% Addendum 183: G1_P171 and the Near-Miss delta approx h(G_2) —
%               A PSLQ Investigation of j(i) - 4*BREATH_PERIOD approx 6
%
%   Investigates G1_P171 = j(i)/(4*BREATH_PERIOD) - 1 approx 0.3456%,
%   the constant introduced in P171 (NOT P179's G1 = 1728/pi^3 - 1).
%   The near-miss delta = j(i) - 4*BREATH_PERIOD approx 5.951 is close
%   to h(G_2) = 6 (Coxeter number of G_2). PSLQ over four bases
%   (Chowla-Selberg ingredients, Gamma(1/4), log, zeta, e^{2pi},
%   e^{-pi*sqrt(163)}) returns NO relation for delta-6 at coeff <= 1000.
%   Verdict: the near-miss is a polynomial coincidence; delta is purely
%   a polynomial in pi with no additional CM or modular structure.
%
% L. F. Vlegels, Independent Researcher
% Copyright: Leon Fernando Vlegels. License: CC BY 4.0. May 2026.

\documentclass[11pt,a4paper]{article}
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\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
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\newtheorem{definition}[theorem]{Definition}
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\newcommand{\Gone}{G_1^{(171)}}
\newcommand{\GoneP}{G_1^{(179)}}
\newcommand{\Js}{J_{\mathrm{short}}}
\newcommand{\BP}{\Omega}
\newcommand{\FBP}{4\Omega}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\Gq}{\Gamma\!\left(\tfrac{1}{4}\right)}
\newcommand{\hG}{h(G_2)}
\newcommand{\wl}{\varpi}

\title{$G_1^{(171)}$ and the Near-Miss $\delta \approx h(G_2)$:\\
A PSLQ Investigation of $j(i) - 4\Omega \approx 6$\\[0.5em]
\large Addendum 183 to the TOE Corpus}

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

\date{May 2026}

\begin{document}
\maketitle
\tableofcontents
\newpage

%==========================================================================
\begin{abstract}
Addendum~171 (P171) introduced the constant
\[
  \Gone \;=\; \frac{j(i)}{4\Omega} - 1
  \;\approx\; 0.3456\%,
\]
where $\Omega = \pi\alpha^{-1}$ is the TOE breath period
($\alpha^{-1} = 4\pi^3 + \pi^2 + \pi$) and $j(i) = 1728$ is the
Klein $j$-invariant at the CM point $\tau = i$.  This is \emph{not} the
constant $\GoneP = 1728/\pi^3 - 1 \approx 54.73$ studied in
Addendum~179~\cite{P179}; the two constants use different TOE denominators
($4\Omega \approx 1722$ versus $\pi^3 \approx 31.006$) and differ by
a factor of roughly $158$.

P171 proved that the absolute gap $\delta = j(i) - 4\Omega
= 1728 - 16\pi^4 - 4\pi^3 - 4\pi^2 \approx 5.9510$ is close to but
strictly less than $\hG = \Js/2 = 6$ --- the Coxeter number of $G_2$ and
half the $G_2$ short-root sector sum $\Js = 12$.  P171 left open the
question of whether this near-miss has additional structure: is the
residual $\delta - 6$ expressible in terms of CM periods, Gamma-function
values, or other modular ingredients at $\tau = i$?

The present addendum provides a definitive PSLQ investigation of this
question.  We run four systematic integer-relation searches at
$\mathtt{mp.dps} = 55$ over bases including $\Gamma(1/4)$, the lemniscate
constant $\wl$, $\log 2$, $\zeta(3)$, $e^{2\pi}$, and the Heegner-related
quantity $e^{-\pi\sqrt{163}}$.  All searches return \textbf{no relation}
for $\delta - 6$ at maximal coefficient $\le 1000$.  The only integer
relation found is the \emph{trivial} polynomial identity
$\delta = 1728 - 16\pi^4 - 4\pi^3 - 4\pi^2$, recovered when maxcoeff
$\ge 1728$.

