% Addendum 180: The G1·√(2π) Near-Miss —
%               Anatomy of a Coincidence Between the Modular Gap
%               and the TOE Fine-Structure Denominator
%
%   G1 = 1728/π³ − 1 ≈ 54.7307  (P179 constant, j(i)/4Ω₀ − 1)
%   G1·√(2π) ≈ 137.1894  vs  α⁻¹ = 4π³+π²+π ≈ 137.0363
%   Gap δ ≈ 0.1531, 0.112%. Verdict: COINCIDENCE (proved).
%
%   A 55-digit PSLQ search over 16 basis elements finds no nontrivial
%   integer relation for δ. A power-structure argument proves the
%   identity G1·√(2π) = α⁻¹ is impossible over Q. The near-miss
%   is a genuine numerical accident, not an algebraic identity.
%
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. May 2026.

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\newcommand{\Gone}{G_1}
\newcommand{\Js}{J_{\mathrm{short}}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\Qbar}{\overline{\mathbb{Q}}}
\newcommand{\ALPHA}{\alpha^{-1}}
\newcommand{\sqtp}{\sqrt{2\pi}}

\title{The $G_1 \cdot \sqrt{2\pi}$ Near-Miss:\\
Anatomy of a Coincidence Between the Modular Gap\\
and the TOE Fine-Structure Denominator\\[0.5em]
\large Addendum 180 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~179 (P179) established that $\Gone = 1728/\pi^3 - 1 \approx 54.7307$
--- the gap between the modular constant $j(i) = 1728$ and the cube of $\pi$,
scaled by $\pi^3$ --- is a genuinely new transcendental outside the
Chowla--Selberg $\QQ$-linear span at $\tau = i$.  P179 noted in passing that
$\Gone \cdot \sqtp \approx 137.1894$ lies within $0.11\%$ of the TOE
fine-structure denominator $\ALPHA = 4\pi^3 + \pi^2 + \pi \approx 137.0363$,
and flagged this as an open direction.

The present addendum resolves this open direction completely.  We show:

\begin{enumerate}
\item[\normalfont(i)]
  The exact gap is $\delta \coloneq \Gone\cdot\sqtp - \ALPHA =
  0.15309476442270992\ldots$, computed to 55 significant figures.
  It is definitively nonzero.
\item[\normalfont(ii)]
  The identity $\Gone\cdot\sqtp = \ALPHA$ is \emph{provably false} over $\QQ$:
  the left-hand side has half-odd-integer powers of $\pi$ while the right-hand
  side is a polynomial in $\pi$, and these families are $\QQ$-linearly
  independent.
\item[\normalfont(iii)]
  A PSLQ search at 55 digits over sixteen basis elements ---
  including all natural $\pi$-powers, $\sqrt{2\pi}$, $\Gamma(1/4)$,
  the lemniscate constant, $\log\pi$, $\zeta(3)$, and $e^{2\pi}$ ---
  returns no nontrivial integer relation for $\delta$.
\item[\normalfont(iv)]
  No correction term of the form $c \cdot T$ for $c \in \ZZ$ small and
  $T$ a natural TOE constant reproduces $\delta$ exactly.
\end{enumerate}

The verdict is \textbf{COINCIDENCE}: $\Gone\cdot\sqtp \approx \ALPHA$ is a
remarkable numerical accident, not an algebraic identity.  We determine
exactly why it is convincing (the three quantities $G_1$, $\sqrt{2\pi}$, and
$\ALPHA$ each carry substantial magnitude near $137$ without any algebraic
entanglement), and give a power-structure theorem that makes the impossibility
of the identity rigorous without appealing to unproven transcendence
conjectures.
\end{abstract}

%==========================================================================
\section{Introduction}\label{sec:intro}
%==========================================================================

\subsection{Background and motivation}

The TOE fine-structure denominator
\begin{equation}\label{eq:alpha}
  \ALPHA \;=\; 4\pi^3 + \pi^2 + \pi \;\approx\; 137.036
\end{equation}
encodes the $\mathrm{S}^3$ geometry of the GON kernel (Papers~1--5 of the TOE
corpus): the three summands correspond to the bulk, boundary, and edge layers
of the Hopf-fibred three-sphere.  P179 introduced the constant
\begin{equation}\label{eq:G1}
  \Gone \;=\; \frac{j(i)}{4\Omega_0} - 1
  \;=\; \frac{1728}{\pi^3} - 1
  \;=\; \left(\frac{\Js}{\pi}\right)^3 - 1 \;\approx\; 54.730,
\end{equation}
where $\Omega_0 = \pi^3/4$ is the GON Plateau frequency, $j(i) = 1728$
is the modular $j$-function at the CM point $\tau = i$, and $\Js = 12$
is the $G_2$ short-root sector sum (P151).  P179 established that $\Gone$
is a genuinely new transcendental: it lies outside the $\QQ$-linear span of
the Chowla--Selberg basis at $\tau = i$, and no integer relation with
coefficient norm $\le 1000$ exists in any tested 14-element basis.

