equation}
\begin{equation}
  \ER^{(\mathrm{II})} = 56.493 \quad \text{(Candidate~II: $\varepsilon_2 = \MU/2$, residual $-0.009$)}.
\end{equation}
The observed value $\ER^{(\mathrm{obs})} = 56.502$ lies precisely midway between
them.  P62 concluded that discrimination requires a $\Dm_{31}$ measurement
with relative precision $\lesssim 0.5\%$ (e.g.\ from JUNO or Hyper-K).

Two questions remain open after P62:
\begin{enumerate}[label=(\alph*)]
  \item Is the midpoint $\ER = 56.502$ itself a formula with a structural
    derivation?  If so, Phase~5b could be closed theoretically even before
    the precision experiments.
  \item Do the $O(\thetaC)$ Wolfenstein corrections from the PMNS sector
    (developed in Addenda 65--68) contribute the missing $\delta \approx 0.009$
    that would select one candidate over the other?
\end{enumerate}
This addendum addresses both questions.

\subsection{The PMNS programme (Addenda 65--68) as context}

Addendum~65 proved the QLC relation $\theta_{12} = \pi/4 - \thetaC$ and the
maximality $\theta_{23} = \pi/4$ from $\mathbb{Z}_3$ symmetry of $\JO$.
Addendum~66 computed the reactor angle $\sin\theta_{13} = \sin\thetaC/\sqrt{2}$
and the CP phase $\delta_{\rm CP} \approx -(\pi/3 + \thetaC/2)$ from the
dihedral modulus of the $(S^6)^3/G_2$ orbit.

Together, these establish the leading-order PMNS structure.  The Wolfenstein
expansion in powers of $\lambda_W = \sin\thetaC = \sin(\pi/14) \approx 0.2225$
gives the subleading corrections:
\begin{align}
  \theta_{12} &= \frac{\pi}{4} - \thetaC + O(\lambda_W^2), \label{eq:th12-W} \\
  \theta_{23} &= \frac{\pi}{4} - \frac{\sin^2\thetaC}{2} + O(\lambda_W^3), \label{eq:th23-W} \\
  \theta_{13} &= \frac{\sin\thetaC}{\sqrt{2}}\left(1 + O(\lambda_W^2)\right). \label{eq:th13-W}
\end{align}
The $O(\lambda_W^2)$ correction in \eqref{eq:th23-W} arises from the
second-order $G_2$ holonomy and was computed in Addendum~68 (P68).  The
question is whether this and related corrections propagate into $\ER$ with
magnitude $\approx 0.009$.

\subsection{Outline}

Section~\ref{sec:midpoint} proves the midpoint numerical identity and
examines whether it has a structural derivation.  Section~\ref{sec:wolfenstein}
computes the $O(\thetaC)$ Wolfenstein correction to $\ER$ order by order and
tests whether any order resolves the Phase~5b degeneracy.
Section~\ref{sec:neutrino-mass} derives the neutrino mass $m_{\nu_3}$
from the midpoint formula.  Section~\ref{sec:verdict} states the Phase~5b
status and what remains open.

%==========================================================================
\section{The midpoint formula}
\label{sec:midpoint}
%==========================================================================

\subsection{Numerical verification}

\begin{proposition}[Midpoint numerical identity]
\label{prop:midpoint-numerical}
Let $\ER^{(\mathrm{I})} = \EPl - 12 + \MU\pi/6$ and
$\ER^{(\mathrm{II})} = \EPl - 12 + \MU/2$ be the two Phase~5b candidates.
Their arithmetic mean is:
\begin{equation}
  \ER^{(\mathrm{mid})} = \frac{\ER^{(\mathrm{I})} + \ER^{(\mathrm{II})}}{2}
  = \EPl - 12 + \MU\cdot\frac{\pi/6 + 1/2}{2}
  = \EPl - 12 + \MU\cdot\frac{\pi + 3}{12}.
  \label{eq:midpoint-def}
\end{equation}
Numerically:
\begin{align}
  \frac{\pi + 3}{12} &= \frac{3.14159265 + 3}{12} = \frac{6.14159265}{12}
                       = 0.511799388, \\
  \MU \cdot \frac{\pi+3}{12} &= 0.793338 \times 0.511799 = 0.406068, \\
  \ER^{(\mathrm{mid})} &= 68.096 - 12 + 0.40607 = 56.502.
\end{align}
This coincides with the observed $\ER^{(\mathrm{obs})} = 56.502$ to three
decimal places, identical with the P62 canonical value.
\end{proposition}

\begin{proof}
Direct arithmetic:
\begin{equation}
  \frac{\pi/6 + 1/2}{2} = \frac{\pi/6 + 3/6}{2} = \frac{(\pi+3)/6}{2}
  = \frac{\pi+3}{12}.
\end{equation}
The numerical evaluation uses $\pi = 3.14159265358979$ and $\MU = 0.793338$
from the P61/P62 canonical table.
\end{proof}

\begin{remark}[Precision]
Using higher precision: $\MU = \muone/\muzero$ with $\muone = 16\pi^3/5
+ 3\pi^2/4 + 2\pi/3$ and $\muzero = 4\pi^3 + \pi^2 + \pi$, the product
$\MU\cdot(\pi+3)/12 = 0.406068$ gives $\ER^{(\mathrm{mid})} = 56.50207$.
The P62 observed value $\ER^{(\mathrm{obs})} = 56.502$ is derived from
$\Dm_{31} = 2.453\times10^{-3}$~eV$^2$ at $1.4\%$ precision.  The match
is exact to the precision available, which is three significant figures after
the decimal point.
\end{remark}

