% Addendum 150: G₂ Angle Uniqueness and the Hopf Measure Pullback.
%   Addendum 132 derived μ_match = μ_conf · exp(2π(6−√3)/23) from the G₂
%   root fan and left as an open item whether θ = π/6 is uniquely forced or a
%   choice.  This addendum closes that item: θ is uniquely determined by the
%   SU(3) ⊂ G₂ embedding together with the G₂ root combinatorics, with no
%   residual freedom.  A secondary computation shows that the G₂-invariant
%   (round) measure on S³, pulled back to the radial interval [0,1] of B⁴
%   via the Hopf fibration, yields the volumetric weight x³ dx, not flat dx.
%   This constrains the open problem CP-3 (P144): the F₄-flat-measure
%   hypothesis cannot be derived from G₂ geometry alone and requires an
%   independent argument from J₃(𝕆).
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. 2026-05-15.

\documentclass[11pt,a4paper]{article}
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\usepackage{amsmath,amssymb,amsthm}
\usepackage{mathtools}
\usepackage{geometry}
\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref}
\geometry{margin=1in}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
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\newcommand{\Gii}{\mathrm{G}_2}
\newcommand{\SUT}{\mathrm{SU}(3)}
\newcommand{\Fiv}{\mathrm{F}_4}
\newcommand{\JO}{J_3(\mathbb{O})}
\newcommand{\mconf}{m_{\mathrm{conf}}}
\newcommand{\mum}{\mu_{\mathrm{match}}}
\newcommand{\rhog}{\rho_{\Gii}}
\newcommand{\PhiG}{|\Phi^+(\Gii)|}

\label{addendum:150}

\title{$\Gii$ Angle Uniqueness and the Hopf Measure Pullback\\[0.5em]
\large Addendum 150 to the TOE Corpus}

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

\date{2026-05-15}

\begin{document}
\maketitle

%==========================================================================
\begin{abstract}
Addendum~132 derived the matching scale
$\mum = \mconf\cdot\exp(2\pi(6-\sqrt{3})/23)$ from the $\Gii$ root fan
and identified $\theta = \pi/6$ as the angle between the short-root
($\SUT$ colour) direction and the long-root ($\Gii$ extra) direction,
leaving uniqueness as an open item.
This addendum proves that $\theta = \pi/6$ is uniquely forced by two
constraints that admit no freedom:\ (i) the $\Gii$ root system has exactly
12 roots, equally spaced at $\pi/6$ in the root plane; and (ii) the
embedding $\SUT\subset\Gii$ identifies the six short roots with the $A_2$
root system, fixing their angular positions.
Together these leave the long-root fan at precisely $\pi/6$ offset from
the short-root fan — no choice is involved.
A second result computes the pullback of the $\Gii$-invariant (round)
measure on $S^3$ to the radial interval $[0,1]$ of $B^4$ via the Hopf
fibration $S^1\to S^3\to S^2$: the pullback weight is $x^3\,dx$
(volumetric), not flat $dx$.
The implication for CP-3 (P144): the $\Gii$ boundary geometry supports the
coefficient~$16$ via the volumetric measure $dV_4 = 2\pi^2 x^3\,dx$, but
does not resolve the P01 inconsistency.
The $\Fiv$-flat-measure hypothesis of CP-3 requires an independent
$J_3(\mathbb{O})$ argument.
\end{abstract}

%==========================================================================
\section{Setup: open items from Addendum~132 and CP-3}
\label{sec:setup}

Addendum~132 showed that every factor in
\begin{equation}
  \mum = \mconf\cdot\exp\!\left(\frac{2\pi(6-\sqrt{3})}{23}\right)
  \label{eq:mum}
\end{equation}
is TOE-native: $\sqrt{3} = \rhog$ is the $\Gii$ long-to-short root ratio,
$6 = \PhiG = 2\rhog^2$ counts the positive roots of $\Gii$, and
$23 = 3b_0$ comes from the $\JO$ Casimir.
The derivation used the single angle $\theta = \pi/6$ for both the
non-perturbative boundary value $\alpha_s(\mconf) = \cot(\pi/6) = \sqrt{3}$
and the matching value $\alpha_s(\mum) = \sin(\pi/6) = 1/2$.
Addendum~132 noted that $\theta$ is the half-opening of the root fan
under the $\SUT$ subgroup, but did not prove this is uniquely forced.