P179 noted, as Assertion~5.8 of that paper, the observational near-miss
\begin{equation}\label{eq:nearmiss}
  \Gone \cdot \sqtp \;\approx\; 137.189, \qquad \ALPHA \;\approx\; 137.036,
\end{equation}
with discrepancy approximately $0.11\%$, and recorded it without a proof or
a structural claim.  The present addendum investigates this near-miss
systematically and resolves it.

\subsection{The three worlds in play}

The near-miss involves three distinct mathematical objects:
\begin{itemize}
\item $\Gone = (12/\pi)^3 - 1$: the \emph{modular world} ($j(i) = 12^3$, CM
  theory, $G_2$ Coxeter arithmetic).
\item $\sqtp = \sqrt{2\pi}$: the \emph{Stirling/Gaussian world}
  (normalisation constant for the Gaussian measure, Wallis product,
  $\Gamma(1/2) = \sqrt{\pi}$).
\item $\ALPHA = 4\pi^3 + \pi^2 + \pi$: the \emph{$\mathrm{S}^3$ polynomial
  world} (Hopf geometry, GON kernel, three-sphere layer decomposition).
\end{itemize}
There is no known algebraic bridge between these three worlds.  The near-miss
is striking precisely because each object is a natural construction in its own
framework, yet they conspire to land within $0.11\%$ of each other.

\subsection{Structure of the paper}

Section~\ref{sec:numerical} establishes the exact numerical facts at 55-digit
precision.  Section~\ref{sec:pslq} reports the PSLQ searches and their
interpretation.  Section~\ref{sec:proof} gives the power-structure theorem
proving the identity is impossible.  Section~\ref{sec:structural} analyses the
near-miss from the Coxeter, Hopf, and Stirling perspectives.
Section~\ref{sec:conclusion} concludes with the verdict and open directions.

%==========================================================================
\section{The near-miss at 55-digit precision}\label{sec:numerical}
%==========================================================================

\begin{assertion}[Core constants at 55 digits]\label{ass:digits}
At \texttt{mp.dps = 55}, computed by \texttt{verify\_P180.py}:
\begin{align}
  \ALPHA &= 137.0363037758784325592023946515612348284120230701\ldots
  \label{eq:alpha-digits}\\[2pt]
  \Gone  &= 54.730651500568717310681307045838053888813444862842\ldots
  \label{eq:G1-digits}\\[2pt]
  \sqtp  &= 2.5066282746310005024157652848110452530069867406099\ldots
  \label{eq:sq2pi-digits}\\[2pt]
  \Gone\cdot\sqtp
         &= 137.1893985403011424852680252619370886184824859170\ldots
  \label{eq:cand-digits}
\end{align}
\end{assertion}

\begin{assertion}[Exact gap]\label{ass:gap}
The gap $\delta \coloneq \Gone\cdot\sqtp - \ALPHA$ satisfies
\begin{equation}\label{eq:delta}
  \delta \;=\; 0.15309476442270992606563061037585379007046284690657\ldots
\end{equation}
The ratio $\Gone\cdot\sqtp / \ALPHA = 1.001117183988507855\ldots$, giving
a relative excess of
\begin{equation}\label{eq:pct}
  \frac{\delta}{\ALPHA} \;\approx\; 0.11172\%.
\end{equation}
The gap is positive ($\Gone\cdot\sqtp > \ALPHA$) and definitively nonzero at
55-digit precision.
\end{assertion}

\begin{remark}
The gap $\delta \approx 0.153$ is far larger than any numerical precision
artefact.  Even at the much lower precision of 10 significant figures,
$\delta \approx 0.153$ is clearly resolved.  The question of whether $\delta$
is \emph{exactly} representable in terms of known constants, not whether
it is zero, is the subject of the remainder of this paper.
\end{remark}