\subsection{Algebraic identity}

\begin{proposition}[Algebraic structure of the midpoint correction]
\label{prop:midpoint-algebra}
The midpoint correction $(\pi+3)/12$ decomposes as:
\begin{equation}
  \frac{\pi+3}{12} = \frac{\pi/6 + 1/2}{2}
  = \frac{\theta_{\mathrm{wall}}(G_2) + \bar{e}(E_6)/h^\vee(E_6)}{2},
  \label{eq:midpoint-decomp}
\end{equation}
where $\theta_{\mathrm{wall}}(G_2) = \pi/h(G_2) = \pi/6$ is the $G_2$ Weyl
chamber wall angle (the source of Candidate~I) and
$\bar{e}(E_6)/h^\vee(E_6) = 6/12 = 1/2$ is the ratio of the mean $E_6$ exponent
$\bar{e}(E_6) = 6$ to the dual Coxeter number $h^\vee(E_6) = 12$ (the source
of Candidate~II).  Equivalently:
\begin{equation}
  \frac{\pi+3}{12} = \frac{\pi + N_{\rm gen}}{h^\vee(E_6)},
  \label{eq:midpoint-Ngen}
\end{equation}
where $N_{\rm gen} = 3$ is the number of Standard Model generations, which
equals the mean exponent $\bar{e}(E_6) \cdot h(G_2)/h^\vee(E_6) = 6\times 6/12$
--- more directly, since the mean of the $E_6$ exponents $\{1,4,5,7,8,11\}$ is
$36/6 = 6 = 2\times N_{\rm gen}$, we have $\bar{e}(E_6)/h^\vee(E_6) = 6/12 = 1/2
= N_{\rm gen}/(2N_{\rm gen}) = 1/2$, so $3 = \bar{e}(E_6)/2 = N_{\rm gen}$.
\end{proposition}

\begin{proof}
The algebraic identity $(\pi+3)/12 = (\pi/6+1/2)/2$ follows from:
$\pi/6 + 1/2 = \pi/6 + 3/6 = (\pi+3)/6$; dividing by 2 gives $(\pi+3)/12$.
The identification $N_{\rm gen} = 3 = \bar{e}(E_6)/2$ follows from
$\bar{e}(E_6) = (1+4+5+7+8+11)/6 = 36/6 = 6 = 2\times 3 = 2N_{\rm gen}$.
\end{proof}

\begin{remark}[Bookkeeping vs.\ structural derivation]
Equation~\eqref{eq:midpoint-Ngen} gives the midpoint correction a suggestive
form: the threshold correction to the $E_6$ spectral gap is $\MU$ times the
sum of the electron energy $E_e = \pi$ (the leading TOE constant) and the
number of generations $N_{\rm gen} = 3$, divided by the $E_6$ dual Coxeter number.
Written out:
\begin{equation}
  \varepsilon_{\rm mid} = \MU \cdot \frac{E_e + N_{\rm gen}}{h^\vee(E_6)},
  \label{eq:eps-narrative}
\end{equation}
with $E_e = \pi$ the spectral coordinate of the electron and
$N_{\rm gen} = 3$ the generation count.  This has a clear narrative: the
threshold correction at the $E_6$ scale involves the electron and the three
generations, normalized by the $E_6$ Coxeter height.

However, this is a \emph{bookkeeping identity}, not a structural derivation.
The identification $N_{\rm gen} = 3 = \bar{e}(E_6)/2$ is a numerical
coincidence of group theory (the mean $E_6$ exponent happens to equal $2N_{\rm gen}$),
and the averaging of Candidate~I and~II lacks a physical principle.  The
midpoint formula does not follow from the $G_2 \subset E_6$ geometry by a
derivation analogous to the holonomy argument for Candidate~I (P62
Theorem~4.4).
\end{remark}

\subsection{The half-splitting}

\begin{proposition}[Half-splitting]
\label{prop:half-split}
Each candidate deviates from the midpoint by:
\begin{equation}
  \delta \equiv \varepsilon_1 - \varepsilon_{\rm mid}
  = \MU\cdot\frac{\pi}{6} - \MU\cdot\frac{\pi+3}{12}
  = \MU\cdot\frac{2\pi - (\pi+3)}{12}
  = \MU\cdot\frac{\pi - 3}{12}.
  \label{eq:half-splitting}
\end{equation}
Numerically:
\begin{equation}
  \delta = \MU\cdot\frac{\pi-3}{12} = 0.793338\times\frac{0.14159}{12}
  = 0.793338\times 0.011799 = 0.009359.
\end{equation}
Candidate~I overshoots by $+\delta$ and Candidate~II undershoots by $-\delta$.
The $1\sigma$ observational uncertainty is $\delta\ER = 0.0088$, so
$\delta/\delta\ER = 0.009359/0.0088 = 1.063$---the half-splitting is
$1.06\sigma$.
\end{proposition}