The open problem CP-3 from Addendum~144 concerns a different but related
inconsistency in Paper~01.
Paper~01 motivates the coefficient $16$ in the cubic phase density
$\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x$
from the $B^4$ volume element $dV_4 = 2\pi^2 r^3\,dr$,
since $\int_0^1 16\pi^3 x^3\,dx = 4\pi^3$ requires the flat integration
$dx$ while the motivating geometry uses $2\pi^2 x^3\,dx$.
Addendum~144 proposed that the $\Fiv$-invariant inner product on $\JO$
might force the radial measure to be flat $dx$, resolving the mismatch.
The $\Gii$ angle $\theta$ couples the $S^3$ boundary geometry to the
$B^4$ interior; if $\theta$ is uniquely determined, it constrains
whether the boundary-to-interior coupling transforms the measure.

This addendum answers both questions.

%==========================================================================
\section{The $\Gii$ root system}
\label{sec:roots}

The exceptional Lie algebra $\Gii$ has rank~2.
Its root system $\Phi(\Gii)$ consists of 12 roots in a two-dimensional
real plane, comprising 6 short roots of length $s=1$ and 6 long roots
of length $l = \sqrt{3}$, giving root-length ratio $\rhog = l/s = \sqrt{3}$.

Choose simple roots $\alpha_1$ (short) and $\alpha_2$ (long).
The Cartan matrix entries are
$\langle\alpha_1,\alpha_2^\vee\rangle = -1$ and
$\langle\alpha_2,\alpha_1^\vee\rangle = -3$,
encoding the angle between the simple roots:
\begin{equation}
  \cos\angle(\alpha_1,\alpha_2) = \frac{\langle\alpha_1,\alpha_2\rangle}
  {|\alpha_1||\alpha_2|}
  = \frac{-3/2}{1\cdot\sqrt{3}} = -\frac{\sqrt{3}}{2},
  \qquad \angle(\alpha_1,\alpha_2) = \frac{5\pi}{6}.
  \label{eq:simple-angle}
\end{equation}

The six positive roots, together with their lengths, are:
\begin{align*}
  &\alpha_1,\;\alpha_1+\alpha_2,\;2\alpha_1+\alpha_2 \quad (\text{short, length }1),\\
  &\alpha_2,\;3\alpha_1+\alpha_2,\;3\alpha_1+2\alpha_2 \quad (\text{long, length }\sqrt{3}).
\end{align*}
Appending their negatives gives all 12 roots.
A direct calculation (using $\langle\alpha_1,\alpha_2\rangle = -3/2$)
verifies each length:
$|\alpha_1+\alpha_2|^2 = 1 - 3 + 3 = 1$,
$|2\alpha_1+\alpha_2|^2 = 4 - 6 + 3 = 1$,
$|3\alpha_1+\alpha_2|^2 = 9 - 9 + 3 = 3$,
$|3\alpha_1+2\alpha_2|^2 = 9 - 18 + 12 = 3$.

The Weyl group $W(\Gii)$ is the dihedral group of order 12, acting by
reflections in the 6 root hyperplanes (lines through the origin perpendicular
to each positive root).
This produces $12$ Weyl chambers, each a sector of angular width
$2\pi/12 = \pi/6$ in the root plane.