Our verdict: \textbf{the near-miss $\delta \approx 6 = \hG$ is a
polynomial coincidence}.  The gap $\delta$ is entirely determined by its
definition as a polynomial in $\pi$; the proximity to the Coxeter number
$\hG = 6$ carries no additional CM, modular, or Gamma-function structure at
coefficient norm $\le 1000$.  All numerical claims are verified to
55~significant figures by \texttt{verify\_P183.py}.
\end{abstract}

%==========================================================================
\section{Background and the two $G_1$ constants}\label{sec:background}
%==========================================================================

\subsection{Two constants sharing a name}

The TOE corpus contains two constants that go by the name $G_1$, arising
from two distinct TOE denominators.  It is essential to keep them separate.

\begin{definition}[The two $G_1$ constants]\label{def:both}
\begin{align}
  \Gone  &\;=\; \frac{j(i)}{4\Omega} - 1,
  \qquad \Omega = \pi\alpha^{-1},
  \quad \alpha^{-1} = 4\pi^3 + \pi^2 + \pi,
  \label{eq:G1-P171}\\
  \GoneP &\;=\; \frac{j(i)}{4\Omega_0} - 1,
  \qquad \Omega_0 = \frac{\pi^3}{4},
  \label{eq:G1-P179}
\end{align}
where $j(i) = 1728 = 12^3$ is the Klein $j$-invariant at $\tau = i$.
\end{definition}

\begin{assertion}[Numerical distinction]\label{ass:distinct}
The two constants are numerically far apart:
\begin{align}
  4\Omega  &= 16\pi^4 + 4\pi^3 + 4\pi^2
             \;\approx\; 1722.049,
  \\
  4\Omega_0 &= \pi^3
              \;\approx\; 31.006,
\end{align}
giving $\Gone \approx 0.003456$ (sub-percent) while
$\GoneP \approx 54.73$ (large).  Their ratio is
$(1 + \GoneP)/(1 + \Gone) = 4\Omega/(pi^3) \approx 55.5$.
The two constants are independent transcendentals.
\end{assertion}

\begin{assertion}[Prior results in the corpus]\label{ass:prior}
P171 \cite{P171} proved:
\begin{enumerate}
  \item[\normalfont(i)] $\Gone$ is transcendental.
  \item[\normalfont(ii)] $j(i) - 4\Omega = 5.9510\ldots < 6 = \hG$
    (strictly non-exact).
  \item[\normalfont(iii)] $4\Omega = \mathrm{vol}(S^2)\cdot\alpha^{-1}$
    (sphere-volume identity).
\end{enumerate}
P179 \cite{P179} proved that $\GoneP$ lies outside the Chowla--Selberg
$\QQ$-linear span at $\tau = i$ (PSLQ: no relation at coeff $\le 1000$).
The present addendum extends the PSLQ investigation to $\Gone$ and to the
near-miss residual $\delta - \hG$.
\end{assertion}

\subsection{Scope of this addendum}

P171 identified the near-miss $\delta = j(i) - 4\Omega \approx 5.951
\approx \hG = 6$ as an open structural question.  The proximity to the
Coxeter number of $G_2$ is intriguing: $j(i) = \Js^3 = 12^3$ already
involves the $G_2$ short-root sum $\Js = 12$, and $\hG = \Js/2 = 6$ is
the Coxeter number of the same Lie algebra.  If $\delta = 6$ were exact,
the identity $j(i) = 4\Omega + \hG$ would express the $j$-invariant as
the $S^2$-volume-weighted coupling constant plus the $G_2$ Coxeter number
--- a beautiful TOE identity.  P171 proved it is not exact.  This addendum
asks: is the near-miss \emph{structured} in some weaker sense?

%==========================================================================
\section{Exact values at 55 digits}\label{sec:exact}
%==========================================================================

\begin{assertion}[$G_1^{(171)}$ at 55 significant figures]\label{ass:exact-G1}
At $\mathtt{mp.dps} = 55$:
\begin{align}
  \alpha^{-1} &= 137.0363037758784325592023946515612348284120230701\ldots
  \label{eq:alphainv}\\
  \Omega       &= 430.5122452173989277400721368217979913364493306638\ldots
  \label{eq:Omega}\\
  4\Omega      &= 1722.0489808695957109602885472871919653457973226555\ldots
  \label{eq:FourOmega}\\
  \Gone        &= 0.003455778085591479075380623365743477970730119293\ldots
  \label{eq:G1val}
\end{align}
Equivalently, $\Gone \approx 0.34558\%$ of $4\Omega$.
Verified by \texttt{verify\_P183.py}.
\end{assertion}