\begin{proof}
By symmetry: $\varepsilon_{\rm mid} = (\varepsilon_1 + \varepsilon_2)/2$,
so $\varepsilon_1 - \varepsilon_{\rm mid} = (\varepsilon_1 - \varepsilon_2)/2
= (\MU\pi/6 - \MU/2)/2 = \MU(\pi/6 - 1/2)/2 = \MU(\pi-3)/12$.
\end{proof}

\begin{remark}[$\pi - 3$ as a fundamental residue]
The quantity $\pi - 3 = 0.14159\ldots$ is the fractional part of $\pi$,
i.e.\ the ``residual'' of $\pi$ after subtracting the number of generations.
It appears in the half-splitting as $\MU(\pi-3)/12 = \MU(\pi-N_{\rm gen})/
h^\vee(E_6)$.  The Phase~5b degeneracy is therefore governed by the
product of $\MU$ and the \emph{irrationality of $\pi$} relative to the integer
generation count, normalized by the $E_6$ Coxeter height.  Any structural
resolution of the degeneracy must either derive the averaging principle
(thereby confirming the midpoint formula) or provide a physical mechanism
for a correction of precisely $\MU(\pi-3)/12$ that selects one candidate.
\end{remark}

%==========================================================================
\section{Wolfenstein corrections to $\ER$}
\label{sec:wolfenstein}
%==========================================================================

\subsection{Setup: how PMNS angles enter $\ER$}

The spectral energy $\ER$ is the Phase~5b seesaw scale defined by
$\ER = 2E_v + |E_{\nu_3}|$ (P53, Theorem~3.1), where $E_v$ is the seesaw
vacuum spectral position and $E_{\nu_3}$ is the spectral coordinate of the
heaviest neutrino mass eigenstate.  The observational input is $\Dm_{31}
= m_{\nu_3}^2 - m_{\nu_1}^2$, which determines $m_{\nu_3}$ (in the normal
hierarchy) via
\begin{equation}
  m_{\nu_3}^2 \approx \Dm_{31}
  \quad \Longrightarrow \quad
  m_{\nu_3} \approx \sqrt{\Dm_{31}},
  \label{eq:mnu3-approx}
\end{equation}
and therefore
\begin{equation}
  E_{\nu_3} = \pi + \frac{\ln(m_{\nu_3}/m_e)}{\MU}
  = \pi + \frac{\ln(\sqrt{\Dm_{31}}/m_e)}{2\MU}\cdot\frac{2}{\,1\,}.
  \label{eq:Enu3}
\end{equation}
The PMNS angles enter because $\Dm_{31} = \Dm_{31}^{(\rm phys)}$ is the
physical (matter-corrected, PMNS-rotated) mass-squared splitting observed
in oscillation experiments; it differs from the vacuum mass-squared
$m_{\nu_3}^2 - m_{\nu_1}^2$ in the basis where the Yukawa matrix is diagonal
by a PMNS rotation.

In the TOE, the Yukawa matrix in the off-diagonal $\JO$ basis has eigenvalues
$y_k = \sin(k\pi/14)$ (P57, Theorem~2.1), and the physical oscillation mass
$\Dm_{31}^{(\rm phys)}$ receives corrections from the PMNS mixing angles:
\begin{equation}
  \Dm_{31}^{(\rm phys)} = \Dm_{31}^{(\rm vac)}\left[1
    + 2\sin^2\theta_{23}\cos^2\theta_{23}\cdot\frac{\Dm_{21}}{\Dm_{31}}
    + O\!\left(\frac{\Dm_{21}}{\Dm_{31}}\right)^2
    + O(\theta_{13}^2)\right].
  \label{eq:Dm31-corr}
\end{equation}
At leading order the correction is suppressed by $\Dm_{21}/\Dm_{31} \approx
\Dm_{21}/\Dm_{31} \approx 0.033$; this is small but the $\theta_{23}$
factor matters.  We compute the $O(\thetaC)$ Wolfenstein correction to
$\theta_{23}$ and track its effect on $\ER$.

\subsection{The $O(\thetaC^2)$ correction from $\theta_{23}$}

Addendum~65 proved $\theta_{23} = \pi/4$ (maximal mixing) from the $\mathbb{Z}_3$
symmetry of $\JO$.  Addendum~68 computed the subleading correction from the
second-order $G_2$ holonomy:
\begin{equation}
  \theta_{23} = \frac{\pi}{4} - \frac{\sin^2\thetaC}{2} + O(\lambda_W^3),
  \label{eq:th23-correction}
\end{equation}
where $\lambda_W = \sin\thetaC = \sin(\pi/14)$.  This is the standard
Wolfenstein-type expansion for the atmospheric angle.

