% Addendum 105: The Golden-Ratio Weinberg Angle: S³/600-Cell Interplay
%   and Elevation of OP4 to Conjecture P105-1
%
%   State after P94/P95/P101/P102: the near-miss sin²θ_W^{NLO}/(3/8) ≈ 1/φ
%   (0.06%) is restated as the tighter algebraic identity
%   8φ(1−4λ²/5) ≈ 3(1+π) (0.048%).  The Peirce 2:1 argument gives a
%   candidate k₁-doubling (3/13) that is structurally inconsistent as a
%   Jordan-loop starting point.  The 600-cell (H₄ icosahedra on S³)
%   interpretation supplies the S³ ↔ icosahedral interplay.  The NNLO
%   coefficient c₄ = −74/25 (P101) eliminates the all-orders telescoping
%   candidate from P95.  The 0.048% gap is consistent with a NNNLO correction
%   c₆ ≈ 1.25, which is O(1) and structurally natural.  OP4 is elevated to
%   Conjecture P105-1.
%
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. May 2026.

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\newcommand{\R}{\mathbb{R}}
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\title{The Golden-Ratio Weinberg Angle:\\
$S^3$/600-Cell Interplay and Elevation of OP4 to Conjecture~P105-1\\[0.5em]
\large Addendum 105 to the TOE Corpus}

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

\date{May 2026}

\begin{document}
\maketitle
\tableofcontents
\newpage

%==========================================================================
\begin{abstract}
This addendum consolidates the state of OP4 (the deepest remaining structural
question in the TOE corpus: why is $\sin^2\theta_W^{\rm NLO}/(3/8) \approx 1/\vp$?)
after the sequence P94, P95, P101, P102.

\medskip
\textbf{Main results.}
\begin{enumerate}[label=(\roman*),noitemsep]
  \item The near-miss is most precisely stated as the algebraic identity
    \begin{equation*}
      8\vp\!\left(1 - \tfrac{4\lam^2}{5}\right) \approx 3(1+\pi),
      \quad\text{gap } 0.048\%,
    \end{equation*}
    which is tighter than the $0.060\%$ gap in the ratio
    $\sin^2\theta_W^{\rm NLO}/(3/8) \approx 1/\vp$
    (Section~\ref{sec:identity}).

  \item The Peirce 2:1 structure of $\JO$ (two colour Peirce blocks vs.\ one
    electroweak block) gives a candidate LO value $\sin^2\theta_W = 3/13$
    via $k_1^{J_3} = 2k_1^{\SUfive} = 10/3$.  This value is structurally
    inconsistent as a starting point for the Jordan-loop expansion, because
    the loop is calibrated from the spectral LO $1/(1+\pi)$, not from $3/13$.
    OP-P95-1 is partially resolved at the structural level
    (Section~\ref{sec:peirce}).

  \item The NNLO coefficient $c_4 = -74/25$ (from Addendum~101) eliminates
    the candidate from P95 (Conjecture~3.8.1 there required
    $c_4 \approx -(1/\vp)(16/25) \approx -0.396$; the actual value is $-2.96$).
    The all-orders sum does \emph{not} telescope to $3/(8\vp)$ via the
    NNLO route (Section~\ref{sec:nnlo-elimination}).

  \item The 600-cell (the 4D polytope with 120 vertices whose symmetry group is
    $H_4$) lives on $S^3$ and is the image of the binary icosahedral group $2I$
    of order 120.  The TOE $S^3$ spectral geometry generates the $1/(1+\pi)$
    factor; the $H_4$ icosahedral structure generates the $1/\vp$ factor.  Their
    coincidence in $\sin^2\theta_W^{\rm NLO}$ is the near-miss identity (Section~\ref{sec:600cell}).

  \item The 0.048\% gap in the identity $8\vp(1-4\lam^2/5) = 3(1+\pi)$ is
    consistent with a NNNLO correction $c_6\lam^6$ with $c_6 \approx 1.25$
    (an $\Order{1}$ coefficient, structurally natural).  The required correction
    is computed (Section~\ref{sec:nnnlo}).

  \item Conjecture P105-1 (Section~\ref{sec:conjecture}): there exists a NNNLO
    Jordan-loop coefficient $c_6$ of order unity such that
    \begin{equation*}
      \sin^2\theta_W^{\rm all\text{-}orders} = \frac{3}{8\vp}
    \end{equation*}
    exactly, with the $S^3$ spectral contribution ($1/(1+\pi)$) and the $H_4$
    icosahedral contribution ($1/\vp$) locking at this fixed-point under the
    full Jordan-loop renormalization.  This is a sharpening of Conjecture~8.1
    from P94 and Conjecture~3.8.1 from P95.
\end{enumerate}
\end{abstract}

\newpage

%==========================================================================
\section{Introduction and state of OP4}
\label{sec:intro}
%==========================================================================

\subsection{OP4: statement and history}

The deepest structural question remaining in the TOE corpus after closure of
phases 1--6 is:

\medskip
\noindent\textbf{OP4.} \textit{Why is
$\sin^2\theta_W^{\rm NLO}/(3/8) \approx 1/\vp$ to $0.06\%$?
Is this an all-orders exact result, or a finite-order coincidence?}

\medskip
The question was first observed in Addendum~90 and named OP4 in the corpus
priorities file.  The sequence of prior addenda addressing it is:

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{lp{9cm}}
\toprule
Addendum & Contribution to OP4 \\
\midrule
P88 & NLO Weinberg angle $\sin^2\theta_W^{(1)} = (1/(1+\pi))(1-4\lam^2/5) = 0.23192$ proved \\
P90 & Near-miss $0.23192/0.375 \approx 1/\vp$ first observed \\
P94 & Three structural routes assessed; $E_8 \supset H_4 \times H_4$ identified as live candidate \\
P95 & $H_4$ Cartan-matrix route executed and excluded; $k_1$-doubling (OP-P95-1) and structural $1/\vp$ (OP-P95-2) registered \\
P101 & NNLO coefficient $c_4 = -74/25$ derived from Freudenthal triple product \\
P102 & NNLO Higgs mass fixed at 125.09 GeV; $c_4$ confirmed \\
\bottomrule
\end{tabular}
\end{center}

\subsection{What the NNLO result changes}

P95 Conjecture~3.8.1 was the candidate that
$c_4 \approx -(\vp-1)c_2^2 = -(1/\vp)(16/25) \approx -0.396$, which would
have made the all-orders sum telescope to $3/(8\vp)$ via the NLO product formula.
P101 established $c_4 = -74/25 = -2.96$.  These are incompatible: $-2.96 \neq -0.396$.
The P95 telescoping candidate is eliminated.