\begin{assertion}[The near-miss gap $\delta$ at 55 digits]\label{ass:delta-exact}
\begin{align}
  \delta \;=\; j(i) - 4\Omega
    &= 5.9510191304042890397114527128080346542026773445\ldots
  \label{eq:delta-exact}\\
  \hG \;=\; 6 \;-\; \delta
    &= 0.0489808695957109602885472871919653457973226554703\ldots
  \label{eq:deficit}
\end{align}
The deficit $6 - \delta > 0$ confirms P171's non-exactness.  In percentage
terms, $|\delta - 6|/|\delta| \approx 0.823\%$.
\end{assertion}

\begin{assertion}[Exact polynomial form of $\delta$]\label{ass:poly-delta}
By the definition $\Omega = \pi\alpha^{-1} = 4\pi^4 + \pi^3 + \pi^2$:
\begin{equation}\label{eq:delta-poly}
  \delta \;=\; 1728 - 16\pi^4 - 4\pi^3 - 4\pi^2.
\end{equation}
The deficit likewise has an exact polynomial form:
\begin{equation}\label{eq:deficit-poly}
  6 - \delta \;=\; 16\pi^4 + 4\pi^3 + 4\pi^2 - 1722
           \;=\; 4\pi\alpha^{-1} - 1722.
\end{equation}
Both are transcendental (nonconstant polynomials in $\pi$ minus rationals).
\end{assertion}

\begin{proof}
Substituting $4\Omega = 16\pi^4 + 4\pi^3 + 4\pi^2$ into $\delta = 1728 - 4\Omega$
gives~\eqref{eq:delta-poly}.  Then $6 - \delta = 6 - (1728 - 16\pi^4 - 4\pi^3
- 4\pi^2) = 16\pi^4 + 4\pi^3 + 4\pi^2 - 1722$.  Transcendence follows from
Lindemann--Weierstrass applied to $\pi$.
\end{proof}

\begin{assertion}[Factored form of $4\Omega$]\label{ass:factor}
The quantity $4\Omega$ admits the factored forms:
\begin{equation}\label{eq:factored}
  4\Omega \;=\; 4\pi \cdot \alpha^{-1}
           \;=\; 4\pi^2\,(4\pi^2 + \pi + 1)
           \;=\; \mathrm{vol}(S^2)\cdot\alpha^{-1}.
\end{equation}
In particular, $4\pi^2(4\pi^2+\pi+1) = 4\Omega$ identically (verified to
55~digits: difference is $0$), confirming the factored form as an identity,
not an approximation.
\end{assertion}

\begin{proof}
$4\Omega = 4\pi\alpha^{-1} = 4\pi(4\pi^3+\pi^2+\pi) = 16\pi^4+4\pi^3+4\pi^2
= 4\pi^2(4\pi^2+\pi+1)$.  The factor $4\pi = \mathrm{vol}(S^2)$ is the
standard surface area of the unit 2-sphere.
\end{proof}

%==========================================================================
\section{PSLQ integer-relation search}\label{sec:pslq}
%==========================================================================

\subsection{Setup}

The PSLQ algorithm~\cite{pslq} searches for integer coefficients
$(c_1,\ldots,c_n)\in\ZZ^n\setminus\{0\}$ such that $\sum c_k x_k = 0$
for a given vector of real numbers $(x_1,\ldots,x_n)$ computed to high
precision.  All runs use \texttt{mpmath.pslq} at $\mathtt{mp.dps}=55$,
which provides approximately 50 guard digits beyond the working precision.
The maximal coefficient bound is stated with each search.

We distinguish between the \emph{trivial polynomial identity} (which any
PSLQ must recover if maxcoeff is large enough) and \emph{additional
structure} (any relation not derivable from the definition of $\delta$).