\begin{theorem}[$O(\thetaC^2)$ Wolfenstein correction to $\ER$ is too large]
\label{thm:wolfenstein-too-large}
The $O(\thetaC^2)$ correction to $\theta_{23}$ from \eqref{eq:th23-correction}
induces a correction to $\ER$ of magnitude:
\begin{equation}
  \delta\ER^{(\mathrm{W},2)} \approx \frac{\sin^2\thetaC}{2\MU}
  = \frac{(0.22252)^2}{2\times 0.793338}
  = \frac{0.04951}{1.58668}
  = 0.03120.
  \label{eq:deltaER-W2}
\end{equation}
This is $0.03120/0.009 = 3.47$ times the required discrimination threshold.
The $O(\thetaC^2)$ atmospheric correction therefore \emph{overshoots} the
Phase~5b degeneracy by a factor of approximately $3.5$, and does not resolve it.
\end{theorem}

\begin{proof}
From \eqref{eq:th23-correction}, the correction $\delta\theta_{23}
= -\sin^2\thetaC/2$ enters the oscillation probability as:
\begin{equation}
  P(\nu_\mu \to \nu_\tau) = \sin^2(2\theta_{23})\sin^2\!\left(\frac{\Dm_{31} L}{4E}\right).
\end{equation}
A shift $\theta_{23} \to \theta_{23} + \delta\theta_{23}$ near maximal mixing
($\theta_{23} = \pi/4$) shifts the effective $\Dm_{31}^{(\rm eff)}$ extracted
from oscillation measurements by:
\begin{equation}
  \frac{\delta\Dm_{31}^{(\rm eff)}}{\Dm_{31}}
  \approx -2\tan(2\theta_{23})\cdot\delta\theta_{23}\big|_{\theta_{23}=\pi/4}
  \cdot\frac{1}{\sin^2(2\theta_{23})}
  = -2\cdot 1 \cdot (-\tfrac{\sin^2\thetaC}{2}) \cdot 1
  = \sin^2\thetaC.
\end{equation}
(At $\theta_{23} = \pi/4$: $\sin^2(2\theta_{23}) = 1$, $d\sin^2(2\theta)/d\theta
= 4\sin(2\theta)\cos(2\theta)|_{\pi/4} = 0$, so the leading correction
enters at quadratic order.  More carefully, $\delta\Dm_{31}^{(\rm eff)}/\Dm_{31}
= \sin^2\thetaC\cdot(1 + O(\sin^2\thetaC))$.)

Translating to a spectral coordinate shift via
$\delta\ER = -\delta\Dm_{31}^{(\rm eff)}/(2\MU\Dm_{31})$:
\begin{equation}
  |\delta\ER^{(\mathrm{W},2)}|
  = \frac{\sin^2\thetaC}{2\MU}
  = \frac{\sin^2(\pi/14)}{2\times 0.793338}.
\end{equation}
Numerically: $\sin(\pi/14) = \sin(12.857°) = 0.22252$,
$\sin^2(\pi/14) = 0.04951$,
$0.04951/(2\times 0.793338) = 0.04951/1.58668 = 0.03120$.
\end{proof}

\begin{remark}
The large magnitude of $\delta\ER^{(\mathrm{W},2)} = 0.031$ compared to the
required $\delta = 0.009$ means that if the $O(\thetaC^2)$ atmospheric
correction were the physical mechanism, it would not select one Phase~5b
candidate over the other --- it would instead shift both candidates by the
same $\delta\ER^{(\mathrm{W},2)} \gg 0.009$, effectively changing the
midpoint value.  This is not what is observed.  The implication is that the
atmospheric $O(\thetaC^2)$ correction is already incorporated in the
observed $\Dm_{31}$ (it affects the extraction of $\Dm_{31}$ from data), so
it should not be applied as a secondary correction to $\ER$.
\end{remark}

\subsection{Order-by-order scan}

\begin{proposition}[Wolfenstein corrections order by order]
\label{prop:wolfenstein-cubic}
Define $\lambda_W = \sin\thetaC = \sin(\pi/14) = 0.22252$ (the Wolfenstein
parameter) and $\thetaC = \pi/14 = 0.22440$~rad.  The induced $\ER$ corrections
at each order in the Wolfenstein expansion are:
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{llll}
\toprule
Order & Expression & Value & vs.\ required $0.009$ \\
\midrule
$O(\lambda_W)$   & $\MU\lambda_W = \MU\sin\thetaC$               & $0.17649$ & $19.6\times$ too large \\
$O(\thetaC)$     & $\MU\thetaC$                                    & $0.17803$ & $19.8\times$ too large \\
$O(\lambda_W^2)$ & $\MU\lambda_W^2/2 = \sin^2\thetaC/(2\MU)^{-1}\cdot\MU$ & see below & \\
                 & $\MU\lambda_W^2 = \MU\sin^2\thetaC$            & $0.039274$ & $4.4\times$ too large \\
                 & $\lambda_W^2/(2\MU)$                            & $0.031199$ & $3.5\times$ too large \\
$O(\thetaC^2)$   & $\MU\thetaC^2$                                  & $0.039932$ & $4.4\times$ too large \\
$O(\lambda_W^3)$ & $\MU\lambda_W^3 = \MU\sin^3\thetaC$            & $0.008749$ & $\mathbf{0.97\times}$ ($-3\%$) \\
$O(\thetaC^3)$   & $\MU\thetaC^3$                                  & $0.008966$ & $\mathbf{1.00\times}$ ($-0.4\%$) \\
$O(\lambda_W^4)$ & $\MU\lambda_W^4$                                & $0.001947$ & $0.22\times$ \\
\bottomrule
\end{tabular}
\end{center}
where the ``required $0.009$'' is the half-splitting $\delta = \MU(\pi-3)/12
= 0.009359$.  Corrections at order $O(\lambda_W^3)$ and $O(\thetaC^3)$ are
numerically close to the required value.
\end{proposition}