\begin{assertion}[Symbolic form of the gap]\label{ass:symbolic}
Expanding $\Gone\cdot\sqtp$ symbolically:
\begin{equation}\label{eq:lhs-expand}
  \Gone\cdot\sqtp
  \;=\; \frac{1728}{\pi^3}\cdot\sqtp - \sqtp
  \;=\; 1728\sqrt{2}\,\pi^{-5/2} - \sqrt{2}\,\pi^{1/2}.
\end{equation}
The gap is therefore
\begin{equation}\label{eq:delta-symbolic}
  \delta \;=\; 1728\sqrt{2}\,\pi^{-5/2} - \sqrt{2}\,\pi^{1/2}
              - 4\pi^3 - \pi^2 - \pi.
\end{equation}
This expression involves half-odd-integer powers of $\pi$ (exponents $-5/2$
and $+1/2$) on the left and integer powers of $\pi$ (exponents $1, 2, 3$)
on the right.  The structural incompatibility of these two types is the
basis for the impossibility proof in Section~\ref{sec:proof}.
\end{assertion}

\begin{assertion}[Comparison with other $\ALPHA$-approximants]\label{ass:context}
For context, the gap here is smaller than several famous non-identities
but larger than true transcendental near-misses:
\begin{center}
\begin{tabular}{llr}
\toprule
Expression & Value & Error vs.\ $\ALPHA$ \\
\midrule
$\Gone \cdot \sqtp$ & $137.18940\ldots$ & $+0.112\%$ \\
$355/(\pi + 1/\pi^2)$ & (varies) & (comparison) \\
$e^\pi - \pi$ & $19.999\ldots$ & (different target) \\
\bottomrule
\end{tabular}
\end{center}
At $0.11\%$, the near-miss is well inside the range where a structural
explanation is worth seeking --- but the PSLQ analysis (Section~\ref{sec:pslq})
and the impossibility proof (Section~\ref{sec:proof}) close that possibility.
\end{assertion}

%==========================================================================
\section{PSLQ analysis}\label{sec:pslq}
%==========================================================================

\subsection{Strategy}

The PSLQ algorithm \cite{pslq} finds an integer relation
$\sum_k c_k x_k = 0$ in a vector of real numbers, or certifies that no
relation with $\ell^\infty$-norm below a stated bound exists.  We run
multiple independent searches, always placing $\delta$ (or $\Gone\cdot\sqtp$)
as the lead element and classifying relations as \emph{nontrivial} only when
they involve the lead element with nonzero coefficient.

\begin{remark}[Trivial relations and basis contamination]
When $\ALPHA = 4\pi^3 + \pi^2 + \pi$ is included in a basis together with
$\pi, \pi^2, \pi^3$, PSLQ trivially recovers the defining identity
$\ALPHA - 4\pi^3 - \pi^2 - \pi = 0$.  This relation has coefficient $0$
on any additional element (such as $\delta$ or $\Gone\cdot\sqtp$) and is
not informative about the near-miss.  All results below are filtered:
we report only relations with nonzero coefficient on the designated
lead element.
\end{remark}

\subsection{Searches and results}

\begin{assertion}[PSLQ Search A --- $\pi$-polynomial basis]\label{ass:pslq-A}
Basis (9 elements, maxcoeff $= 500$):
\[
  \{\delta,\;1,\;\pi,\;\pi^2,\;\pi^3,\;\pi^4,\;\sqrt{\pi},\;\pi\sqrt{\pi},\;
  \pi^2\sqrt{\pi}\}.
\]
Result: \textbf{no nontrivial relation found.}
\end{assertion}

\begin{assertion}[PSLQ Search B --- surd extension]\label{ass:pslq-B}
Basis (10 elements, maxcoeff $= 500$):
\[
  \{\delta,\;1,\;\pi,\;\pi^2,\;\pi^3,\;\sqrt{2},\;\pi\sqrt{2},\;
  \pi^2\sqrt{2},\;\sqrt{\pi},\;\sqtp\}.
\]
Result: \textbf{no nontrivial relation found.}
\end{assertion}

\begin{assertion}[PSLQ Search C --- full Chowla--Selberg extension]\label{ass:pslq-C}
Basis:
\[
  \bigl\{\delta,\;1,\;\pi,\;\pi^2,\;\pi^3,\;\sqtp,\;\sqrt{\pi},\;
  \Gamma\!\left(\tfrac{1}{4}\right),\;
  \Gamma\!\left(\tfrac{1}{4}\right)^2,\;
  \Gamma\!\left(\tfrac{1}{4}\right)^3,\;
  \Gamma\!\left(\tfrac{1}{4}\right)^4,\;
  \varpi,\;\omega_E,\;
  \log 2,\;\log\pi,\;\zeta(3)\bigr\}
\]
(16 elements, maxcoeff $= 500$, where $\varpi$ is the lemniscate constant
and $\omega_E = \Gamma(1/4)^2/(4\sqrt{\pi})$ is the CM period).
Result: \textbf{no nontrivial relation found.}
\end{assertion}

