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\title{\textbf{Addendum P155}\\[4pt]
SU(3)-Orbit Average of the G\textsubscript{2} Coroot Pairing\\[2pt]
and Gap G1}
\author{L\'eon Fernando Vlegels}
\date{May 2026}

\begin{document}
\maketitle

\begin{abstract}
Addendum P154 exhausted six Laplace-type smearings of the discrete
G\textsubscript{2} root spectral measure; none closes gap G1 without tautological
construction.  The one genuinely untried direction identified there is the
\emph{SU(3)-orbit average} of the squared coroot pairing: replacing each root's
discrete contribution to the second moment by the Haar average of
$B_{g_2}(\mathrm{Ad}_g H_s,H_s)^2$ over $g\in\mathrm{SU}(3)\subset G_2$.
We compute this integral in two independent ways---analytic Schur orthogonality
and a $N=200\,000$ Monte Carlo over SU(3)---and compare the result with the
G\textsubscript{2}-Haar analogue and the target $\Omega/9$.  The SU(3) orbit average
evaluates to the exact rational $18 = 12^2/8$, undershooting the target
$\Omega/9\approx 47.835$ by $62\%$, a discrepancy $180\times$ larger than G1.
The G\textsubscript{2} Haar average gives $48^2/14\approx 164.6$, overshooting by
$3.4\times$.  The discrete value $48$ (P151) remains the unique approach to $\Omega/9$
within G1.  A structural identity $48 = B_{\mathfrak{su}(3)}(H_s,H_s)^2/|\Phi^+(A_2)|$
is recorded.  G1 is not closed.
\end{abstract}

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\section{Setup}
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Throughout, the TOE constants are
\begin{equation}
  \alpha^{-1} = 4\pi^3+\pi^2+\pi,\qquad
  \Omega = \pi\alpha^{-1}\approx 430.512 \quad(\text{breath period}),
\end{equation}
and the target second moment is
\begin{equation}
  \mathcal{T} \;=\; \frac{\Omega}{9}\;\approx\;47.835.
\end{equation}
The discrete coroot Killing value from P151 is $B_{G_2}(H_s,H_s)=48$, where $H_s$ is
the coroot of the short simple root of G\textsubscript{2}, normalised so that its
projections onto the 12 G\textsubscript{2} roots produce the root integer
$\{-3,-3,-2,-1,-1,0,0,1,1,2,3,3\}$ with $\sum_\alpha\langle\alpha,H_s\rangle^2=48$.
The gap is
\begin{equation}
  \mathrm{G1} \;=\;\frac{432-\Omega}{\Omega}\;\approx\;3.456\times10^{-3}.
\end{equation}

The SU(3) embedding is $A_2\hookrightarrow G_2$ via the short-root subgroup.  The
Lie algebra decomposes as $\mathfrak{g}_2 = \mathfrak{su}(3)\oplus W$ where
$W\cong\mathbf{3}\oplus\bar{\mathbf{3}}$ as an $\mathfrak{su}(3)$-module.
$H_s$ lies in the Cartan subalgebra of $\mathfrak{su}(3)$; we represent it as
$H_s = \mathrm{diag}(1,-1,0)\in\mathfrak{su}(3)\subset\mathfrak{g}_2$.

%%─────────────────────────────────────────────────────────────────
\section{The three integrals}
%%─────────────────────────────────────────────────────────────────

\subsection*{Integral I: G\textsubscript{2} Haar average}

By Schur orthogonality for the 14-dimensional real adjoint representation of
G\textsubscript{2}:
\begin{equation}
  I_{G_2} \;=\; \int_{G_2} B_{g_2}(\mathrm{Ad}_g H_s,H_s)^2\,dg
  \;=\; \frac{B_{G_2}(H_s,H_s)^2}{\dim G_2}
  \;=\; \frac{48^2}{14} \;=\; \frac{1152}{7}
  \;\approx\;164.571.
\end{equation}
This exceeds $\mathcal{T}$ by a factor of $3.44$.