\begin{proof}
All values follow from $\lambda_W = \sin(\pi/14)$, $\thetaC = \pi/14$, and
$\MU = 0.793338$:
\begin{align}
  \MU\lambda_W^3 &= 0.793338\times(0.22252)^3
                  = 0.793338\times 0.011020 = 0.008749, \\
  \MU\thetaC^3   &= 0.793338\times(\pi/14)^3
                  = 0.793338\times\pi^3/2744
                  = 0.793338\times 0.011299 = 0.008966.
\end{align}
The proximity to $\delta = 0.009359$ is:
$(0.009359 - 0.008749)/0.009359 = 6.5\%$ for $\MU\lambda_W^3$;
$(0.009359 - 0.008966)/0.009359 = 4.2\%$ for $\MU\thetaC^3$.
\end{proof}

\begin{remark}[Why cubic and not lower orders?]
The coincidence at $O(\lambda_W^3)$ follows from the numerical proximity of
three quantities:
\begin{equation}
  \frac{\pi - 3}{12} \approx \frac{\pi^3}{14^3} \approx \frac{\sin^3(\pi/14)}{1}.
  \label{eq:three-way}
\end{equation}
Numerically:
$(\pi-3)/12 = 0.011799$,
$\pi^3/2744 = 0.011299$,
$\sin^3(\pi/14) = 0.011020$.
These three quantities span a $7\%$ range and do not coincide exactly.
The proximity is a consequence of $\pi \approx 3 + (something small)$ and
$\pi/14$ being a small angle, so that $\theta \approx \sin\theta \approx \tan\theta$
at this scale.  It is not a structural identity.
\end{remark}

\subsection{Does the cubic Wolfenstein correction resolve the degeneracy?}

\begin{proposition}[No structural resolution at $O(\lambda_W^3)$]
\label{prop:no-cubic-resolution}
The proximity $\MU\sin^3\thetaC \approx \delta$ (where $\delta = \MU(\pi-3)/12$
is the half-splitting) is numerical but not exact.  Specifically:
\begin{equation}
  \frac{\MU\sin^3\thetaC}{\delta}
  = \frac{\sin^3(\pi/14)}{(\pi-3)/12}
  = \frac{12\sin^3(\pi/14)}{\pi-3}
  = \frac{12\times 0.011020}{0.14159}
  = \frac{0.13224}{0.14159}
  = 0.934,
  \label{eq:cubic-ratio}
\end{equation}
a $6.6\%$ discrepancy.  No structural identity of the form
$12\sin^3(\pi/14) = \pi - 3$ holds; the two sides differ by $0.009$.
\end{proposition}

\begin{proof}
Evaluate $12\sin^3(\pi/14)$:
$\sin(\pi/14) = 0.222521$ (6 significant figures),
$\sin^3(\pi/14) = 0.011020$,
$12\times 0.011020 = 0.13224$.
The right-hand side: $\pi - 3 = 0.14159$.
$0.14159 - 0.13224 = 0.00935 \neq 0$.
\end{proof}

\begin{remark}[A possible approximate identity]
The relation $12\sin^3(\pi/14) \approx \pi - 3$ holds to $6.6\%$.  This
is better than a random approximation but falls well short of being a
TOE-level identity.  By contrast, the coincidences that drive the Phase~5b
candidates are accurate to $\lesssim 2.5\%$ (P62, Section~2.3).  A $6.6\%$
approximation would not be accepted as a structural result at the standard
of this corpus.
\end{remark}

\subsection{Summary: Wolfenstein corrections do not resolve Phase~5b}

\begin{corollary}[PMNS Wolfenstein corrections do not break the degeneracy]
\label{cor:no-resolution}
No Wolfenstein correction of the form $\MU\cdot f(\thetaC)$ with $f$ a
trigonometric or polynomial function of $\thetaC = \pi/14$ at order $k \leq 3$
equals the half-splitting $\delta = \MU(\pi-3)/12$ exactly.  The closest
is $\MU\thetaC^3$ at $4.2\%$ discrepancy and $\MU\sin^3\thetaC$ at $6.6\%$.
Neither constitutes a structural derivation.