%==========================================================================
\section{Uniqueness of $\theta = \pi/6$}
\label{sec:uniqueness}

\begin{theorem}[Uniqueness of $\theta$]
\label{thm:unique}
Let $\Gii$ be the rank-$2$ exceptional root system with the embedding
$\SUT\subset\Gii$ that identifies the six short roots of $\Gii$ with the
$A_2$ ($\SUT$) root system.
Then the angle between any short root and its nearest long root is
uniquely $\theta = \pi/6$.
No continuous freedom remains; $\theta$ is not a parameter.
\end{theorem}

\begin{proof}
\textbf{Step 1: the 12 roots are equally spaced.}
The Weyl group $W(\Gii) \cong \mathrm{Dih}_{12}$ is the dihedral group of
order 12, acting faithfully on the root plane.
Its 12 elements map roots to roots and include a rotation by $2\pi/12 = \pi/6$.
Since the rotation by $\pi/6$ is an element of $W(\Gii)$ and the root system
is $W(\Gii)$-stable, the 12 roots must be permuted transitively (up to the
short/long distinction) by this rotation.
Concretely, the 12 roots occupy positions at angles
$\phi_k = \phi_0 + k\pi/6$ for $k=0,1,\ldots,11$
in the root plane, where $\phi_0$ is the angle of any fixed root.

\textbf{Step 2: SU(3) fixes the short-root positions.}
The six short roots of $\Gii$ form an $A_2$ root system with
$|W(A_2)| = 6$; they are equally spaced at $\pi/3$.
The embedding $\SUT\subset\Gii$ identifies these six roots with the
colour-sector root system and fixes their angular positions:
\begin{equation}
  \phi_k^{\mathrm{short}} = \phi_0 + k\cdot\frac{\pi}{3}, \qquad k=0,\ldots,5.
  \label{eq:short-fan}
\end{equation}
The orientation $\phi_0$ is a global frame choice (rotation of the entire
root diagram), which is immaterial for the angle $\theta$ between
adjacent roots of different type.

\textbf{Step 3: the long roots fill the remaining positions.}
From Step~1, the 12 positions are $\{\phi_0 + k\pi/6 : k=0,\ldots,11\}$.
The six short roots occupy the even positions $k=0,2,4,6,8,10$
(i.e.\ spacing $\pi/3$ as in Step~2).
The six long roots therefore occupy the odd positions
$k=1,3,5,7,9,11$, giving angular positions
\begin{equation}
  \phi_k^{\mathrm{long}} = \phi_0 + \frac{\pi}{6} + k\cdot\frac{\pi}{3},
  \qquad k=0,\ldots,5.
  \label{eq:long-fan}
\end{equation}
The offset between adjacent short and long positions is
$\phi_k^{\mathrm{long}} - \phi_k^{\mathrm{short}} = \pi/6$.

\textbf{Step 4: no freedom.}
The angle $\theta = \pi/6$ between consecutive short and long roots is
completely fixed by (a) the 12-fold $\mathrm{Dih}_{12}$ symmetry of $W(\Gii)$
and (b) the six-fold placement of the short roots forced by $\SUT\subset\Gii$.
There is no continuous parameter to choose; any deformation of $\theta$
would violate the $W(\Gii)$ root axioms or the $\SUT$ embedding.
\end{proof}

\begin{remark}
The uniqueness is stronger than just fixing an angle: it fixes the entire
interleaved short/long root fan.
The angle $\theta = \pi/6$ is the angular width of a $\Gii$ Weyl chamber,
and each such chamber is bounded on one side by a short-root hyperplane
and on the other by a long-root hyperplane.
The Weyl chamber width $\pi/6$ is a discrete invariant of the root system,
not a metric parameter.
\end{remark}

\begin{corollary}
The exponent in equation~\eqref{eq:mum} inherits the full rigidity of the
$\Gii$ root system.
The two trigonometric values at $\theta = \pi/6$ used in Addendum~132 —
$\cot(\pi/6) = \sqrt{3}$ and $\sin(\pi/6) = 1/2$ — are forced by the
same geometric constraint, with no remaining freedom.
\end{corollary}

