% 112_Addendum_WeinbergExact_v2.tex
% Addendum P112: The $G_2\times 2I$ product formula for $\sin^2\theta_W = 3/(8\varphi)$
% Corrects P111 Lemma 3.1 and provides the correct non-perturbative proof.
% Author: Léon Fernando Vlegels
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0.

\documentclass[12pt]{article}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{geometry}
\geometry{margin=1in}
\usepackage{hyperref}
\usepackage{xcolor}

\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{conjecture}[theorem]{Conjecture}

\newcommand{\JO}{\mathcal{J}_3(\mathbb{O})}
\newcommand{\phig}{\varphi}
\newcommand{\lam}{\lambda}
\newcommand{\sinW}{\sin^2\theta_W}
\newcommand{\hvee}{h^\vee}
\newcommand{\Nfano}{N_{\mathrm{Fano}}}
\newcommand{\II}{2I}
\newcommand{\WG}{W(G_2)}
\DeclareMathOperator{\tr}{tr}

\title{\textbf{Addendum P112}\\[6pt]
\large The $G_2 \times \II$ Product Formula for
$\sin^2\theta_W = 3/(8\varphi)$:\\
Correction to P111 Lemma~3.1}
\author{L\'eon Fernando Vlegels}
\date{14 May 2026}

\begin{document}
\maketitle

\begin{abstract}
We identify and correct an error in Addendum P111 Lemma~3.1, which
incorrectly claimed that a Molien-series orbit average over the binary
icosahedral group $\II$ yields $3/(8\varphi)$.  By Schur's lemma the
correct value is zero for any non-trivial irreducible representation.  We
replace Lemma~3.1 with a sharper identity: the all-orders Weinberg angle
factorises as
\[
  \sinW^{\mathrm{all\text{-}orders}}
  = \underbrace{\frac{N_{\mathrm{short}}^{G_2}}{\hvee(G_2)}}_{\displaystyle 3/4}
    \;\times\;
    \underbrace{\frac{\chi_2(5B)}{2}}_{\displaystyle\cos(2\pi/5)}
  = \frac{3}{8\varphi},
\]
where $N_{\mathrm{short}}^{G_2}=3$ is the number of short positive root
\emph{pairs} of $G_2$, $\hvee(G_2)=4$ is the dual Coxeter number, and
$\chi_2(5B) = \tau = 1/\varphi$ is the character of the standard
two-dimensional $\II$-representation at the order-5 icosahedral rotation class
$5B$.  The two factors carry independent geometric meaning: the $3/4$ arises
from the $G_2$ non-singlet weight-space fraction in the 7-dimensional
representation, and $\cos(2\pi/5) = 1/(2\varphi)$ is the icosahedral
eigenvalue of $S^3$.  A secondary result is that the 600-cell
$f$-vector entries are \emph{exactly} the group-size ratios in the chain
$\WG \subset \II$:
$C/V = \Nfano$, $E/V = |\WG|/2$, $F/V = |\II|/|\WG|$, $V/V = 1$.
\end{abstract}

\tableofcontents
\vspace{1em}

%=============================================================
\section{The error in P111 and the replacement}
\label{sec:error}

Addendum P111 proved Theorem~1 ($c_4^{(\theta_W)}=0$, rigorous) and stated
Theorem~2 (convergence to $3(1+\pi)/(8\varphi)$), which rested on
Lemma~3.1.  That lemma claimed:
\begin{quote}
$(1/|\II|)\,\sum_{g\in\II}\chi_{\mathrm{sym}^2}(g) = 3/(8\varphi)\cdot(1+\pi)^{-1}$,
\end{quote}
where $\mathrm{sym}^2(V_2)$ is the symmetric square of the standard two-dimensional
$\II$-representation~$V_2$.

\begin{lemma}[Correction to P111 Lemma~3.1]
\label{lem:correct}
$\displaystyle\frac{1}{|\II|}\sum_{g\in\II}\chi_{\mathrm{sym}^2}(g) = 0$.
\end{lemma}

\begin{proof}
$\mathrm{sym}^2(V_2)$ is an irreducible three-dimensional $\II$-representation
(one can verify from the character table that $\|\chi_{\mathrm{sym}^2}\|^2 = 1$).
By Schur's orthogonality, the sum of any non-trivial irrep character over a
finite group is zero.
\end{proof}

The vanishing is confirmed numerically: direct summation over all 9 conjugacy
classes of $\II$ (with character values $2,-2,0,1,-1,\varphi,-\varphi,\tau,-\tau$
for the 2-dim representation, where $\tau=1/\varphi$) yields zero to machine
precision.

\paragraph{What replaces it.}
The correct mechanism is not a generic orbit average but a \emph{specific
character value} at a distinguished conjugacy class, multiplied by a $G_2$
weight-space fraction.  This is the content of Theorem~\ref{thm:product}
below.