Consequently, the Phase~5b degeneracy between Candidate~I and Candidate~II
is \emph{not} resolved by the PMNS Wolfenstein corrections at any order $\leq 3$.
The two candidates remain indistinguishable at the current $\Dm_{31}$ precision
of $1.4\%$, and the observation $\ER^{(\rm obs)} = 56.502$ coinciding with
the midpoint is not explained structurally by the PMNS sector.
\end{corollary}

\begin{proof}
Proposition~\ref{prop:wolfenstein-cubic} established that no Wolfenstein
correction at orders $k = 1, 2, 3$ matches $\delta$ exactly.
Proposition~\ref{prop:no-cubic-resolution} confirmed the closest case
($O(\lambda_W^3)$) fails by $6.6\%$.
\end{proof}

%==========================================================================
\section{Neutrino mass from the midpoint formula}
\label{sec:neutrino-mass}
%==========================================================================

\subsection{Deriving $m_{\nu_3}$ from $\ER = 56.502$}

If the midpoint formula is taken as exact --- i.e.\ if we assume
$\ER = 56.502$ is the correct Phase~5b value --- then the neutrino mass
$m_{\nu_3}$ can be derived as follows.

From P53 (Theorem~3.1), the seesaw spectral relation gives:
\begin{equation}
  |E_{\nu_3}| = \ER - 2E_v,
  \label{eq:Enu3-from-ER}
\end{equation}
where $E_v$ is the seesaw vacuum spectral position.  Using the P53 canonical
value $E_v = 19.636$ (derived from the right-handed neutrino mass scale):
\begin{align}
  |E_{\nu_3}| &= 56.502 - 2\times 19.636 = 56.502 - 39.272 = 17.230.
  \label{eq:Enu3-value}
\end{align}
Since $E_{\nu_3}$ is negative (neutrinos are far below the electron mass),
we have $E_{\nu_3} = -17.230$, and the TOE mass formula gives:
\begin{align}
  m_{\nu_3} &= m_e \cdot \exp\!\bigl(\MU(E_{\nu_3} - \pi)\bigr)
             = m_e \cdot \exp\!\bigl(\MU(-17.230 - \pi)\bigr) \notag \\
             &= m_e \cdot \exp\!\bigl(0.793338 \times (-17.230 - 3.14159)\bigr) \notag \\
             &= m_e \cdot \exp\!\bigl(0.793338 \times (-20.372)\bigr) \notag \\
             &= m_e \cdot \exp(-16.162).
  \label{eq:mnu3-calc}
\end{align}
Numerically:
\begin{align}
  \exp(-16.162) &= 9.524\times 10^{-8}, \\
  m_{\nu_3} &= 0.511\text{ MeV} \times 9.524\times 10^{-8}
             = 4.867\times 10^{-8}\text{ MeV} = 48.67\text{ meV}.
  \label{eq:mnu3-value}
\end{align}

\begin{proposition}[Neutrino mass from the midpoint formula]
\label{prop:mnu3}
If $\ER = \EPl - h^\vee(E_6) + \MU(\pi+3)/12 = 56.502$ exactly, then the
heaviest neutrino mass eigenstate has spectral mass:
\begin{equation}
  m_{\nu_3} = m_e\cdot\exp\!\bigl(\MU(-(|E_{\nu_3}|) - \pi)\bigr)
  = 0.511\,\mathrm{MeV}\times\exp(-16.162)
  \approx 48.7\,\mathrm{meV}.
  \label{eq:mnu3-prediction}
\end{equation}
This corresponds to $\sqrt{\Dm_{31}} \approx m_{\nu_3} \approx 48.7$~meV,
giving $\Dm_{31} \approx 2.37\times10^{-3}$~eV$^2$.

\textbf{Comparison with observation}: PDG $\Dm_{31} = 2.453\times10^{-3}$~eV$^2$
($\sqrt{\Dm_{31}} = 49.5$~meV).  The midpoint-formula prediction is $1.6\%$
below the central PDG value --- consistent with the $1.4\%$ measurement
uncertainty but not a confirmation.
\end{proposition}

\begin{proof}
Equations~\eqref{eq:Enu3-from-ER}--\eqref{eq:mnu3-value} above.
The comparison: $49.5^2\times10^{-6}$~eV$^2 = 2.450\times10^{-3}$~eV$^2$
(midpoint formula), vs.\ PDG $2.453\times10^{-3}$~eV$^2$.  Discrepancy
$0.12\%$---but this is circular since $\ER^{(\rm obs)} = 56.502$ was
\emph{derived from} the PDG $\Dm_{31}$ in P62.
\end{proof}

\begin{remark}[The circularity]
The midpoint formula $\ER^{(\rm mid)} = 56.502$ is derived by taking
the arithmetic mean of $\ER^{(\rm I)} = 56.511$ and $\ER^{(\rm II)} = 56.493$.
These candidates were themselves fitted to the observed $\ER^{(\rm obs)}
= 56.502$ from $\Dm_{31}$ in P62.  Therefore the prediction $m_{\nu_3}
\approx 48.7$~meV from the midpoint formula is not an independent prediction ---
it recovers (approximately) the input.  The midpoint formula can make a
genuine prediction only if it is derived from structure, not from averaging
the two candidates that were already tuned to the data.
\end{remark}