\subsection{Basis A: the trivial polynomial identity}

\begin{assertion}[PSLQ Basis A --- polynomial identity recovered]\label{ass:basis-A}
Over the basis $\{1,\,\delta,\,\pi^2,\,\pi^3,\,\pi^4\}$ with
$\mathtt{maxcoeff}=2000$, PSLQ returns the unique relation
\begin{equation}\label{eq:basis-A-rel}
  -1728 + \delta + 4\pi^2 + 4\pi^3 + 16\pi^4 \;=\; 0,
\end{equation}
equivalently $\delta = 1728 - 16\pi^4 - 4\pi^3 - 4\pi^2$.
Verification residual: $|{-2.09\times 10^{-53}}|$ (numerical round-off only).
Note: $\mathtt{maxcoeff}=200$ fails to recover this relation because the
coefficient $1728$ exceeds the search bound; the coefficient $1728 = j(i)$
is forced by the definition of~$\delta$.
\end{assertion}

This relation carries no information beyond the definition~\eqref{eq:delta-poly}.
The question is whether $\delta - 6$ or $\Gone$ has \emph{additional} structure
beyond this polynomial form.

\subsection{Basis B: the residual $\delta - 6$ against CM/Gamma ingredients}

\begin{assertion}[PSLQ Basis B --- no CM/Gamma relation]\label{ass:basis-B}
Over the basis
\[
  \bigl\{\delta - 6,\;\pi,\;\pi^2,\;\pi^3,\;\pi^4,\;
  \Gamma\!\left(\tfrac{1}{4}\right),\;\log 2,\;\zeta(3)\bigr\}
\]
with $\mathtt{maxcoeff} = 500$, PSLQ returns \textbf{no relation}.
If any $\QQ$-linear combination of these elements equals $\delta - 6$,
all integer coefficients exceed $500$ in absolute value.
\end{assertion}

\subsection{Basis C: $G_1^{(171)}$ against a transcendental basis}

\begin{assertion}[PSLQ Basis C --- no relation for $G_1^{(171)}$]\label{ass:basis-C}
Over the basis
\[
  \bigl\{\Gone,\;\pi,\;\pi^2,\;\pi^3,\;
  \Gamma\!\left(\tfrac{1}{4}\right),\;\log 2,\;\zeta(3),\;
  \sin^4\!\left(\tfrac{\pi}{14}\right)\bigr\}
\]
(the last element drawn from the P175 $\delta_{\mathrm{QLC}}$ basis)
with $\mathtt{maxcoeff} = 500$, PSLQ returns \textbf{no relation}.
The constant $\Gone$ has no expression over this basis at coefficient
norm $\le 500$.
\end{assertion}

\subsection{Basis D: extended search including $e^{2\pi}$ and the Heegner number}

\begin{assertion}[PSLQ Basis D --- no relation over extended basis]\label{ass:basis-D}
Over the ten-element basis
\[
  \bigl\{\delta - 6,\;\pi,\;\pi^2,\;\pi^3,\;\pi^4,\;\alpha^{-1},\;
  \Gamma\!\left(\tfrac{1}{4}\right),\;\log 2,\;e^{2\pi},\;
  e^{-\pi\sqrt{163}}\bigr\}
\]
with $\mathtt{maxcoeff} = 1000$, PSLQ returns \textbf{no relation} for $\delta - 6$.
Remark: the element $\alpha^{-1}$ is included but PSLQ finds no relation
for $\delta - 6$; the only linear relation among the basis elements
themselves (with $\alpha^{-1}$ present) is the trivial
$\alpha^{-1} - 4\pi^3 - \pi^2 - \pi = 0$, which PSLQ correctly recovers
but which does not involve $\delta - 6$.
\end{assertion}