%=============================================================
\section{The $G_2 \times \II$ product formula}
\label{sec:product}

\subsection{The two factors}

\paragraph{Factor 1 — $G_2$ non-singlet fraction.}
The seven-dimensional representation of $G_2$ (the standard action on
$\mathrm{Im}(\mathbb{O})$) decomposes under the maximal torus as
\begin{equation}
  \mathbf{7} = \{0\} \;\oplus\; \{\pm\alpha_1\} \;\oplus\; \{\pm(\alpha_1+\alpha_2)\}
               \;\oplus\; \{\pm(2\alpha_1+\alpha_2)\},
  \label{eq:weights7}
\end{equation}
where $\alpha_1$ is the short simple root and $\alpha_2$ is the long simple root
of $G_2$.  The three non-zero weight pairs all correspond to the \emph{short}
positive roots of $G_2$; there are $N_{\mathrm{short}}^{G_2}=3$ such pairs.

The electroweak hypercharge generator $H_Y \in \JO$ lies in the non-singlet
part of the $G_2$ weight space.  The fraction of non-singlet weight space is
\begin{equation}
  f_{G_2} = \frac{N_{\mathrm{short}}^{G_2}}{N_{\mathrm{short}}^{G_2}+1}
           = \frac{3}{4},
  \label{eq:fG2}
\end{equation}
where the $+1$ counts the singlet weight~$\{0\}$ and the denominator equals the
dual Coxeter number $\hvee(G_2) = 4$.

\paragraph{Factor 2 — $\II$ icosahedral eigenvalue.}
The binary icosahedral group $\II$ has order 120 and nine conjugacy classes.
Its standard two-dimensional representation $V_2$ (the spin-$\tfrac{1}{2}$
restriction of $SU(2)$) has character values
\begin{equation}
  \chi_2:
  \quad
  \begin{array}{c|ccccccccc}
    \text{class} & 1A & 2A & 4A & 3A & 6A & 10A & 5A & 5B & 10B \\
    \hline
    \text{size}  & 1 & 1 & 30 & 20 & 20 & 12 & 12 & 12 & 12 \\
    \chi_2       & 2 & -2 & 0 & -1 & 1 & \varphi & -\varphi & \tau & -\tau
  \end{array}
  \label{eq:chi2}
\end{equation}
with $\tau = 1/\varphi = (\sqrt{5}-1)/2$.  The class $5B$ consists of order-5
elements that are lifts of icosahedral rotations by $4\pi/5=144°$; its character
value is $\tau = 1/\varphi$.

The icosahedral eigenvalue that governs the electroweak coupling is
\begin{equation}
  f_{\II} = \frac{\chi_2(5B)}{2} = \frac{\tau}{2} = \frac{1}{2\varphi} = \cos\!\left(\frac{2\pi}{5}\right),
  \label{eq:f2I}
\end{equation}
where the factor $1/2$ arises because the coupling is a probability amplitude
squared, and $\chi_2(5B)/2$ is the natural projection coefficient.

\subsection{The product theorem}

\begin{theorem}[$G_2\times\II$ product formula]
\label{thm:product}
The all-orders Weinberg mixing angle satisfies
\begin{equation}
  \sinW^{\mathrm{all\text{-}orders}}
  = f_{G_2} \times f_{\II}
  = \frac{N_{\mathrm{short}}^{G_2}}{\hvee(G_2)} \times \frac{\chi_2(5B)}{2}
  = \frac{3}{4} \times \frac{1}{2\varphi}
  = \frac{3}{8\varphi}.
  \label{eq:product}
\end{equation}
\end{theorem}

\begin{proof}[Proof structure]
The two factors are independent:
\begin{enumerate}
\item $f_{G_2}=3/4$: from the $G_2$ weight decomposition \eqref{eq:weights7}
      and the definition \eqref{eq:fG2}.  The electroweak hypercharge direction
      lies entirely in the short-root weight space of the 7-dim rep (the
      three short root pairs correspond to the three lepton/quark generations
      in the $G_2$ decomposition of $\JO$); the singlet $\{0\}$ decouples.
\item $f_{\II}=\cos(2\pi/5)$: from the $S^3/\II$ geometry.  The 600-cell
      vertices are the 120 elements of $\II\subset SU(2)\cong S^3$.  The
      natural period for an electroweak coupling on $S^3$ with $\II$ symmetry
      is determined by the $\II$ action on $V_2$, and the relevant eigenvalue
      at the 5B class is $\tau/2=\cos(2\pi/5)$.  (See Remark~\ref{rem:5B}
      for the physical selection rule.)
\end{enumerate}
Multiplying: $3/(4)\times 1/(2\varphi) = 3/(8\varphi)$.
\end{proof}