%==========================================================================
\section{Measure pullback via Hopf}
\label{sec:measure}

\begin{theorem}[Hopf pullback of the $\Gii$-induced $S^3$ measure]
\label{thm:measure}
Let $S^3 = \partial B^4$ carry the round (SO(4)-invariant) Riemannian
measure $d\sigma$, which is the unique $\Gii$-compatible measure on $S^3$.
Let $\pi_H: S^3\to S^2$ be the Hopf fibration with $S^1$ fibers.
Identify the radial interval $[0,1]$ of $B^4$ with the orbit space of
the Hopf-fibered boundary via the radial coordinate $x\in[0,1]$.
Then the measure on $[0,1]$ induced from the $B^4$ volume form
$dV_4 = 2\pi^2 x^3\,dx$ is proportional to $x^3\,dx$.
\end{theorem}

\begin{proof}
The $B^4$ volume form in radial coordinates is
$dV_4 = \mathrm{vol}(S^3_x)\,dx = 2\pi^2 x^3\,dx$,
where $S^3_x$ is the sphere of radius $x$ with round area $2\pi^2 x^3$.
The $\Gii$ root geometry is discrete (a finite set of reflection
hyperplanes in the 2D root plane) and does not deform the Riemannian
metric on $S^3\subset\mathbb{R}^4$.
The $\Gii$-compatible measure on $S^3$ is therefore the round measure
$d\sigma = 2\pi^2\,d(\text{normalised area})$, which is the unique
$\mathrm{SO}(4)$-invariant probability measure on $S^3$
(the $\mathrm{SU}(3)\subset\Gii$ subgroup acts by isometries that
preserve this measure).
Under the Hopf fibration $\pi_H: S^3\to S^2$, the round measure decomposes
as
\begin{equation}
  d\sigma_{S^3} = d\sigma_{S^2}\otimes d\theta_{S^1}
  \quad (\text{Riemannian submersion}),
  \label{eq:hopf-decomp}
\end{equation}
where $d\sigma_{S^2}$ is the Fubini-Study measure on $S^2$ and $d\theta_{S^1}$
is the Haar measure on the $S^1$ fibre.
Integrating over the $S^1$ fibre (which has length $2\pi$ in unit $S^3$) and
then pulling back to the radial $B^4$ coordinate $x\in[0,1]$ yields a
measure weighted by the $S^3$ area $2\pi^2 x^3$, giving
\begin{equation}
  d\mu_{\mathrm{radial}}
  = \underbrace{2\pi^2 x^3}_{{\rm area}(S^3_x)}\,dx
  \;\propto\; x^3\,dx.
  \label{eq:radial-measure}
\end{equation}
This is the volumetric measure, not flat $dx$.
\end{proof}

\begin{remark}
The key step is that the $\Gii$ root angle $\theta = \pi/6$ is a property
of the discrete symmetry group acting on the root plane, not a curvature
parameter of the $S^3$ metric.
The round metric on $S^3\subset\mathbb{R}^4$ is determined by the embedding
and is $\Gii$-invariant; it is not distorted by the root fan geometry.
Hence the pullback measure is $x^3\,dx$ regardless of whether one
specifies the $\Gii$ root angle or not.
\end{remark}

%==========================================================================
\section{Implication for CP-3}
\label{sec:cp3}

CP-3 (Addendum~144) identifies the following internal inconsistency in
Paper~01.
The coefficient $16$ in $\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x$
satisfies $\int_0^1 16\pi^3 x^3\,dx = 4\pi^3$, which is the correct
$B^4$ bulk contribution to $\alpha^{-1}$.
Paper~01 motivates this coefficient from $dV_4 = 2\pi^2 r^3\,dr$
(volumetric), but the integral
$\int_0^1\rho(x)\,dx = 4\pi^3+\pi^2+\pi = \alpha^{-1}$ uses flat $dx$.
Addendum~144 proposed that the $\Fiv$-invariant inner product on $\JO$
might force the radial measure to be flat $dx$.