\subsection{Main theorem}

\begin{theorem}[The near-miss $\delta \approx 6$ is a polynomial coincidence]
\label{thm:main}
Let
\[
  \mathcal{B} \;=\;
  \bigl\{1,\;\pi,\;\pi^2,\;\pi^3,\;\pi^4,\;
  \Gamma\!\left(\tfrac{1}{4}\right),\;\log 2,\;\zeta(3),\;e^{2\pi},\;
  e^{-\pi\sqrt{163}}\bigr\}.
\]
If $\delta - 6$ satisfies any $\QQ$-linear relation
$\sum_{b\in\mathcal{B}} c_b\,b = c_0\,(\delta - 6)$ with
$c_0,(c_b)\in\ZZ$, then
\[
  \bigl\|(c_0,\,(c_b)_{b\in\mathcal{B}})\bigr\|_\infty \;>\; 1000.
\]
Consequently, the proximity of $\delta = j(i) - 4\Omega$ to
$\hG = 6$ is not explained by any combination of Chowla--Selberg
periods, Gamma-function values, logarithms, $e^{2\pi}$, or the
Ramanujan--Heegner exponential at coefficient norm $\le 1000$.
The only exact identity involving $\delta$ is the trivial polynomial
$\delta = 1728 - 16\pi^4 - 4\pi^3 - 4\pi^2$.
\end{theorem}

\begin{proof}
Assertions~\ref{ass:basis-B}--\ref{ass:basis-D} cover all four searches.
Basis B covers $\Gamma(1/4)$, $\log 2$, $\zeta(3)$ at coeff $\le 500$.
Basis D extends to include $e^{2\pi}$ and $e^{-\pi\sqrt{163}}$ at coeff
$\le 1000$.  Together, the searches span the Chowla--Selberg basis for
$D=-4$ and key transcendentals associated with the CM point $\tau = i$.
Since PSLQ is certified to find any relation at the stated coefficient
bound given sufficient precision~\cite{pslq}, and $55$-digit precision
is sufficient for coefficient bounds well below $10^{10}$, the stated
lower bound on the coefficient norm holds.
\end{proof}

\begin{corollary}[$\delta - 6$ lies outside the CM period world]
\label{cor:outside-CM}
The residual $\delta - 6 = 16\pi^4 + 4\pi^3 + 4\pi^2 - 1722$ is not
expressible as a rational-coefficient linear combination of the real
period $\omega_E = \Gamma(1/4)^2/(4\sqrt{\pi})$ of the CM elliptic
curve $E: y^2 = x^3 - x$, powers of $\pi$, logarithms, or $e^{2\pi}$
at coefficient norm $\le 1000$.
\end{corollary}

\begin{proof}
The CM period $\omega_E$ lies in the $\QQ$-linear span of $\Gamma(1/4)^2$
and $\pi^{1/2}$ by the Chowla--Selberg formula~\cite{chowla}.  Both
$\Gamma(1/4)$ and $\pi^{1/2}$ appear in Basis D.  The claim follows from
Theorem~\ref{thm:main}.
\end{proof}

%==========================================================================
\section{Structural interpretation}\label{sec:interpretation}
%==========================================================================

\subsection{Why the near-miss is a polynomial accident}

\begin{assertion}[The deficit is itself a polynomial in $\pi$]
\label{ass:deficit-poly}
The deficit
\begin{equation}\label{eq:deficit2}
  6 - \delta \;=\; 16\pi^4 + 4\pi^3 + 4\pi^2 - 1722
           \;=\; 4\pi\alpha^{-1} - 1722
\end{equation}
is a nonconstant polynomial in $\pi$ with integer coefficients, evaluated
at $\pi \approx 3.14159$.  Its value $\approx 0.04899$ is an accident
of the specific numerical value of $\pi$: there is no number-theoretic
mechanism forcing this quantity to be small relative to $\hG = 6$.
Numerically, $|6 - \delta|/|6| \approx 0.82\%$, which while small is
not exceptionally so by the standards of polynomial-in-$\pi$ coincidences.
\end{assertion}

\begin{remark}
The deficit $4\pi\alpha^{-1} - 1722$ can be bounded by noting that
$\alpha^{-1} \approx 137.036$ while $1722/(4\pi) \approx 136.992$.
The difference $\alpha^{-1} - 1722/(4\pi) \approx 0.044$ is entirely
determined by the specific coefficients in $\alpha^{-1} = 4\pi^3+\pi^2+\pi$
and the integer $1722$.  There is no reason from Lie theory, CM theory,
or modular arithmetic why this difference should be small; it merely is.
\end{remark}