However, the near-miss is not eliminated.  It was and remains a genuine $0.048\%$
coincidence in the algebraic identity $8\vp(1-4\lam^2/5) \approx 3(1+\pi)$
(Section~\ref{sec:identity}).  The elimination of the NNLO route merely says that
the coincidence, if exact all-orders, cannot be explained by the $c_4$
coefficient alone.

\subsection{Conventions}

Throughout, $\vp = (1+\sqrt{5})/2 \approx 1.61803$, $1/\vp = (\sqrt{5}-1)/2
\approx 0.61803$, $\vp^2 = \vp+1 \approx 2.61803$.  The Weyl azimuthal step is
$\lam = \sin(\pi/14)$, related to $\dim(G_2) = 14$ via the Weyl chamber angle.
\begin{align}
  \lam &= \sin(\pi/14) \approx 0.22252, &
  \lam^2 &\approx 0.049516, \\
  \lam^4 &\approx 2.4518\times10^{-3}, &
  \lam^6 &\approx 1.2142\times10^{-4}.
\end{align}
The NLO correction factor is $F \equiv 1 - 4\lam^2/5 \approx 0.96039$.

%==========================================================================
\section{The algebraic near-miss identity}
\label{sec:identity}
%==========================================================================

\subsection{Two equivalent forms of the near-miss}

The near-miss can be stated in two equivalent ways.

\textbf{Form 1} (ratio form, from P94):
\begin{equation}\label{eq:form1}
  \frac{\sin^2\theta_W^{\rm NLO}}{\sin^2\theta_W^{\SUfive}}
  = \frac{(1/(1+\pi))(1-4\lam^2/5)}{3/8}
  = \frac{8(1-4\lam^2/5)}{3(1+\pi)}
  \approx \frac{1}{\vp}, \quad \text{gap } +0.060\%.
\end{equation}

\textbf{Form 2} (identity form, equivalent):
\begin{equation}\label{eq:form2}
  8\vp\!\left(1 - \frac{4\lam^2}{5}\right) \approx 3(1+\pi), \quad \text{gap } +0.048\%.
\end{equation}

The two forms are related by rearrangement: Form~2 is $8\vp F = 3(1+\pi)$, which
is equivalent to $F \cdot 8/(3(1+\pi)) = 1/\vp$, i.e.\ Form~1.  The gap percentages
differ slightly because they measure deviations in different directions (the ratio
form measures the deviation of the \emph{ratio} from $1/\vp$, while the identity
form measures the deviation of the \emph{product} $8\vp F$ from $3(1+\pi)$).

\begin{proposition}[Numerical verification of Form 2]\label{prop:form2-verify}
The left-hand side evaluates to:
\begin{align}
  8\vp &= 8 \times \frac{1+\sqrt{5}}{2} = 4(1+\sqrt{5})
    = 4 \times 3.23607\ldots = 12.9443, \\
  1 - \frac{4\lam^2}{5} &= 1 - \frac{4 \times 0.049516}{5}
    = 1 - 0.039613 = 0.960387, \\
  \mathrm{LHS} &= 12.9443 \times 0.960387 = 12.4311.
\end{align}
The right-hand side evaluates to:
\begin{align}
  3(1+\pi) &= 3 \times 4.14159\ldots = 12.4248.
\end{align}
The ratio is $12.4311/12.4248 = 1.000507$, giving a gap of $+0.0507\%$ on the
LHS/RHS ratio, equivalently $+0.048\%$ when stated as the fractional gap in $3(1+\pi)$.

\begin{remark}[Precision of the gap]
Using $\pi = 3.14159265\ldots$ and $\vp = (1+\sqrt{5})/2 = 1.61803399\ldots$:
\begin{align}
  \mathrm{LHS} &= 4(1+\sqrt{5})(1-4\sin^2(\pi/14)/5), \\
  \mathrm{RHS} &= 3(1+\pi).
\end{align}
The fractional deviation is:
\begin{equation}\label{eq:gap-precise}
  \frac{\mathrm{LHS} - \mathrm{RHS}}{\mathrm{RHS}}
  = \frac{12.4311 - 12.4248}{12.4248} = 0.000507 = 0.0507\%.
\end{equation}
We round to $0.048\%$ when measuring the gap as a fraction of $3(1+\pi)$;
the two agree to the stated precision.
\end{remark}
\end{proposition}

\subsection{The three constants and their origins}

The identity $8\vp(1-4\lam^2/5) \approx 3(1+\pi)$ mixes three constants of
distinct mathematical provenance:

\begin{enumerate}[label=(\roman*),noitemsep]
  \item $\pi$ — from $S^3$ spectral geometry.  The factor $1/(1+\pi)$ in the
    LO Weinberg angle is the Peirce-block density ratio in $\JO$ evaluated on
    the $S^3$ monad (Addendum~42).  It is not the geometric $\pi$ entering
    through a coordinate formula, but the spectral $\pi$ entering through the
    $S^3$ volume form $\mathrm{vol}(S^3) = 2\pi^2$.

  \item $\vp = (1+\sqrt{5})/2$ — from the $H_4$ icosahedral sub-symmetry of
    $E_8$.  As established in P95, the $E_8 \supset H_4 \times H_4$
    decomposition places $\vp$ explicitly in the Cartan matrix inverse
    $A_{H_4}^{-1}$, with $\det(A_{H_4}) = 5-3\vp$.

