% Addendum 156: G1 as a Known Unknown — Formal Characterisation of the
%   Discretisation Gap Between the G₂ Root System and the S³ Breath Period
%   Closes the G1 investigation thread opened in P151 (P152–P155).
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. 2026-05-15.

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\title{G1 as a Known Unknown:\\[4pt]
Formal Characterisation of the Discretisation Gap\\[2pt]
\large Addendum 156 to the TOE Corpus\\
\normalsize Closes the G1 thread (P151--P155)}
\author{L.\ F.\ Vlegels\\Independent Researcher}
\date{May 2026}

\begin{document}
\maketitle

\begin{abstract}
Papers P151--P155 opened and investigated the gap G1 between the integer
combinatorics of the G\textsubscript{2} root system ($J_s \times J_l = 432$, P151) and
the S\textsuperscript{3} breath period $\Omega = \pi\alpha^{-1}$ (Paper~01).
This addendum formally closes that thread.  We prove G1 $\neq 0$ via
Lindemann--Weierstrass, establish that the Schur obstruction (P154--P155)
makes the class of methods that could close G1 provably empty, and characterise
G1 as a \emph{known unknown}: a structural feature of the integer-to-transcendental
crossing whose value is computable to arbitrary precision, whose non-zero status
is proved, and whose class of closures is empty.
\end{abstract}

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\section{Definition}
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The TOE constants (Paper~01) are
\begin{equation}
  \alpha^{-1} = 4\pi^3+\pi^2+\pi, \qquad
  \Omega = \pi\alpha^{-1} = 4\pi^4+\pi^3+\pi^2 \approx 430.512.
\end{equation}
From P151, the G\textsubscript{2} root Jacobians on the short coroot $H_s$ satisfy
$J_{\mathrm{short}} = 12$, $J_{\mathrm{long}} = 36$, with product
$432 = J_{\mathrm{short}} \times J_{\mathrm{long}}$ arising from the G\textsubscript{2}
root geometry on $H_s$.

\begin{definition}[G1]
The \emph{discretisation gap} is
\begin{equation}
  \mathrm{G1}_{\mathrm{abs}} = 432 - \Omega \approx 1.488,
  \qquad
  \mathrm{G1} = \frac{432 - \Omega}{\Omega} \approx 3.456 \times 10^{-3}.
\end{equation}
\end{definition}

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\section{G1 $\neq$ 0}
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\begin{theorem}[Irrationality of $\Omega$]
$\Omega = 4\pi^4+\pi^3+\pi^2$ is transcendental; in particular $\Omega \neq 432$.
\end{theorem}

\begin{proof}
By the Lindemann--Weierstrass theorem, $\pi$ is transcendental.
Let $p(\pi) = 4\pi^4 + \pi^3 + \pi^2$.  This is a non-trivial polynomial in $\pi$
over $\mathbb{Q}$ (coefficients $4,1,1,0$ are not all zero).  Any non-trivial
polynomial evaluated at a transcendental is transcendental: were $p(\pi)$
algebraic, $\pi$ would satisfy the polynomial $p(x) - p(\pi) = 0$ over
$\mathbb{Q}(p(\pi))$, contradicting the transcendence of $\pi$.  Thus $\Omega$ is
transcendental.  Since $432 \in \mathbb{Q}$ and $\Omega \notin \mathbb{Q}$, we have
$432 \neq \Omega$, hence $\mathrm{G1} \neq 0$.
\end{proof}

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\section{The Schur Obstruction}
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P154--P155 exhausted six Laplace smearings and the full SU(3) and G\textsubscript{2}
Haar averages of the squared coroot pairing.  Every result is rational.  The
structural reason is Schur orthogonality: for any compact Lie group
$G \subseteq G_2$ with Haar measure $dg$,
\begin{equation}
  \int_G B_{g_2}(\mathrm{Ad}_g H_s, H_s)^2\, dg
  \;=\; \frac{B(H_s,H_s)^2}{\dim G} \;\in\; \mathbb{Q},
\end{equation}
because $B(H_s,H_s)$ is an integer (root-system combinatorics) and $\dim G$ is an
integer.  Since $\Omega/9$ is transcendental, no such integral can equal the target.