\subsection{The Coxeter-number interpretation is observational}

\begin{assertion}[The Coxeter identity fails to be exact]\label{ass:coxeter}
The identity $j(i) = 4\Omega + \hG$ --- which would express the
$j$-invariant at $\tau = i$ as the $S^2$-weighted coupling constant
plus the $G_2$ Coxeter number --- is \emph{not} exact.  The precise
statement (P171~\cite{P171}) is:
\[
  j(i) \;=\; 4\Omega + \delta,
  \qquad
  \delta \;=\; 5.9510\ldots \;\ne\; 6 \;=\; \hG.
\]
The near-miss $\delta \approx \hG$ is therefore an observation, not a
theorem.  The $G_2$ Coxeter number appears in the TOE through the
root system and the short-root sector sum $\Js = 12$ (P151); its
relationship to $\delta$ is numerical rather than algebraic.
\end{assertion}

\begin{assertion}[No Coxeter mechanism connects $\delta$ and $6$]
\label{ass:no-coxeter}
The Lie-theoretic interpretation of $h(G_2) = 6$ is that $G_2$ has
the Coxeter number $h = 2m + 2$ where $m = 2$ is the number of short
positive roots; equivalently $h(G_2) = |\Phi|/\mathrm{rank}(G_2) = 12/2
= 6$.  This combinatorial quantity is an integer, while $\delta = 1728 -
16\pi^4 - 4\pi^3 - 4\pi^2$ is transcendental.  No algebraic mechanism
can make a transcendental polynomial in $\pi$ equal to an integer, so
the near-miss is irreducibly a coincidence.
\end{assertion}

\subsection{Geometric reading via $S^2$}

\begin{assertion}[$S^2$ interpretation of the near-miss]\label{ass:S2}
Using $4\Omega = \mathrm{vol}(S^2)\cdot\alpha^{-1}$ (P171,
Assertion~\ref{ass:factor} above), the near-miss reads:
\begin{equation}\label{eq:S2-nearid}
  j(i) \;\approx\; \mathrm{vol}(S^2)\cdot\alpha^{-1} + \hG.
\end{equation}
In words: the $j$-invariant at the CM point $\tau = i$ is
\emph{approximately} equal to the product of the $S^2$ surface area and
the TOE fine-structure denominator, plus the $G_2$ Coxeter number.
The approximation is off by $\approx 0.049$, roughly $0.003\%$ of $j(i)$.
This is a striking numerical coincidence but, by Theorem~\ref{thm:main},
not a structural identity.
\end{assertion}

\subsection{Contrast with P179}

\begin{assertion}[Contrast with P179's negative PSLQ result]\label{ass:contrast}
P179~\cite{P179} found that $\GoneP = 1728/\pi^3 - 1 \approx 54.73$
lies outside the Chowla--Selberg $\QQ$-linear span at coefficient
norm $\le 1000$.  The present addendum finds the same conclusion for
$\Gone \approx 0.003456$ and for $\delta - 6$: neither has additional
structure over the CM/Gamma/Heegner basis.  Both constants sit at the
interface between the algebraic CM world ($j(i) = 1728 \in \ZZ$) and
the transcendental $\pi$-polynomial world (the TOE geometric constants).
The Chowla--Selberg formula mediates between modular arithmetic and
Gamma values but cannot reach purely polynomial expressions in $\pi$.
\end{assertion}

\begin{remark}[Why the Chowla--Selberg formula cannot reach $\delta - 6$]
The Chowla--Selberg formula for $D=-4$ expresses the real period
$\omega_E = \Gamma(1/4)^2/(4\sqrt{\pi})$ of the CM elliptic curve
$E: y^2 = x^3 - x$ in terms of $\Gamma(1/4)$ and $\pi^{1/2}$.
The basis elements $\Gamma(1/4)^k$ and $\pi^{k/2}$ are periods or
quasi-periods of $E$.  In contrast, $\delta - 6 = 16\pi^4 + 4\pi^3
+ 4\pi^2 - 1722$ involves $\pi^2$, $\pi^3$, $\pi^4$ but not
$\pi^{1/2}$ or $\Gamma(1/4)$; the polynomial structure in integer
powers of $\pi$ is not accessible from the period world.  By
Nesterenko's theorem~\cite{nesterenko}, $\pi$, $e^\pi$, and
$\Gamma(1/4)$ are algebraically independent over $\QQ$, which
implies that no polynomial in $\pi$ with rational coefficients
can equal any element of the $\QQ[\Gamma(1/4)]$ ring.
\end{remark}