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

\subsection{Status of each result}

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{p{7cm}p{3cm}l}
\toprule
\textbf{Statement} & \textbf{Status} & \textbf{Ref.} \\
\midrule
Midpoint $(\ER^{(\rm I)}+\ER^{(\rm II)})/2 = 56.502$ & Proved & Prop.~\ref{prop:midpoint-numerical} \\
Midpoint correction $= \MU(\pi+N_{\rm gen})/h^\vee(E_6)$ & Proved (algebraic identity) & Prop.~\ref{prop:midpoint-algebra} \\
Half-splitting $\delta = \MU(\pi-3)/12 = 0.009359$ & Proved & Prop.~\ref{prop:half-split} \\
$O(\thetaC^2)$ Wolfenstein correction is $3.5\times$ too large & Proved & Thm.~\ref{thm:wolfenstein-too-large} \\
$O(\thetaC^3)$ proximity ($4$--$7\%$) to $\delta$ & Proved (numerical) & Prop.~\ref{prop:wolfenstein-cubic} \\
$12\sin^3(\pi/14) \neq \pi-3$ & Proved & Prop.~\ref{prop:no-cubic-resolution} \\
PMNS Wolfenstein $\leq O(\lambda_W^3)$ does not resolve degeneracy & Proved & Cor.~\ref{cor:no-resolution} \\
$m_{\nu_3} \approx 48.7$~meV from midpoint formula & Computed & Prop.~\ref{prop:mnu3} \\
Midpoint formula: structural derivation & \textbf{Open} & \S\ref{sec:conjecture} \\
Phase~5b closure & \textbf{Conjecture} & Conj.~\ref{conj:midpoint} \\
\bottomrule
\end{tabular}
\end{center}

\subsection{The midpoint conjecture}
\label{sec:conjecture}

\begin{conjecture}[Phase~5b closure via the midpoint formula]
\label{conj:midpoint}
The correct Phase~5b formula for the seesaw scale spectral energy is:
\begin{equation}
  \boxed{
  \ER = \EPl - h^\vee(E_6) + \MU\cdot\frac{\pi + N_{\mathrm{gen}}}{h^\vee(E_6)}
  = \EPl - 12 + \MU\cdot\frac{\pi + 3}{12},
  }
  \label{eq:midpoint-conjecture}
\end{equation}
where $\pi = E_e$ is the spectral coordinate of the electron, $N_{\rm gen} = 3$
is the number of Standard Model generations, and $h^\vee(E_6) = 12$.
Numerically: $\ER^{(\rm pred)} = 68.096 - 12 + 0.40607 = 56.502$, coinciding
with the observed value to three decimal places.

The conjecture is \emph{structural} in the following sense: the threshold
correction $\MU(\pi+3)/12$ is the mean of the two geometrically-motivated
corrections in P62 (the $G_2$ Weyl chamber angle $\MU\pi/6$ and the mean
$E_6$ exponent $\MU/2$).  Both corrections arise from the $G_2 \subset E_6$
breaking at the seesaw scale.  The mean is the correction when both geometric
sources contribute equally --- i.e.\ when the $E_6 \to G_2$ spectral flow
does not prefer the angular correction over the exponent correction, but averages them.

The conjecture is \emph{not a theorem} because:
\begin{enumerate}[label=(\alph*)]
  \item The averaging principle (equal weight to $G_2$ Weyl angle and
    $E_6$ mean exponent) lacks a derivation from the $\JO$ trilinear structure.
  \item The coincidence $\ER^{(\rm mid)} = \ER^{(\rm obs)}$ holds to three
    decimal places but the observed value carries a $1.4\%$ uncertainty;
    a change in the PDG $\Dm_{31}$ central value within this band would
    shift $\ER^{(\rm obs)}$ by up to $\pm 0.009$ (to $56.493$ or $56.511$),
    which are the Candidate~II or Candidate~I values respectively.
  \item The identification $N_{\rm gen} = 3 = \bar{e}(E_6)/2$ is a numerical
    coincidence of Lie theory, not a structural relation imposed by the TOE.
\end{enumerate}
\end{conjecture}

\subsection{What would close Phase~5b}

\begin{enumerate}[label=(\alph*)]
  \item \textbf{Theoretical (averaging principle)}: Derive the equal weighting of
    the $G_2$ Weyl angle and $E_6$ mean exponent from the
    $J_3(\OO)$ trilinear structure or from the conformal field theory of the
    $\widehat{E}_6$ level-1 WZW model.  Specifically: the holonomy correction
    at the trinification breaking scale involves an integral over the $G_2$-orbit
    in the $E_6$ root space; if this integral has equal contributions from the
    ``angular'' ($\pi/h(G_2)$) and ``spectral'' ($\bar{e}(E_6)/h^\vee(E_6)$)
    directions, the midpoint formula follows.  This would upgrade
    Conjecture~\ref{conj:midpoint} to a theorem.

  \item \textbf{Observational}: A $\Dm_{31}$ measurement with precision
    $\lesssim 0.5\%$ that places $\ER^{(\rm obs)}$ clearly above $56.502$
    (confirming Candidate~I: $\ER = 56.511$), or below (confirming
    Candidate~II: $\ER = 56.493$), or precisely at $56.502$ (consistent
    with the midpoint formula).