\subsection*{Integral II: SU(3) Haar average}

\paragraph{Key formula.}
For $g\in\mathrm{SU}(3)$, write $h = g\,H_s\,g^\dagger$ and let
$h_1,h_2,h_3=h_{11},h_{22},h_{33}$ be the (real) diagonal entries of $h$.  The six
roots of $A_2$ are $\pm(e_i-e_j)$ with projections
\begin{equation}
  \alpha(H)\;=\;\langle e_i-e_j,H\rangle \;=\; H_{ii}-H_{jj}.
\end{equation}
A direct root-sum calculation gives
\begin{equation}
  B_{\mathfrak{su}(3)}\!\bigl(\mathrm{Ad}_g H_s,\,H_s\bigr)
  \;=\;\sum_{\alpha\in\Phi(A_2)}\alpha\!\bigl(\mathrm{Ad}_g H_s\bigr)\,\alpha(H_s)
  \;=\; 6(h_1-h_2),
  \label{eq:Bformula}
\end{equation}
since $\alpha_1(H_s)=2$, $\alpha_2(H_s)=-1$, $(\alpha_1+\alpha_2)(H_s)=1$ and the
total contracts to $2(2h_1-2h_2+h_1-h_3-h_2+h_3)=6(h_1-h_2)$.

\paragraph{Schur result.}
Let $f(g)=B_{\mathfrak{su}(3)}(\mathrm{Ad}_g H_s,H_s)$.  By Schur orthogonality for
the adjoint representation of $\mathrm{SU}(3)$ (dimension $d=8$, real orthogonal):
\begin{equation}
  \int_{\mathrm{SU}(3)}\bigl(\mathrm{Ad}_g X\bigr)_i\bigl(\mathrm{Ad}_g X\bigr)_j\,dg
  \;=\;\frac{B_{\mathfrak{su}(3)}(X,X)}{d}\,\delta_{ij},
\end{equation}
so
\begin{equation}
  I_{\mathrm{SU}(3)} \;=\; \int_{\mathrm{SU}(3)} f(g)^2\,dg
  \;=\;\frac{B_{\mathfrak{su}(3)}(H_s,H_s)^2}{d}.
  \label{eq:Schur}
\end{equation}
The A\textsubscript{2} root projections onto $H_s$ are
$\{+2,-2,+1,-1,-1,+1\}$, giving
$B_{\mathfrak{su}(3)}(H_s,H_s)=4+4+1+1+1+1=12$, and therefore
\begin{equation}
  \boxed{I_{\mathrm{SU}(3)} \;=\; \frac{12^2}{8} \;=\; \frac{144}{8} \;=\; 18.}
  \label{eq:I_SU3}
\end{equation}

\paragraph{Monte Carlo verification.}
We generate $N=200\,000$ Haar-distributed $\mathrm{SU}(3)$ matrices via QR
decomposition of a complex Gaussian ensemble (Mezzadri 2007), compute
$[6(h_1-h_2)]^2$ for each sample, and average.  The result $18.053\pm 0.048$
agrees with~\eqref{eq:I_SU3} to within one standard error.

\subsection*{Integral III: flag-variety fibration}

The flag variety $G_2/T$ carries an $\mathrm{SU}(3)$-equivariant fibration
$G_2/T\to G_2/\mathrm{SU}(3)$ with fibre $\mathrm{SU}(3)/T$.
Under the Liouville measure, the fibre-average of the integrand
$B_{g_2}(\mathrm{Ad}_g H_s,H_s)^2$ over each fibre is exactly $I_{\mathrm{SU}(3)}=18$
by translation-invariance of the Haar measure.  The base $G_2/\mathrm{SU}(3)$ carries
no additional modulation from $H_s$ in the $\mathfrak{su}(3)$ direction.  Thus the
flag-variety second moment reduces to the SU(3) Haar average computed above.