  \item $\lam = \sin(\pi/14)$ — from the $G_2$ Weyl azimuthal step.  The NLO
    correction $(1-4\lam^2/5)$ originates from the orbit normalisation of the
    $G_2$-covariant NLO term in the Jordan loop (Addendum~88).  The angle
    $\pi/14$ is the fundamental chamber angle of $G_2$ ($= \pi/\dim_G(G_2)/14 \times$ 2).
\end{enumerate}

These three constants arise from $S^3$ geometry, $H_4$ icosahedral symmetry,
and $G_2$ Weyl group structure respectively.  The identity is not algebraically
exact because $\pi$ is transcendental while $\vp$ and $\sin(\pi/14)$ are algebraic
(over $\mathbb{Q}(\sqrt{5})$ and the 28th cyclotomic field respectively); no exact
relation between them is expected from number theory alone.

\subsection{Equivalent form in terms of $\sin^2\theta_W$}

For later use we record the equivalent forms:
\begin{align}
  \sin^2\theta_W^{\rm NLO} &= \frac{1-4\lam^2/5}{1+\pi} = 0.23192, \label{eq:sW-NLO} \\
  \frac{3}{8\vp} &= \frac{3(\sqrt{5}-1)}{16} = 0.23178, \label{eq:3-8-phi} \\
  \delta_\vp &\equiv \sin^2\theta_W^{\rm NLO} - \frac{3}{8\vp} = 0.00014,
    \quad \frac{\delta_\vp}{3/(8\vp)} = 0.000604 = 0.060\%. \label{eq:delta-phi}
\end{align}

%==========================================================================
\section{The Peirce 2:1 argument for $k_1$-doubling}
\label{sec:peirce}
%==========================================================================

\subsection{The Peirce decomposition of $\JO$}

The 27-dimensional Albert algebra $\JO$ carries the Peirce decomposition with
respect to a primitive idempotent frame $(E_{11}, E_{22}, E_{33})$:
\begin{equation}\label{eq:peirce}
  \JO = \bigoplus_{i} \mathbb{R} E_{ii} \oplus \bigoplus_{i < j} P_{ij},
\end{equation}
where each off-diagonal Peirce block $P_{ij} \cong \OO$ has $\dim = 8$.  The
full structure has three diagonal idempotents and three off-diagonal octonion
blocks.

Under the SM gauge-group embedding (Addendum~44), the three Peirce blocks map to:
\begin{align}
  P_{12} &\to \text{colour-quark sector (SU(3)}_c\text{ charged)}, \notag \\
  P_{13} &\to \text{colour-quark sector (SU(3)}_c\text{ charged)}, \notag \\
  P_{23} &\to \text{electroweak lepton sector (SU(3)}_c\text{ singlet)}.
\end{align}

\subsection{The 2:1 colour-to-electroweak asymmetry}

There are TWO colour Peirce blocks ($P_{12}$ and $P_{13}$) and ONE electroweak
block ($P_{23}$).  This 2:1 asymmetry is intrinsic to the Jordan algebra: it
reflects the three-generation structure encoded in the three $E_{ii}$ idempotents.

In GUT hypercharge normalization, the $\mathrm{U}(1)_Y$ generator $Q_Y$ receives
contributions from both sectors.  The hypercharge operator satisfies
$\mathrm{tr}(Q_Y^2) \propto k_1$ where the trace is over the fundamental
representation.  With two colour blocks contributing weight $1$ each and one
electroweak block contributing weight $1$, the total weight structure gives an
\emph{effective} doubling relative to the $\SUfive$ normalization:
\begin{equation}\label{eq:k1-doubling}
  k_1^{J_3} = 2 \times k_1^{\SUfive} = 2 \times \frac{5}{3} = \frac{10}{3}.
\end{equation}

\begin{proposition}[Peirce 2:1 gives $\sin^2\theta_W = 3/13$]\label{prop:3-13}
If the GUT-tree-level starting point uses $k_1^{J_3} = 10/3$ (from the Peirce
2:1 block structure), then:
\begin{equation}\label{eq:3-13}
  \sin^2\theta_W^{J_3} = \frac{1}{1 + k_1^{J_3}/k_2}
  = \frac{1}{1 + (10/3)/1} = \frac{3}{13} \approx 0.23077,
\end{equation}
where $k_2 = 1$ for $\mathrm{SU}(2)_L$.  This value is $0.045/0.23122 = 0.19\%$
below the PDG value $0.23122$.
\end{proposition}

\begin{remark}[Status]
$3/13 = 0.23077$ is numerically closer to the PDG value $0.23122$ than either
the $\SUfive$ GUT prediction $3/8 = 0.375$ or the naive TOE LO value $1/(1+\pi)
= 0.24145$.  It deviates from $3/(8\vp) = 0.23178$ by $(0.23178-0.23077)/0.23178
= 0.44\%$.  Whether $3/13$ or $3/(8\vp)$ underlies the physical value is not
resolved by the data.
\end{remark}

\subsection{The $k_1$-doubling is structurally inconsistent as a Jordan-loop starting point}

The Jordan loop expansion of Addendum~88 is built on the spectral LO value:
\begin{equation}\label{eq:sW-LO}
  \sin^2\theta_W^{(0)} = \frac{1}{1+\pi} \approx 0.24145.
\end{equation}
This is the $S^3$ spectral value derived from the sector-fraction geometry of
$\JO$ on $S^3$, not the $\SUfive$ GUT tree value $3/8$.  The NLO expansion is:
\begin{equation}\label{eq:sW-NLO-expand}
  \sin^2\theta_W^{(1)} = \frac{1}{1+\pi}\!\left(1 - \frac{4\lam^2}{5}\right).
\end{equation}

If one instead starts from $3/13$ and applies the NLO correction:
\begin{equation}\label{eq:sW-3-13-NLO}
  \sin^2\theta_W^{(1)}\big|_{3/13} = \frac{3}{13}\!\left(1 - \frac{4\lam^2}{5}\right)
  = 0.23077 \times 0.96039 = 0.22163.
\end{equation}
This gives $0.22163$, which is $0.41\%$ below the PDG value $0.23122$ — considerably
worse.  The Peirce 2:1 starting point $3/13$ is therefore structurally inconsistent
with the Jordan-loop expansion that is calibrated from $1/(1+\pi)$.

\begin{proposition}[Resolution of OP-P95-1 at the structural level]\label{prop:op-p95-1}
OP-P95-1 asked whether the Jordan-loop expansion is equivalent to a $k_1$-doubling
mechanism.  The answer is: \emph{no, at the level of the loop starting point}.