  \item \textbf{Ruling out the midpoint conjecture}: If the precision
    measurement yields $\ER^{(\rm obs)} \neq 56.502$ by more than $0.003$
    (i.e.\ $|\ER^{(\rm obs)} - 56.502| > 0.003$), the midpoint conjecture
    is falsified at $> 3\sigma$ and one of the P62 candidates is confirmed.

  \item \textbf{Fourth candidate}: If the Wolfenstein O($\lambda_W^3$) term
    has the structural derivation
    \begin{equation}
      \delta_{\rm W3} = \frac{1}{2}\MU\sin^3\thetaC \cdot (\text{group theory factor}),
    \end{equation}
    with the group theory factor from the $G_2$ off-diagonal root structure,
    then the Phase~5b picture acquires a fourth candidate whose value can be
    computed from P66 data.  We compute:
    \begin{equation}
      \ER^{(\rm W3)} = \EPl - 12 + \MU\cdot(\tfrac{\pi+3}{12} + \sin^3\thetaC)
      = 56.502 + \MU\sin^3\thetaC = 56.502 + 0.008749 = 56.511 = \ER^{(\rm I)}.
    \end{equation}
    Up to the $6.6\%$ discrepancy between $\sin^3(\pi/14)$ and $(\pi-3)/12$,
    the Wolfenstein O($\lambda_W^3$) shifted midpoint reproduces Candidate~I.
    This coincidence ($\ER^{(\rm I)} \approx \ER^{(\rm mid)} + \MU\lambda_W^3$)
    is a further numerical observation warranting investigation but not
    constituting a derivation.
\end{enumerate}

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

\subsection{What was proved}

\textbf{1. The midpoint identity.}
The arithmetic mean of the two Phase~5b candidates from P62 is
$\ER^{(\rm mid)} = \EPl - 12 + \MU(\pi+3)/12 = 56.502$, coinciding with
the observed $\ER$ to three decimal places.  This is a proved numerical identity
(Proposition~\ref{prop:midpoint-numerical}).

\textbf{2. The $G_2/E_6$ bookkeeping.}
The correction $(\pi+3)/12$ equals $(\pi + N_{\rm gen})/h^\vee(E_6)$ with
$N_{\rm gen} = 3$.  The factor $(\pi + 3)$ is the sum of the electron
spectral energy $E_e = \pi$ and the number of generations.  This is a
proved algebraic identity (Proposition~\ref{prop:midpoint-algebra}) but not
a structural derivation.

\textbf{3. Wolfenstein corrections do not resolve the degeneracy.}
The $O(\thetaC^2)$ atmospheric correction is $3.5\times$ the required
$\delta = 0.009359$ (Theorem~\ref{thm:wolfenstein-too-large}).  No
Wolfenstein correction at order $k \leq 3$ matches $\delta$ exactly
(Corollary~\ref{cor:no-resolution}).  The Phase~5b degeneracy is not broken
by the PMNS sector at these orders.

\textbf{4. Cubic proximity.}
The $O(\lambda_W^3)$ term $\MU\sin^3\thetaC \approx 0.00875$ and
$O(\thetaC^3)$ term $\MU\thetaC^3 \approx 0.00897$ bracket the half-splitting
$\delta = 0.00936$ at the $4$--$7\%$ level
(Proposition~\ref{prop:wolfenstein-cubic}).  This is numerically suggestive
of a connection at cubic order but not structurally derivable.

\textbf{5. The neutrino mass.}
Assuming the midpoint formula is exact, $m_{\nu_3} \approx 48.7$~meV
(Proposition~\ref{prop:mnu3}).  This is not an independent prediction because
the midpoint formula was derived from the observed $\Dm_{31}$.

\subsection{Phase 5b status}

\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{p{5cm}p{5cm}p{2.5cm}}
\toprule
\textbf{Path to closure} & \textbf{What it requires} & \textbf{Status} \\
\midrule
Candidate I confirmed & $\Dm_{31}$ at $\lesssim 0.5\%$ giving $\ER > 56.502$ & Pending JUNO/Hyper-K \\
Candidate II confirmed & $\Dm_{31}$ at $\lesssim 0.5\%$ giving $\ER < 56.502$ & Pending JUNO/Hyper-K \\
Midpoint formula proved & Structural derivation of the averaging principle from $\JO$ geometry & Open \\
Wolfenstein O($\lambda_W^3$) resolves it & Exact identity $12\sin^3(\pi/14) = \pi - 3$ & Fails ($6.6\%$ off) \\
\bottomrule
\end{tabular}
\end{center}

Phase~5b is \emph{not yet closed}.  The midpoint formula
$\ER = \EPl - 12 + \MU(\pi+3)/12$ is a compelling conjecture
(Conjecture~\ref{conj:midpoint}) supported by a numerical coincidence
and a natural bookkeeping identity in $G_2/E_6$ data.  The structural derivation
of the averaging principle, and the precision $\Dm_{31}$ measurement, remain
the two gates to closure.

%==========================================================================
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\end{document}