%==========================================================================
\section{mpmath.identify probes}\label{sec:identify}
%==========================================================================

\begin{assertion}[No closed form found by \texttt{identify}]\label{ass:identify}
The \texttt{mpmath.identify} function (tolerance $10^{-14}$ at $55$~digits)
returns no closed-form expression for any of the following quantities:
$\Gone$, $\delta$, $\delta - 6$, $\Gone\cdot\pi$, $\Gone\cdot\pi^2$,
$\delta/\pi$, $(\delta-6)/\pi^2$, $(\delta-6)/\Gamma(1/4)$, and $6\cdot\Gone$.
The only spurious matches (using low-denominator exponents over small
integers) appear for $\Gone\cdot\pi$ and $(\delta-6)\cdot\alpha^{-1}$,
which \texttt{identify} associates with expressions of the form
$(p/q)$-rational exponents of $2,3,5,7$; these do not survive cross-checking
and are artefacts of the numerical search algorithm.
\end{assertion}

%==========================================================================
\section{Summary and open directions}\label{sec:summary}
%==========================================================================

\subsection{Main results}

The central result of this addendum is:

\begin{equation}\label{eq:main-box}
\boxed{
  \Gone \;=\; \frac{j(i)}{4\Omega} - 1
  \;\approx\; 3.456\times 10^{-3},
  \qquad
  \delta \;=\; j(i) - 4\Omega \;=\; 1728 - 16\pi^4 - 4\pi^3 - 4\pi^2
  \;\approx\; 5.9510,
}
\end{equation}

\noindent with the PSLQ verdict:

\begin{equation}\label{eq:verdict-box}
\boxed{
  \text{VERDICT: COINCIDENCE.}\quad
  \delta \approx 6 = \hG \text{ is a polynomial accident in } \pi;
  \text{ no additional CM structure at coeff} \le 1000.
}
\end{equation}

The supporting results are:
\begin{align}
  \delta &= 1728 - 16\pi^4 - 4\pi^3 - 4\pi^2
  \quad\text{(only exact relation, Assertion~\ref{ass:basis-A}),}
  \\
  \delta - 6 &\notin \QQ\text{-span}\bigl(\{1,\pi,\pi^2,\pi^3,\pi^4,
    \Gamma({\textstyle\frac{1}{4}}),\log 2,\zeta(3),e^{2\pi},
    e^{-\pi\sqrt{163}}\}\bigr),\ \|\text{coeff}\|_\infty \le 1000
  \nonumber\\&\qquad\text{(Theorem~\ref{thm:main}),}
  \\
  j(i) &= 4\Omega + \delta
  \;\approx\; \mathrm{vol}(S^2)\cdot\alpha^{-1} + \hG
  \quad\text{(near-identity, not exact, Assertion~\ref{ass:S2}).}
\end{align}

\subsection{Open directions}

\begin{enumerate}
  \item \emph{Coefficient lower bound.}  Theorem~\ref{thm:main} gives
    $\|\text{coeff}\|_\infty > 1000$.  Improving this to an analytic
    lower bound --- e.g., by Gel'fond--Schneider or Baker-type results
    on linear forms in logarithms --- would convert the computational
    result into a theorem.  Nesterenko's algebraic independence of
    $\pi$, $e^\pi$, $\Gamma(1/4)$ implies that $\delta - 6$, being a
    polynomial in $\pi$, is algebraically independent of $\Gamma(1/4)$
    and $e^\pi$ over $\QQ$, giving a partial step in this direction.

  \item \emph{The deficit $6 - \delta \approx 0.04898$.}  This quantity
    equals $16\pi^4 + 4\pi^3 + 4\pi^2 - 1722 = 4\pi\alpha^{-1} - 1722$.
    P171~\cite{P171} remarked that $6 - \delta \approx \pi^2/200$
    (with $\pi^2/200 \approx 0.04935$), a further near-miss within the
    near-miss.  Whether this sub-near-miss has any combinatorial
    interpretation is open.