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\section{Script output (verbatim)}
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\begin{verbatim}
=================================================================
P155 FLAG-VARIETY ORBIT AVERAGE VERIFICATION
=================================================================
ALPHA_INV        = 137.0363037759
BREATH_PERIOD    = 430.5122452174
TARGET (Omega/9) = 47.8346939130
DISCRETE         = 48
G1 = (432-Omega)/Omega = 3.45577809e-03  (0.345578%)

--- INTEGRAL I: G2 HAAR AVERAGE  (Schur, analytic) ---
I_G2 = B(H_s,H_s)^2 / dim(G2) = 48^2/14 = 164.571429
TARGET                                   = 47.834694
ratio I_G2 / TARGET = 3.440420  (overshoots by 244.04%)

--- INTEGRAL II: SU(3) HAAR AVERAGE  (Schur, analytic) ---
A2 root projections onto H_s = diag(1,-1,0): [2,-2,1,-1,-1,1]
B_{su3}(H_s,H_s) = Sum alpha(H_s)^2 = 12
Schur formula: I_SU3 = B^2/dim(su3) = 12^2/8 = 18.000000
TARGET                                          = 47.834694
ratio I_SU3 / TARGET = 0.376296  (undershoots by 62.37%)

Structural identity: B_{su3}(H_s,H_s)^2/|Phi+(A2)| = 12^2/3 = 48.0
DISCRETE = 48  (match: True)

--- INTEGRAL II: SU(3) HAAR AVERAGE  (Monte Carlo, N=200,000) ---
I_SU3 (MC)       = 18.052622  +/-  0.047835  (1-sigma)
I_SU3 (analytic) = 18.000000
MC vs analytic discrepancy: 0.0526  (0.292%)
TARGET (Omega/9) = 47.834694
ratio I_SU3_MC / TARGET = 0.377396
gap I_SU3 - TARGET      = -29.7821

=================================================================
SUMMARY TABLE
=================================================================
Quantity                                    Value   Ratio/TARGET
-----------------------------------------------------------------
  Discrete B_{G2}(H_s,H_s)  [P151]        48.0000       1.003456
  G2 Haar avg  48^2/14       [analytic]   164.5714       3.440420
  SU(3) Haar avg  12^2/8     [analytic]    18.0000       0.376296
  SU(3) Haar avg             [MC]         18.0526       0.377396
  TARGET  Omega/9                         47.8347       1.000000

--- INTERPOLATION ---
  t* = (Omega/9 - 18) / (48 - 18) = 0.9944897971
  1 - t* = 0.0055102029
  (1-t*) as multiple of G1: 1.594490
  (1-t*) = (432-Omega)/30  [no recognised geometric constant]

=================================================================
G1 CLOSING STATUS
=================================================================
  G1                            = 0.345578%
  |I_SU3 - TARGET| / TARGET     = 62.3704%
  Ratio (orbit gap) / G1        = 180.48x

CONCLUSION: I_SU3 = 18 undershoots TARGET = 47.8347 by 62.37%,
which is 180.5x G1. DISCRETE 48 remains the unique value within
G1 of Omega/9. G1 is NOT closed by orbit averaging.

STRUCTURAL NOTE:
  I_SU3 x dim(su3) / |Phi+(A2)| = 18 x 8/3 = 48.0 = DISCRETE
  DISCRETE = B_su3(H_s,H_s)^2 / |Phi+(A2)| = 144/3 = 48
\end{verbatim}

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\section{Findings}
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\subsection*{Summary of integral values}

\begin{center}
\begin{tabular}{lrr}
\hline
Quantity & Value & Ratio to $\mathcal{T}=\Omega/9$ \\
\hline
Discrete $B_{G_2}(H_s,H_s)$ \quad [P151]    & $48$       & $1.003456$ \\
$G_2$ Haar average $48^2/14$                  & $164.571$  & $3.4404$   \\
SU(3) Haar average $12^2/8$ \quad [analytic] & $18$       & $0.3763$   \\
SU(3) Haar average \quad [MC]                & $18.053$   & $0.3774$   \\
Target $\Omega/9$                             & $47.835$   & $1$        \\
\hline
\end{tabular}
\end{center}

The SU(3) orbit average is the exact rational $18$.  It undershoots the target by
$62.37\%$, a gap that is $180.5\times$ the size of G1.  The G\textsubscript{2}
Haar average overshoots by $3.44\times$.  The discrete value $48$ is the only
entry in the table that lies within G1 of the target.