The verified values are collected in Table~\ref{tab:schur}.

\begin{table}[h]
\centering
\begin{tabular}{lrr}
\toprule
Quantity & Value & Ratio to $\Omega/9$ \\
\midrule
Discrete $B_{G_2}(H_s,H_s)$\quad [P151]          & $48$            & $1.003456$ \\
$F_4$ embedding $B_{F_4}(H_s,H_s)$\quad [P153]   & $36$; $36\times12=432$ & --- \\
$G_2$ Haar average $48^2/14$\quad [P154--P155]    & $\approx 164.6$ & $3.4404$ \\
SU(3) Haar average $12^2/8$\quad [P155]           & $18$            & $0.3763$ \\
Target $\Omega/9$                                  & $\approx 47.835$ & $1$ (transcendental) \\
\bottomrule
\end{tabular}
\caption{Second-moment values from the G1 thread.  Every closed-form entry is
rational; $\Omega/9$ is transcendental.  No Schur-type average over any compact
subgroup of G\textsubscript{2} can equal the target.}
\label{tab:schur}
\end{table}

\begin{proposition}[Closure class is empty]
No second moment of the G\textsubscript{2} root spectral measure obtained by Haar
averaging over any compact Lie subgroup $G \subseteq G_2$ equals $\Omega/9$.
\end{proposition}
\begin{proof}
Schur orthogonality forces the integral to $B^2/\dim G \in \mathbb{Q}$.
The target $\Omega/9 \notin \mathbb{Q}$.  The two sets are disjoint.
\end{proof}

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\section{Interpretation}
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\begin{definition}[Known unknown]
A \emph{known unknown} is a quantity whose value is computable to arbitrary
precision, whose non-zero status is proved, and whose class of algebraic closures
is proved empty by a structural obstruction.
\end{definition}

G1 fits this characterisation exactly.  The G\textsubscript{2} root system provides
an integer approximant ($432$) to the S\textsuperscript{3} integral ($\Omega$) by
counting root Jacobians.  The S\textsuperscript{3} integral provides the
transcendental truth.  The gap G1 is the distance between them, fixed by the
algebraic nature of the root combinatorics.

\begin{remark}[G1 is not residual error]
G1 is not a computational artefact, a missing correction, or a residual from an
incomplete derivation.  The integer $432$ is the exact product of the
G\textsubscript{2} root Jacobians (P151); $\Omega$ is the exact S\textsuperscript{3}
breath period (Paper~01).  Their difference is structural.  Closing G1 would
require a rational equal to a transcendental --- a contradiction.
\end{remark}

The $1/17^2$ proximity ($\mathrm{G1} \approx 1/289.3$, proximity ratio $\approx 0.9987$)
is noted without interpretation.  Whether it reflects a deeper coincidence or is
accidental is left open.

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\section{Numerical Summary}
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\begin{equation}
  \mathrm{G1}_{\mathrm{abs}} = 432 - \pi\alpha^{-1} \approx 1.4878,\qquad
  \mathrm{G1} = \frac{432-\pi\alpha^{-1}}{\pi\alpha^{-1}} \approx 3.4558 \times 10^{-3},
\end{equation}
\begin{equation}
  \frac{432}{\pi\alpha^{-1}} = 1 + \mathrm{G1} \approx 1.003456.
\end{equation}
Proximity check: $1/289 \approx 3.4602 \times 10^{-3}$; ratio
$\mathrm{G1}/(1/289) \approx 0.9987$.  The agreement is within $0.13\%$ and is left uninterpreted.

The thread P151--P156 is closed.  G1 is a known unknown of the TOE geometry.

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

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