  \item \emph{Extending PSLQ to $\Gamma(1/3)$, $\Gamma(1/8)$, and
    other CM discriminants.}  The present analysis is restricted to
    the $D=-4$ Chowla--Selberg basis.  CM points at discriminants
    $D = -3$, $-7$, $-11$, $-163$ produce different Gamma-function
    ingredients.  Whether $\delta - 6$ is reachable from any of
    these is not tested here.

  \item \emph{The Hopf fibration interpretation.}  Assertion~\ref{ass:S2}
    expresses the near-miss as $j(i) \approx \mathrm{vol}(S^2)\cdot\alpha^{-1}
    + \hG$.  The $S^3$ geometry carries a Hopf fibration with $S^2$ base.
    Whether the $G_2$ Coxeter number $\hG = 6$ appears naturally as a
    degree or winding number in this fibration is an open geometric question.

  \item \emph{Relation to P179.}  The ratio
    $(\GoneP + 1)/(\Gone + 1) = 4\Omega/\pi^3 = \Omega/\Omega_0$
    is the ratio of the two TOE periods.  Whether the PSLQ negative
    results for $\Gone$ and $\GoneP$ are related --- e.g., whether one
    being outside the CM frame forces the other to be --- is unknown.
\end{enumerate}

%==========================================================================
\section{Conclusion}\label{sec:conclusion}
%==========================================================================

The TOE near-miss $j(i) - 4\Omega \approx 6 = h(G_2)$ was identified in
P171~\cite{P171} as a compelling but unproven structural observation.  The
present addendum closes the computational investigation: PSLQ over four
independent bases covering the Chowla--Selberg ingredients for the CM
elliptic curve $E: y^2 = x^3 - x$ at $\tau = i$, together with
$\log 2$, $\zeta(3)$, $e^{2\pi}$, and $e^{-\pi\sqrt{163}}$, returns no
integer relation for the residual $\delta - 6$ at coefficient norm
$\le 1000$.

The verdict is unambiguous: \textbf{the near-miss $\delta \approx \hG$ is
a polynomial coincidence}.  The gap $\delta = 1728 - 16\pi^4 - 4\pi^3 -
4\pi^2$ is entirely defined by the polynomial structure of the TOE breath
period $\Omega = \pi\alpha^{-1}$; the proximity to the $G_2$ Coxeter
number $\hG = 6$ is a numerical accident.  No Chowla--Selberg period, no
Gamma-function value, and no modular quantity at $\tau = i$ explains the
discrepancy from exactness.

This places $\Gone$ alongside P179's $\GoneP$ as a genuinely transcendental
structural constant of the TOE that sits irreducibly outside the CM period
world.  Both constants arise from the ratio of the algebraic CM value
$j(i) = 1728$ to a $\pi$-polynomial TOE quantity; in neither case does the
Chowla--Selberg framework absorb the gap.  The TOE thus exhibits a
systematic transcendental interface at $\tau = i$: its geometric constants
($\Omega$ and $\Omega_0$) approach the CM modular slot $j(i)$ closely but
never reach it, and the shortfall is irreducible to any CM or Gamma-function
structure at the tested coefficient scales.

\begin{thebibliography}{9}

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L.\ F.\ Vlegels,
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Addendum~171 to the TOE Corpus, May~2026.

\bibitem{P179}
L.\ F.\ Vlegels,
\emph{$G_1$ and the Chowla--Selberg Formula: Does the CM Period at
$\tau=i$ Resolve the Modular Gap in $j(i)/(4\Omega_0)$?},
Addendum~179 to the TOE Corpus, May~2026.

\bibitem{pslq}
H.\ R.\ P.\ Ferguson, D.\ H.\ Bailey and S.\ Arno,
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\bibitem{chowla}
S.\ Chowla and A.\ Selberg,
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\bibitem{shimura}
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\bibitem{silverman}
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Springer, 1994.

\bibitem{nesterenko}
Y.\ V.\ Nesterenko,
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\emph{Mat.\ Sb.} \textbf{187} (1996), 65--96.

\bibitem{lang}
S.\ Lang,
\emph{Elliptic Functions}, 2nd ed., Springer, 1987.

\bibitem{gelfond}
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\end{thebibliography}

\end{document}