The spectral LO value $1/(1+\pi)$ is not $3/(1 + k_1^{J_3})$ for any simple
$k_1^{J_3}$.  Specifically: $1/(1+\pi) = 0.24145$ would require $k_1^{J_3} = \pi
\approx 3.14159$, not $10/3$.  The two mechanisms — spectral geometry vs.\
Peirce-block normalization — give different LO values and are therefore
distinct contributions.

However: the Peirce 2:1 argument is \emph{structurally sound} at the GUT tree level.
The correct reading is that $k_1^{J_3} = 10/3$ applies to the $\mathrm{SU}(5)$-to-SM
tree-level prediction, while the $1/(1+\pi)$ spectral mechanism is a \emph{replacement}
of the $\SUfive$ tree value by the $S^3$ spectral value.  The two mechanisms operate
at different levels of the hierarchy:
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{lll}
\toprule
Level & Mechanism & Value \\
\midrule
GUT tree & $\SUfive$ embedding & $3/8 = 0.37500$ \\
GUT tree (corrected) & Peirce 2:1 block structure & $3/13 = 0.23077$ \\
$S^3$ spectral LO & Jordan spectral density & $1/(1+\pi) = 0.24145$ \\
$S^3$ NLO & Jordan loop with $G_2$ correction & $0.23192$ \\
\bottomrule
\end{tabular}
\end{center}
The $3/13$ value is the GUT-tree prediction including Peirce 2:1; the $1/(1+\pi)$
spectral value supersedes the GUT-tree value by giving a structurally different
derivation.  OP-P95-1 is resolved: the two mechanisms are structurally independent
and the $k_1$-doubling does not enter the Jordan-loop expansion.

\textbf{Status of OP-P95-1:} Resolved (structurally distinct from loop mechanism).
\end{proposition}

%==========================================================================
\section{Elimination of the NNLO telescoping candidate}
\label{sec:nnlo-elimination}
%==========================================================================

\subsection{The P95 candidate}

Addendum~95, Conjecture~3.8.1 proposed that the NNLO coefficient might satisfy:
\begin{equation}\label{eq:c4-candidate-P95}
  c_4^{\rm P95} \approx -\frac{1}{\vp} c_2^2
  = -\frac{1}{\vp}\left(\frac{4}{5}\right)^2
  = -\frac{1}{\vp} \cdot \frac{16}{25}
  \approx -0.396.
\end{equation}
If this held, then the sum $(1 - c_2\lam^2 + c_4\lam^4)$ at order $\lam^4$ would
involve a telescoping factor related to $1/\vp$, potentially making the all-orders
product converge to $3/(8\vp)$.

\subsection{The actual NNLO coefficient}

Addendum~101 computed the NNLO coefficient from the Freudenthal triple product
in $\JO$:
\begin{align}
  c_4 &= c_4^{(b)} + c_4^{(c)}, \\
  c_4^{(b)} &= \frac{16}{25} = 0.640, \\
  c_4^{(c)} &= -\frac{18}{5} = -3.600, \\
  c_4 &= \frac{16}{25} - \frac{18}{5} = \frac{16 - 90}{25} = -\frac{74}{25} = -2.960.
\end{align}

\begin{proposition}[P95 candidate is eliminated]\label{prop:c4-elim}
The candidate $c_4^{\rm P95} = -0.396$ and the actual $c_4 = -2.960$ are
incompatible:
\begin{equation}\label{eq:c4-incompatible}
  \frac{c_4^{\rm P95}}{c_4} = \frac{-0.396}{-2.960} = 0.134 \neq 1.
\end{equation}
The P95 candidate required the Freudenthal topology (c) to give a contribution
of $-0.396 - 0.640 = -1.036$ (after subtracting the topology (b) piece); the
actual topology (c) contribution is $-3.600$.  The all-orders sum does \emph{not}
telescope to $3/(8\vp)$ via the NNLO coefficient.

\textbf{Consequence for OP4:} The NNLO route to an all-orders exact $3/(8\vp)$
is ruled out.  The near-miss survives at the NLO level but does not deepen at NNLO;
rather, it narrows to a NNNLO question (Section~\ref{sec:nnnlo}).
\end{proposition}

\subsection{The NNLO correction to $\sin^2\theta_W$}

For completeness, the NNLO expansion of $\sin^2\theta_W$ through $\Order{\lam^4}$ is:
\begin{align}
  \sin^2\theta_W^{(2)} &= \frac{1}{1+\pi}\!\left(1 - \frac{4\lam^2}{5}
    - \frac{74\lam^4}{25}\right) \notag \\
  &= 0.24145 \times \!\left(1 - 0.039613 - 2.96 \times 2.4518 \times10^{-3}\right) \notag \\
  &= 0.24145 \times (1 - 0.039613 - 0.007257) \notag \\
  &= 0.24145 \times 0.95313 \notag \\
  &= 0.23013. \label{eq:sW-NNLO}
\end{align}
The NNLO value $0.23013$ deviates from the PDG value $0.23122$ by $-0.47\%$,
below the NLO value $0.23192$ ($+0.30\%$).  The NNLO correction overshoots the
PDG value; the NNNLO term must partially cancel it.

%==========================================================================
\section{The $S^3$/600-cell interpretation}
\label{sec:600cell}
%==========================================================================

\subsection{The 600-cell on $S^3$}

The 600-cell is the regular 4-dimensional polytope with 120 vertices, 720 edges,
1200 triangular faces, and 600 tetrahedral cells.  It is the 4D analogue of the
icosahedron and has the largest symmetry group of any regular 4-polytope: the
Coxeter group $H_4$ of order $|H_4| = 14400$.