Theorem~\ref{thm:measure} settles the $\Gii$ side of this question:
the $\Gii$ boundary geometry gives the volumetric weight $x^3\,dx$,
in agreement with $dV_4 = 2\pi^2 x^3\,dx$.
This has two consequences.

\smallskip\noindent\textbf{(a)~The coefficient~16 is consistent with the volumetric measure.}
The $\Gii$ root geometry supports the chain
\begin{equation}
  \text{short roots} \xrightarrow{\;\SUT\subset\Gii\;}
  S^3\;\text{round measure}
  \xrightarrow{\;\text{Hopf}\;}
  x^3\,dx
  \xrightarrow{\;\int_0^1\!16\pi^3 x^3\,dx\;}
  4\pi^3,
  \label{eq:chain}
\end{equation}
with no inconsistency at the $\Gii$ level.

\smallskip\noindent\textbf{(b)~The flat-measure hypothesis requires a separate argument.}
For $\rho(x)$ to be integrated with flat $dx$ self-consistently,
the $\Fiv$-invariant inner product on $\JO$ would have to
\emph{override} the $\Gii$ boundary measure and replace $x^3\,dx$ with $dx$.
This would require an independent $\Fiv\supset\Gii$ geometric argument
showing that the $\JO$ inner product modifies the radial measure.
The $\Gii$ uniqueness result in Section~\ref{sec:uniqueness} shows there
is no residual freedom in $\theta$ that could absorb such a modification:
the $\Gii$ geometry is fully determined, so any flat-measure resolution
must come from the $\Fiv$ level.
CP-3 therefore remains open, with the constraint that its resolution
cannot invoke $\Gii$ freedom.

%==========================================================================
\section{Open items}
\label{sec:open}

\paragraph{F$_4$ geometry and the flat-measure question.}
$\Fiv$ has rank 4 and contains $\Gii\times\SUT(3)$ as a maximal subgroup.
The relevant question is whether the $\Fiv$-invariant inner product on
$\JO$ — the unique (up to scale) positive definite $\Fiv$-equivariant form —
replaces the volumetric $x^3\,dx$ with flat $dx$ when restricted to the
radial $B^4$ direction.
This requires an explicit computation of the $\Fiv$ action on the
radial coordinate of $B^4\subset\JO$, which depends on how $B^4$ is
embedded in the 27-dimensional Jordan algebra.
No such computation is available within the current corpus.

\paragraph{Higher-order Weyl chambers.}
Theorem~\ref{thm:unique} establishes $\theta = \pi/6$ as the
inter-fan angle in the 2D $\Gii$ root plane.
A finer question is whether the $\Gii$ Weyl chamber angle has a
distinguished role in the 7D ambient space (the imaginary octonions)
that $\Gii$ acts on as the automorphism group of $\mathbb{O}$.
In particular, the 7D analogue of the root fan (the set of $\Gii$-orbit
hyperplanes) might impose further constraints on the $S^3\hookrightarrow S^6$
embedding.

\paragraph{Hopf lapse interpretation of $\theta$.}
Addendum~132 noted that the factor $2\pi$ in equation~\eqref{eq:mum}
corresponds to the Hopf latitude $u = \cos(2\beta)$ double-winding.
Now that $\theta = \pi/6$ is uniquely fixed, a natural question is:
what is the Hopf latitude $u = \cos(2\beta)$ corresponding to the
$\Gii$ fan boundary at $\theta = \pi/6$?
Setting $\beta = \theta = \pi/6$ gives $u = \cos(\pi/3) = 1/2$,
which is precisely the GON half-lapse.
Whether this coincidence has a structural explanation — connecting
the matching condition $\alpha_s(\mum) = \sin(\pi/6) = 1/2 = u$ to
a specific $S^3$ locus — is an open geometric question.

\end{document}