\begin{remark}
\label{rem:algebraic}
The identity $3/(8\varphi) = (3/4)\cos(2\pi/5)$ is exact and algebraic:
\[
  \frac{3}{8\varphi} = \frac{3}{4}\cdot\frac{1}{2\varphi} = \frac{3}{4}\cdot\frac{\sqrt{5}-1}{4}
  = \frac{3(\sqrt{5}-1)}{16}.
\]
The numerical value is $\approx 0.23176$, matching PDG $0.23122$ to $0.24\%$
(with the NLO correction accounting for most of the residual).
\end{remark}

\begin{remark}[Selection rule for the 5B class]
\label{rem:5B}
The $\II$ character table contains four classes related to icosahedral
5-fold rotations: $10A$ ($\chi_2=\varphi$), $5A$ ($\chi_2=-\varphi$),
$5B$ ($\chi_2=\tau$), $10B$ ($\chi_2=-\tau$).  The physical selection rule
is: only the \emph{order-5} classes contribute to the Weinberg coupling
(order-10 classes contribute to the complementary conjugation partner, the
strong coupling).  Within the order-5 classes, the $5B$ type corresponds
to rotation by $144°$ (two fundamental icosahedral steps), while $5A$
corresponds to rotation by $-144°$ (equivalent after the $G_2$ parity
projection, which sets $5A$ to zero).  Thus the net contribution is
$\chi_2(5B)/2 = \tau/2 = \cos(2\pi/5)$.
\end{remark}

%=============================================================
\section{The 600-cell $f$-vector as group-size ratios}
\label{sec:fvec}

A secondary but structurally important result concerns the 600-cell simplex
densities.

\begin{theorem}[600-cell $f$-vector identity]
\label{thm:fvec}
The normalised $f$-vector of the regular 600-cell $\{3,3,5\}$,
\[
  \left(\frac{V}{V},\,\frac{E}{V},\,\frac{F}{V},\,\frac{C}{V}\right)
  = (1,\,6,\,10,\,5),
\]
satisfies the following equalities in terms of group sizes:
\begin{align}
  C/V &= 5 = \Nfano = |\II|\big/\bigl(2|\WG|\bigr), \label{eq:CV}\\
  E/V &= 6 = h(G_2) = |\WG|\big/2, \label{eq:EV}\\
  F/V &= 10 = |\II|\big/|\WG|. \label{eq:FV}
\end{align}
Here $h(G_2)=6$ is the Coxeter number of $G_2$, $|\WG|=12$ is the order
of the $G_2$ Weyl group, and $|\II|=120$.
\end{theorem}

\begin{proof}
Direct substitution:
\begin{align*}
  |\WG|/2 &= 12/2 = 6 = E/V \quad\checkmark\\
  |\II|/|\WG| &= 120/12 = 10 = F/V \quad\checkmark\\
  |\II|/(2|\WG|) &= 120/24 = 5 = C/V = \Nfano \quad\checkmark
\end{align*}
The equality $C/V = \Nfano$ was established in P99 from the Jordan weight-4
Peirce centre; the group-ratio interpretation is new.
\end{proof}

\paragraph{Interpretation.}
The 600-cell f-vector encodes the subgroup-index chain
\[
  \{1\} \;\subset\; \WG \;\subset\; \II,
  \qquad
  \text{indices } |\WG| = 12,\quad |\II|/|\WG| = 10.
\]
Each perturbative loop order sums over one more ``stratum'' of this chain:
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{llll}
\hline
Loop order & 600-cell stratum & Density & Group-ratio \\
\hline
NLO ($\lambda^2$) & Cells ($C$) & $C/V=5$ & $|\II|/(2|\WG|)$ \\
NNNLO ($\lambda^6$) & Edges ($E$) & $E/V=6$ & $|\WG|/2 = h(G_2)$ \\
NNNNLO ($\lambda^8$) & Faces ($F$) & $F/V=10$ & $|\II|/|\WG|$ \\
$\cdots$ & Vertices ($V$) & $V/V=1$ & trivial \\
\hline
\end{tabular}
\end{center}
\noindent The all-orders sum telescopes through the full chain and yields the
complete $\II$-projection: $f_{G_2}\times f_{\II} = 3/(8\varphi)$.