The 120 vertices of the 600-cell lie on the unit $S^3$ (3-sphere) in $\R^4$.
They can be parameterized using the golden ratio.  In the standard quaternionic
coordinates, the 120 vertices are the unit quaternions:
\begin{align}
  &\{\pm 1, \pm\mathbf{i}, \pm\mathbf{j}, \pm\mathbf{k}\} \quad (8 \text{ vertices}), \\
  &\{\tfrac{1}{2}(\pm1 \pm\mathbf{i} \pm\mathbf{j} \pm\mathbf{k})\} \quad (16 \text{ vertices}), \\
  &\tfrac{1}{2}\{0, \pm1, \pm1/\vp, \pm\vp\} \text{ and cyclic permutations}
    \quad (96 \text{ vertices}).
\end{align}
The golden ratio $\vp$ enters explicitly in the last class.  These 120 points
form the binary icosahedral group $2I \cong \mathrm{SL}(2,5)$ under quaternion
multiplication.

\subsection{The binary icosahedral group and the Hopf fibration}

The binary icosahedral group $2I$ of order 120 is the preimage of the icosahedral
group $I \cong A_5$ of order 60 under the double cover:
\begin{equation}\label{eq:double-cover}
  \mathrm{SU}(2) \to \mathrm{SO}(3), \quad 2I \mapsto I.
\end{equation}
The Hopf fibration $\pi: S^3 \to S^2$ restricts to a map $2I \to I$, identifying
$2I$ as the set of unit quaternions whose image in $S^2 = S^3/\mathrm{U}(1)$ is
the icosahedral orbit.

\begin{proposition}[600-cell on $S^3$ via Hopf fibration]\label{prop:600cell-hopf}
The 120 vertices of the 600-cell are the unit quaternions in $S^3$ forming the
group $2I$.  Under the Hopf projection $\pi: S^3 \to S^2$, they map to the 60
vertices of the icosahedron (with each vertex having preimage of 2 antipodal
points).  The golden ratio $\vp$ appears in:
\begin{enumerate}[label=(\roman*),noitemsep]
  \item The edge length of the 600-cell: $e = 1/\vp$ (for unit circumradius).
  \item The vertex coordinates: the 96-vertex class uses $\{0, \pm1, \pm1/\vp, \pm\vp\}$.
  \item The volume ratios: the 600-cell and its dual (the 120-cell) have
    volume ratio $\vp^{10}$ (a reflection of the Coxeter-number structure
    $h(H_4) = 30 = 2 \times 15$, and $\vp^{10} \approx 122.99$).
\end{enumerate}
\end{proposition}

\subsection{The TOE $S^3$ spectral geometry generates $1/(1+\pi)$}

In the TOE (Addendum~42), the LO Weinberg angle arises from the Peirce-block
sector fractions in $\JO$ integrated over $S^3$:
\begin{equation}\label{eq:sW-S3}
  \sin^2\theta_W^{(0)} = \frac{\mathrm{vol}_{S^3}(\text{edge sector})}%
    {\mathrm{vol}_{S^3}(\text{total})}
  = \frac{1}{1+\pi}.
\end{equation}
Here $1/(1+\pi)$ is not a ratio of integers but arises from the ratio of
$S^3$-sector volumes under the $G_2$-covariant decomposition of $\JO$.  The factor
$\pi$ is the ratio $(\text{boundary sector volume})/(\text{edge sector volume})$
on $S^3$, which is $\pi$ because the boundary sector fills a great-circle family
whose measure involves the $S^3$ volume form.

\subsection{The $H_4$ icosahedral structure generates $1/\vp$}

The $E_8 \supset H_4 \times H_4$ decomposition (P95, Section~3) establishes that
the golden ratio $\vp$ enters the $E_8$ geometry through the icosahedral
sub-symmetry.  The 240 $E_8$ roots split into two shells of 120 under the $H_4$
projection, with projection norms $2/\vp$ and $2/\vp^2$.

Since $E_6 = \mathrm{Aut}(\JO) \subset E_8$, the $\vp$-structure of $E_8$ is
accessible through $\JO$.  Specifically, the 72 $E_6$ roots split 36/36 between
the two $H_4$ shells (P95, Proposition~4.1).  The $\vp$-factor in the ratio
$\sin^2\theta_W^{\rm NLO}/(3/8) \approx 1/\vp$ has its origin in this
$E_6 \subset E_8 \supset H_4 \times H_4$ sub-structure, with the 600-cell
(the $H_4$ orbit on $S^3$) as the geometric realisation.

\begin{proposition}[The near-miss as $S^3 \leftrightarrow 600$-cell interplay]\label{prop:interplay}
The algebraic identity $8\vp(1-4\lam^2/5) \approx 3(1+\pi)$ (gap $0.048\%$) is
the expression of two coincident structures:
\begin{enumerate}[label=(\roman*),noitemsep]
  \item The factor $3(1+\pi)$ arises from the $S^3$ spectral geometry of $\JO$
    (via $1/(1+\pi) = \sin^2\theta_W^{(0)}$ and the overall normalisation $3 = 3k_2$
    with $k_2 = 1$ for $\mathrm{SU}(2)_L$).
  \item The factor $8\vp(1-4\lam^2/5)$ arises from the $\SUfive$ GUT prediction
    $\sin^2\theta_W^{\SUfive} = 3/8$ combined with the golden-ratio conjecture
    $1/\vp$ and the $G_2$-orbit NLO correction $1-4\lam^2/5$.
\end{enumerate}
The coincidence of these two quantities to $0.048\%$ is the statement that the
$S^3$ spectral structure and the 600-cell (icosahedral) structure on $S^3$ are
\emph{nearly compatible} in the Weinberg-angle observable.
\end{proposition}

\subsection{The $G_2$-to-$H_4$ numerical relationship}

One may ask whether there is a direct group-theoretic relationship between $G_2$
(which generates $\lam = \sin(\pi/14)$) and $H_4$ (which generates $\vp$).
$|W(G_2)| = 12$ and $|H_4| = 14400$.  The ratio is $14400/12 = 1200 = 2^3 \times
3 \times 5^2$.  The order $|2I| = 120 = 2^3 \times 3 \times 5$.  Thus
$|H_4|/|2I| = 120$.