\subsection*{Why the SU(3) orbit average does not close G1}

Equation~\eqref{eq:Schur} gives $I_{\mathrm{SU}(3)}=B^2/d$ where $B=12$ is an integer
(the A\textsubscript{2} Killing value) and $d=8$ is an integer (the dimension of
$\mathfrak{su}(3)$).  The result $18$ is therefore rational.  The target
$\mathcal{T}=\pi(4\pi^3+\pi^2+\pi)/9$ is transcendental.  No rational equals a
transcendental: the orbit average cannot close the gap regardless of the smearing
strategy, because Schur collapses the entire Haar integral to an algebraic
combination of the Killing norm and the representation dimension.

This is the same obstruction as in P154, now at the level of the full SU(3) Haar
measure rather than a Laplace transform.  Maximal smearing (Haar) does not help.

\subsection*{Structural identity}

The computation reveals
\begin{equation}
  48 \;=\; \frac{B_{\mathfrak{su}(3)}(H_s,H_s)^2}{\lvert\Phi^+(A_2)\rvert}
       \;=\; \frac{12^2}{3} \;=\; 48.
  \label{eq:struct}
\end{equation}
Equivalently, $I_{\mathrm{SU}(3)}\times\dim(\mathfrak{su}(3))/|\Phi^+(A_2)| = 18\times 8/3 = 48 =
\text{DISCRETE}$.  Equation~\eqref{eq:struct} provides a third representation of the
P151 discrete value: it equals the A\textsubscript{2} Schur second moment scaled by
the dimension-to-positive-root ratio.  This is an algebraic identity connecting
the G\textsubscript{2} Killing norm to the SU(3) subgroup data; it does not
introduce transcendental content.

\subsection*{Interpolation}

The unique linear interpolation
$I(t) = (1-t)\,I_{\mathrm{SU}(3)} + t\cdot48$ that hits $\mathcal{T}$ satisfies
\begin{equation}
  t^* \;=\; \frac{\mathcal{T}-18}{30} \;=\; \frac{\Omega/9-18}{30}
       \;\approx\; 0.99449.
\end{equation}
The deviation from 1 is $1-t^* = (432-\Omega)/30\approx 5.51\times10^{-3}$, a
multiple of G1 with proportionality constant $\approx 1.59$.  The denominator $30 =
48 - 18$ carries no recognised G\textsubscript{2} invariant.  There is no natural
geometric interpolation parameter with a recognisable value.

\subsection*{What remains}

P154 (item~1 of its ``what would be needed'' list) called for a continuous measure on
the G\textsubscript{2} root system whose second moment evaluates to $\Omega/9$
\emph{from first principles}.  The SU(3) Haar measure is the most natural such
candidate; it evaluates to $18$, not $\Omega/9$.  The direction is exhausted.
Any resolution of G1 via spectral methods must originate from a structure that
connects the G\textsubscript{2} root lattice to the S\textsuperscript{3} holonomy
geometry at the TOE scale --- a layer of geometry not accessible through Schur
orthogonality on any subgroup of G\textsubscript{2}.

\subsection*{Status of G1}

\textbf{G1 remains open.}  The SU(3)-orbit average of
$B_{g_2}(\mathrm{Ad}_g H_s,H_s)^2$ equals the rational $18$, falling short of
$\Omega/9\approx 47.835$ by $62\%$ ($180.5\times$ G1).  The G\textsubscript{2}
Haar average equals $48^2/14\approx 164.6$, overshooting by $3.44\times$.  The
discrete P151 value $48$ remains the unique approach to $\mathcal{T}$ within G1.
Schur orthogonality guarantees that any Haar integral over a compact subgroup of
G\textsubscript{2} returns a rational multiple of Killing-form data; no such
rational can equal the transcendental $\Omega/9$.

\bigskip
\hrule
\bigskip
{\small \textcopyright\ L\'eon Fernando Vlegels. MIT License.}

\end{document}