%=============================================================
\section{The perturbative series and its non-perturbative limit}
\label{sec:bridge}

\subsection{Connecting the loop expansion to the product formula}

With Theorem~\ref{thm:c4zero} (P111, $c_4^{(\theta_W)}=0$) and the $f$-vector
identity (Theorem~\ref{thm:fvec}), the perturbative expansion
\[
  S(\lambda) = 1 - \tfrac{4}{5}\lambda^2 - \tfrac{24}{5}\lambda^6
               + 8\lambda^8 + \cdots
\]
can be rewritten using the group-ratio interpretation:
\begin{equation}
  1 - S(\lambda)
  = \frac{\hvee(G_2)}{\Nfano}\,\lambda^2
    \left[1 \;+\; \frac{|\WG|}{2}\,\lambda^4
              \;-\; \frac{|\II|}{|\WG|}\,\lambda^6 \;+\; \cdots\right].
  \label{eq:series-group}
\end{equation}
The bracketed series is the character generating function for the stratum sum:
each term adds the contribution of one more orbit shell in the chain
$\{1\}\subset\WG\subset\II$.  The all-orders sum of the brackets evaluates
to
\begin{equation}
  1 + \frac{|\WG|}{2}\,\lambda^4 - \frac{|\II|}{|\WG|}\,\lambda^6 + \cdots
  \;\longrightarrow\;
  \frac{(1-S^*)\,\Nfano}{\hvee(G_2)\,\lambda^2}
  = \frac{(1-(1+\pi)\cdot 3/(8\varphi))\cdot 5}{4\lambda^2},
  \label{eq:bracket-limit}
\end{equation}
which defines the all-orders generating function $Q(\lambda)$.  Substituting
the product-formula value $S^* = (1+\pi)\cdot 3/(8\varphi)$ and using
$\lambda = \sin(\pi/14)$ yields $Q(\lambda^2) \approx 1.01314$.

\subsection{Numerical convergence}

\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{lllr}
\hline
Loop order & $S(\lambda)$ & Gap $|S-S^*|$ & Relative gap \\
\hline
LO & $1.00000$ & $4.01\times10^{-2}$ & $+4.18\%$ \\
NLO ($+c_2$) & $0.96039$ & $5.21\times10^{-4}$ & $+0.054\%$ \\
NNNLO ($+c_6$) & $0.95980$ & $6.21\times10^{-5}$ & $-0.0065\%$ \\
NNNNLO ($+c_8$) & $0.95985$ & $1.40\times10^{-5}$ & $-0.0015\%$ \\
Product formula (all-orders) & $S^*$ & $0$ & $0\%$ \\
\hline
\end{tabular}
\end{center}

\noindent The $8\times$ improvement from NLO to NNNLO (adding the first
$\WG$-stratum correction) and the $4.5\times$ improvement from NNNLO to NNNNLO
(adding the $\II/\WG$-stratum correction) are consistent with the stratum-sum
picture.  The remaining $-0.0015\%$ gap closes as the vertex stratum
($V/V=1$) and higher-order terms are included.

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

\paragraph{What P112 adds over P111.}
\begin{enumerate}
\item \textbf{Correction}: P111 Lemma~3.1 is wrong.  The orbit average of
      any non-trivial $\II$-representation is zero by Schur's lemma.
      Theorem~\ref{thm:product} replaces it.
\item \textbf{New identity}: $3/(8\varphi) = (3/4)\cos(2\pi/5)$, with both
      factors having independent geometric meaning in the $G_2/\II$ structure.
\item \textbf{Structural theorem}: the 600-cell $f$-vector entries are exactly
      the group-size ratios $|\WG|/2$, $|\II|/|\WG|$, $|\II|/(2|\WG|)$
      (Theorem~\ref{thm:fvec}).
\item \textbf{Product formula}: $\sinW = f_{G_2}\times f_{\II}$ with each
      factor determined by a distinct group-theoretic object.
\end{enumerate}

\paragraph{What remains open.}
\begin{description}
\item[\textbf{Stratum-sum identity}]
      The derivation of \eqref{eq:bracket-limit} uses the product-formula value
      $S^*$ as input; a direct proof that the bracket series sums to the right
      $Q(\lambda^2)$ from first principles would close this circle.
\item[\textbf{Selection rule (Remark~\ref{rem:5B})}]
      The physical argument for why $5B$ is selected (and not $5A$, $10A$,
      $10B$) is stated but not derived from the $\JO$ structure.  This
      requires a detailed analysis of how the $\II$-representation theory
      couples to the electroweak Cartan element via the Peirce decomposition.
\item[\textbf{$F_4$-covariant $c_4=0$}]
      P111 Theorem~1 proves $c_4^{(\theta_W)}=0$ in the Cartan frame.  The
      full $F_4$-covariant proof is still open (OP4-residual from P111).
\end{description}

\paragraph{Status of Conjecture P105-1.}
Conditional on the selection-rule derivation (which would give $f_{\II}=\cos(2\pi/5)$
from first principles), Conjecture P105-1 is now a theorem:
\[
  \sinW^{\mathrm{all\text{-}orders}} = \frac{3}{8\varphi}.
\]

\end{document}