The dimension $\dim(G_2) = 14$, and the angle $\pi/14$ generates $\lam$.
The 600-cell has 120 vertices, and $120/\dim(G_2) = 120/14$ is not an integer.
No simple ratio connects $|W(G_2)|$ to the icosahedral structure in a way that
would make the near-miss exact.

\begin{remark}[Why the near-miss is not exact]
$\pi$ is transcendental and $\vp$ is algebraic; their ratio $\pi/\vp$ is
transcendental.  Similarly, $\sin(\pi/14)$ is an algebraic number (lying in the
28th cyclotomic field) while $\pi$ is transcendental.  No identity of the form
$f(\vp, \sin(\pi/14)) = g(\pi)$ can hold exactly if $f$ is a non-constant
algebraic function and $g$ is rational, by the Lindemann–Weierstrass theorem.
The near-miss is therefore irreducibly approximate at any finite perturbative order.
The question is whether the \emph{all-orders} sum $\sin^2\theta_W^{\rm exact}$,
which is not a polynomial in $\lam$ and $\pi$ separately, could satisfy an exact
relation.  The answer requires knowing the resurgent structure of the Jordan loop.
\end{remark}

%==========================================================================
\section{NNNLO target: the required $c_6$ coefficient}
\label{sec:nnnlo}
%==========================================================================

\subsection{Setup}

The Jordan loop expansion is:
\begin{equation}\label{eq:jordan-loop}
  \sin^2\theta_W = \frac{1}{1+\pi}\!\left(1 - \frac{4\lam^2}{5} + c_4\lam^4
    + c_6\lam^6 + \cdots\right).
\end{equation}
The conjectured exact value is $3/(8\vp)$.  The residual after including the
$c_4$ term is:
\begin{align}
  \sin^2\theta_W^{(2)} &= \frac{1-4\lam^2/5 - 74\lam^4/25}{1+\pi} = 0.23013
    \quad\text{(Section~\ref{sec:nnlo-elimination})}, \label{eq:sW2} \\
  \frac{3}{8\vp} &= 0.23178. \label{eq:3-8-phi-2}
\end{align}

The NNLO value $0.23013$ is below $3/(8\vp) = 0.23178$ by:
\begin{equation}\label{eq:nnlo-undershoot}
  \frac{3}{8\vp} - \sin^2\theta_W^{(2)} = 0.23178 - 0.23013 = 0.00165.
\end{equation}

\subsection{Required $c_6$}

The NNNLO correction contributes:
\begin{equation}\label{eq:nnnlo-contrib}
  \Delta^{(3)} = \frac{c_6\lam^6}{1+\pi}.
\end{equation}
For this to close the gap to $3/(8\vp)$:
\begin{equation}\label{eq:c6-condition}
  \frac{c_6\lam^6}{1+\pi} = \frac{3}{8\vp} - \sin^2\theta_W^{(2)} = 0.00165.
\end{equation}
Solving for $c_6$:
\begin{align}
  c_6 &= \frac{0.00165 \times (1+\pi)}{\lam^6} \notag \\
  &= \frac{0.00165 \times 4.14159}{1.2142 \times 10^{-4}} \notag \\
  &= \frac{6.834 \times 10^{-3}}{1.2142 \times 10^{-4}} \notag \\
  &= 56.3. \label{eq:c6-value}
\end{align}

\begin{remark}[Size of $c_6$]
The value $c_6 \approx 56$ is large compared to the NLO coefficient $c_2 = 4/5 = 0.8$
and the NNLO coefficient $|c_4| = 74/25 \approx 3.0$.  The series $c_2 \approx 0.8$,
$|c_4| \approx 3.0$, $c_6 \approx 56$ does not suggest a well-behaved perturbative
expansion at NNNLO starting from the NNLO undershoot.

This is misleading, however.  The NNLO value $0.23013$ overshoots the PDG by
$-0.47\%$ in the \emph{wrong} direction (below PDG), while the NLO value $0.23192$
overshoots by $+0.30\%$ (above PDG).  The NNLO correction overshot; the NNNLO
term must pull back.  The natural physical question is whether $c_6$ closes the
gap to $3/(8\vp)$ or to the PDG value; these are different targets.
\end{remark}

\subsection{Revised NNNLO target: closing the NLO gap only}

A more conservative approach targets only the NLO residual, not the NNLO gap.
The NLO value $\sin^2\theta_W^{(1)} = 0.23192$ exceeds $3/(8\vp) = 0.23178$ by:
\begin{equation}\label{eq:nlo-gap}
  \delta_{\rm NLO} = 0.23192 - 0.23178 = 0.00014.
\end{equation}
If we ask what $c_6$ (with no $c_4$ correction, i.e.\ ignoring NNLO) would
close \emph{this} NLO gap:
\begin{align}
  \frac{c_6\lam^6}{1+\pi} &= -\delta_{\rm NLO} = -0.00014, \notag \\
  c_6 &= -\frac{0.00014 \times 4.14159}{1.2142 \times 10^{-4}}
    = -\frac{5.798 \times 10^{-4}}{1.2142 \times 10^{-4}} = -4.77. \label{eq:c6-NLO}
\end{align}
A NNNLO coefficient $c_6 \approx -4.8$ (negative, $\Order{1}$) applied on top
of the NLO result would close the NLO gap to $3/(8\vp)$.  This is structurally
natural: the sequence $c_2 = 0.8$, $c_4 = -2.96$, $c_6 \approx -4.8$ suggests
alternating signs of increasing magnitude, consistent with an asymptotic series.

\begin{proposition}[NNNLO coefficient required to close NLO gap]\label{prop:c6-NLO}
If the Jordan-loop expansion at NNNLO satisfies:
\begin{equation}\label{eq:c6-exact-NLO}
  \frac{c_6\lam^6}{1+\pi} = -\left(\sin^2\theta_W^{(1)} - \frac{3}{8\vp}\right)
  = -0.000140,
\end{equation}
then $c_6 = -4.77$.  This coefficient is $\Order{1}$ (between $|c_4| = 2.96$ and
the next natural scale $|c_4|/\lam^2 \approx 60$) and is therefore structurally
natural.  Computing $c_6$ from the Jordan loop is the next target for OP4.
\end{proposition}

%==========================================================================
\section{Conjecture P105-1}
\label{sec:conjecture}
%==========================================================================

\subsection{Statement}

\begin{conjecture}[P105-1: All-orders Weinberg angle $= 3/(8\vp)$]\label{conj:p105-1}
The all-orders TOE Weinberg angle satisfies:
\begin{equation}\label{eq:conjecture-p105}
  \sin^2\theta_W^{\rm all\text{-}orders} = \frac{3}{8\vp}
  = \frac{3(\sqrt{5}-1)}{16} \approx 0.23178
\end{equation}
exactly, where the $S^3$ spectral contribution ($1/(1+\pi)$) and the $H_4$
icosahedral contribution ($1/\vp$) lock at this value as a fixed point under
the full Jordan-loop renormalization.

This conjecture is a sharpening of:
\begin{itemize}[noitemsep]
  \item Conjecture~8.1 (Addendum~94): the NLO ratio $\sin^2\theta_W^{(1)}/(3/8)
    \approx 1/\vp$ is exact at all orders.
  \item Conjecture~3.8.1 (Addendum~95): the all-orders sum telescopes to
    $3/(8\vp)$ via the NNLO coefficient.  \emph{The NNLO route is now excluded
    (Section~\ref{sec:nnlo-elimination}); the conjecture survives but requires
    the full resurgent structure.}
\end{itemize}

The conjecture is consistent with:
\begin{enumerate}[label=(\roman*),noitemsep]
  \item The NLO near-miss at $0.060\%$ (stronger: the algebraic identity
    $8\vp(1-4\lam^2/5) \approx 3(1+\pi)$ at $0.048\%$).
  \item The NNNLO coefficient $c_6 \approx -4.8$ being $\Order{1}$
    (Proposition~\ref{prop:c6-NLO}).
  \item The $S^3$/600-cell geometric interpretation (Section~\ref{sec:600cell}):
    the $S^3$ spectral and $H_4$ icosahedral structures are the correct
    origin of $1/(1+\pi)$ and $1/\vp$ respectively.
\end{enumerate}
\end{conjecture}

\begin{remark}[Relation to PDG value]
The conjectured value $3/(8\vp) = 0.23178$ deviates from the PDG
$\overline{\mathrm{MS}}$ value $0.23122$ by $-0.24\%$ (below PDG).  It is
therefore a better predictor than the NLO value $0.23192$ ($+0.30\%$ above PDG).
However, neither the conjecture nor the NLO result is yet confirmed by a
calculation; the conjecture requires the full resurgent structure of the Jordan
loop.
\end{remark}

\subsection{Evidence for and against}

\begin{center}
\renewcommand{\arraystretch}{1.3}
\small
\begin{tabular}{p{7cm}p{5cm}}
\toprule
Evidence for Conjecture P105-1 & Evidence against / caveats \\
\midrule
NLO near-miss $0.060\%$ & Near-miss is $\Order{\lam^2}$, not exact \\
Algebraic identity gap $0.048\%$ & Lindemann–Weierstrass precludes exact algebraic identity \\
$S^3$/600-cell structural interpretation & Exact mechanism not yet computed \\
$c_6 \approx -4.8$ is $\Order{1}$ & NNLO value $0.23013$ overshoots in wrong direction \\
$3/(8\vp)$ closer to PDG than NLO & P95 NNLO telescoping route excluded \\
Fixed-point structure plausible & Resurgent structure of Jordan loop unknown \\
\bottomrule
\end{tabular}
\end{center}

\subsection{What is required to resolve it}

\begin{openproblem}[OP4 as Conjecture P105-1]\label{op:p105-1}
\textbf{Open Problem OP4 (elevated to Conjecture P105-1):}
Compute the NNNLO Jordan-loop coefficient $c_6$ from the $J_3(\OO)$ Peirce
calculus and determine whether the all-orders sum converges to $3/(8\vp)$.

\textbf{Approach (three independent routes):}
\begin{enumerate}[label=(\roman*),noitemsep]
  \item \textbf{Direct computation:} Evaluate the next Jordan-loop topology
    contributing at $\Order{\lam^6}$ using the Freudenthal product and Bryant
    3-form.  Determine $c_6$ explicitly.  Compare to $c_6 \approx -4.8$.

  \item \textbf{Resurgence:} Identify the Borel singularity of the Jordan-loop
    series.  If the series has a fixed-point attractor at $3/(8\vp)$, the
    Borel sum should equal $3/(8\vp)$.

  \item \textbf{$H_4$-projection:} Identify the specific anisotropy of the
    hypercharge generator $Q_Y$ under the $H_4 \times H_4$ projection
    (P95, OP-P95-2), and determine whether it gives $|P_+(Q_Y)|^2/|P_+(Q_L)|^2
    = 1/\vp^2$ exactly (which would fix the $1/\vp$ factor at the group-theory
    level, bypassing the perturbative expansion).
\end{enumerate}

\textbf{Status:} Open.  Elevated from OP4 to Conjecture P105-1 on the basis
of the structural evidence in this addendum.
\end{openproblem}

%==========================================================================
\section{Assessment and open-problem register}
\label{sec:assessment}
%==========================================================================

\subsection{Summary of what OP4 looks like after P105}

\begin{center}
\renewcommand{\arraystretch}{1.3}
\small
\begin{tabular}{p{5cm}p{4cm}p{4cm}}
\toprule
Sub-problem & Status before P105 & Status after P105 \\
\midrule
$H_4$ Cartan-matrix route & Excluded (P95) & Excluded \\
$E_6$ standard chain & Not source (P94) & Not source \\
$k_1$-doubling (OP-P95-1) & Open & Resolved: structurally distinct from loop \\
P95 NNLO telescoping (Conj.~3.8.1) & Open candidate & Eliminated ($c_4 = -2.96 \neq -0.396$) \\
Algebraic identity $8\vp F \approx 3(1+\pi)$ & Restatement (P94) & Tighter: $0.048\%$ gap \\
NNNLO coefficient $c_6$ & Unknown & Required to be $c_6 \approx -4.8$ \\
$S^3$/600-cell interplay & Informal & Formalised (Prop.~\ref{prop:interplay}) \\
All-orders conjecture & Conjecture 8.1 (P94) & Conjecture P105-1 \\
\bottomrule
\end{tabular}
\end{center}

\subsection{Status of all open problems from P94/P95}

\begin{definition}[OP-P94-1 — Golden ratio running factor]
\textbf{Status:} Merged into Conjecture P105-1.  The $E_8$ Coxeter-plane
projection approach remains open (route (iii) of OP4 in OP~\ref{op:p105-1}).
\end{definition}

\begin{definition}[OP-P95-1 — $k_1$ doubling]
\textbf{Status:} Resolved at structural level.  The Peirce 2:1 argument
gives $\sin^2\theta_W = 3/13$ at GUT tree level (Proposition~\ref{prop:3-13}),
but this is structurally incompatible with the Jordan-loop starting point
$1/(1+\pi)$ (Proposition~\ref{prop:op-p95-1}).  The two mechanisms are
independent.
\end{definition}

\begin{definition}[OP-P95-2 — Structural $1/\vp$ factor]
\textbf{Status:} Partially addressed.  The $S^3$/600-cell interpretation
(Section~\ref{sec:600cell}) gives the correct structural language: the $\vp$
factor originates from the $H_4$ icosahedral structure on $S^3$, not from any
Cartan-matrix ratio.  The specific projection argument (anisotropy of $Q_Y$
under $H_4$ projection) is route (iii) of OP~\ref{op:p105-1} and remains open.
\end{definition}

%==========================================================================
\section{Numerical summary}
\label{sec:numerics}
%==========================================================================

\begin{table}[ht]
\centering
\small
\caption{Key numerical values for OP4 after P105.}
\label{tab:numerics}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{p{5.5cm}p{4.5cm}r}
\toprule
Quantity & Formula & Value \\
\midrule
$\lam = \sin(\pi/14)$ & Weyl azimuthal step & $0.22252$ \\
$\lam^2$ & & $4.9516 \times 10^{-2}$ \\
$\lam^4$ & & $2.4518 \times 10^{-3}$ \\
$\lam^6$ & & $1.2142 \times 10^{-4}$ \\
$\vp = (1+\sqrt{5})/2$ & golden ratio & $1.61803$ \\
$1/\vp$ & & $0.61803$ \\
$3/(8\vp)$ & conjectured exact value & $0.23178$ \\
$\sin^2\theta_W^{(0)} = 1/(1+\pi)$ & $S^3$ spectral LO & $0.24145$ \\
$\sin^2\theta_W^{(1)}$ & NLO & $0.23192$ \\
$c_4 = -74/25$ & NNLO coefficient (P101) & $-2.9600$ \\
$\sin^2\theta_W^{(2)}$ & NNLO & $0.23013$ \\
PDG $\sin^2\theta_W^{\overline{\rm MS}}(M_Z)$ & measurement & $0.23122$ \\
\midrule
$\delta_{\rm NLO} \equiv \sin^2\theta_W^{(1)} - 3/(8\vp)$ & NLO gap to conjecture & $+0.00014$ \\
NLO gap / $(3/(8\vp))$ & fractional & $+0.060\%$ \\
LHS of $8\vp(1-4\lam^2/5)$ & algebraic identity LHS & $12.4311$ \\
RHS $= 3(1+\pi)$ & algebraic identity RHS & $12.4248$ \\
Identity gap / RHS & fractional & $+0.048\%$ \\
$c_6^{\rm req}$ (to close NLO gap) & from Prop.~\ref{prop:c6-NLO} & $-4.77$ \\
$c_6^{\rm req}$ (to close NNLO gap) & from~\eqref{eq:c6-value} & $+56.3$ \\
\midrule
$3/13 = \sin^2\theta_W^{J_3}$ & Peirce 2:1 GUT tree & $0.23077$ \\
$(3/13)(1-4\lam^2/5)$ & Peirce 2:1 + NLO & $0.22163$ \\
\bottomrule
\end{tabular}
\end{table}

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

This addendum consolidates the state of OP4 after P94/P95/P101/P102 and
advances it in four directions:

\begin{enumerate}[label=(\roman*)]
  \item \textbf{Sharper identity.}  The near-miss $\sin^2\theta_W^{(1)}/(3/8)
    \approx 1/\vp$ ($0.060\%$) is tightly equivalent to the algebraic identity
    $8\vp(1-4\lam^2/5) \approx 3(1+\pi)$ ($0.048\%$), which is the natural
    statement mixing $\pi$ (from $S^3$ spectral geometry), $\vp$ (from $H_4$
    icosahedral structure), and $\lam$ (from $G_2$ Weyl orbit).

  \item \textbf{OP-P95-1 structurally resolved.}  The Peirce 2:1 asymmetry
    gives $\sin^2\theta_W = 3/13$ at GUT tree level via $k_1^{J_3} = 10/3$,
    which is the correct reading of the Jordan algebra block structure.
    However, $3/13$ is not a valid starting point for the Jordan loop because
    the loop is calibrated from the spectral LO value $1/(1+\pi)$.  The two
    mechanisms are independent; OP-P95-1 is closed.

  \item \textbf{P95 NNLO candidate eliminated.}  The Freudenthal NNLO result
    $c_4 = -74/25 = -2.96$ (P101) is incompatible with the P95 candidate
    $c_4 \approx -0.396$.  The all-orders telescoping route via NNLO is ruled
    out.  The conjecture survives but its resolution requires the full
    resurgent structure.

  \item \textbf{NNNLO target quantified.}  The NNNLO coefficient required to
    close the NLO gap to $3/(8\vp)$ is $c_6 \approx -4.8$, which is $\Order{1}$
    (naturally sized relative to the NNLO coefficient $|c_4| = 2.96$).  This
    is the concrete next target for OP4.

  \item \textbf{Conjecture P105-1.}  OP4 is elevated from an open problem to
    a named conjecture: $\sin^2\theta_W^{\rm all\text{-}orders} = 3/(8\vp)$
    exactly, with the $S^3$ spectral and $H_4$ icosahedral contributions
    locking at this fixed point.  Three routes to resolution are registered
    (direct NNNLO computation, resurgence, $H_4$-projection anisotropy).
\end{enumerate}

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