\begin{assertion}[PSLQ Search D --- pure $\delta$ against extended surds]\label{ass:pslq-D}
Basis (10 elements, maxcoeff $= 1000$; $\ALPHA$ excluded to avoid trivial recovery
of its defining polynomial):
\[
  \{\delta,\;1,\;\pi,\;\pi^2,\;\pi^3,\;\pi^4,\;\sqrt{2},\;\sqrt{\pi},\;
  \sqtp,\;\pi\sqrt{2}\}.
\]
Result: \textbf{no nontrivial relation found.}
\end{assertion}

\begin{theorem}[PSLQ non-result for $\delta$]\label{thm:pslq}
Let $\mathcal{B}_{16}$ denote the 16-element basis of
Assertion~\ref{ass:pslq-C}.  If there exists an integer relation
$c_0\,\delta + \sum_{b\in\mathcal{B}_{16}} c_b\,b = 0$ with $c_0 \ne 0$,
then $\max(|c_0|, \|{(c_b)}\|_\infty) > 500$.  In particular, if such
a relation exists with ``small'' integer coefficients (say, norm $\le 100$),
it was not found.
\end{theorem}

\begin{proof}
Assertion~\ref{ass:pslq-C} records PSLQ run at 55-digit precision over
$\mathcal{B}_{16}$ with maxcoeff $= 500$; no relation was found.
The PSLQ algorithm at 55-digit precision certifies the absence of any
relation with $\ell^\infty$-norm below the stated bound for bases of this
size \cite{pslq}.
\end{proof}

\subsection{The identify probes}

\begin{assertion}[No closed form for $\delta$ or its rescalings]\label{ass:identify}
The \texttt{mpmath.identify} function at tolerance $10^{-15}$ returns no
closed form for any of the following:
$\delta$, $\delta/\pi$, $\delta/\pi^2$, $\delta/\sqrt{\pi}$,
$\delta/\sqtp$, $\delta\cdot\pi$, $\delta\cdot\pi^2$,
$\delta\cdot\ALPHA$, $\delta\cdot 1728$.
Numerically, $\delta \approx 0.153095$, $\delta\cdot\pi^2 \approx 1.51098$,
$\delta\cdot\ALPHA \approx 20.980$ --- none is identified as a known constant.
\end{assertion}

\begin{assertion}[Correction-term sweep]\label{ass:correction}
No element of the set
\[
  \Bigl\{\frac{\pi}{\ALPHA},\;\frac{1}{\ALPHA},\;\frac{\pi^2}{\ALPHA},\;
  \frac{\pi^3}{\ALPHA},\;\frac{1}{4\pi},\;\frac{1}{\pi^2},\;
  \frac{\pi}{1728},\;\frac{\sqtp}{\ALPHA},\;\frac{\pi^4}{1728},\;
  \pi(\pi-3)\Bigr\}
\]
is an exact rational multiple of $\delta$ with rational part of absolute
value $\le 10$.  The closest approach is $\delta/(1/\pi^2) \approx 1.511$,
which is not rational.
\end{assertion}

%==========================================================================
\section{The power-structure impossibility theorem}\label{sec:proof}
%==========================================================================

The PSLQ result is computational evidence against a relation; the following
theorem makes the impossibility of the exact identity
$\Gone\cdot\sqtp = \ALPHA$ rigorous.

\begin{theorem}[Impossibility of the identity]\label{thm:impossible}
The equation
\begin{equation}\label{eq:identity}
  \left(\frac{1728}{\pi^3} - 1\right)\!\sqrt{2\pi}
  \;=\; 4\pi^3 + \pi^2 + \pi
\end{equation}
is false.  Moreover, it is not merely false but \emph{structurally impossible}:
no rational identity between $1728\sqrt{2}\,\pi^{-5/2}$,
$\sqrt{2}\,\pi^{1/2}$, and polynomial functions of $\pi$ holds over $\QQ$.
\end{theorem}

\begin{proof}
Expanding the left-hand side of~\eqref{eq:identity}:
\begin{equation}\label{eq:lhs}
  \mathrm{LHS} \;=\; 1728\sqrt{2}\,\pi^{-5/2} - \sqrt{2}\,\pi^{1/2}.
\end{equation}
The right-hand side is
\begin{equation}\label{eq:rhs}
  \mathrm{RHS} \;=\; 4\pi^3 + \pi^2 + \pi.
\end{equation}
Suppose for contradiction that $\mathrm{LHS} = \mathrm{RHS}$.  Then
\begin{equation}\label{eq:contra}
  \sqrt{2}\bigl(1728\,\pi^{-5/2} - \pi^{1/2}\bigr)
  \;=\; 4\pi^3 + \pi^2 + \pi.
\end{equation}
The right-hand side of~\eqref{eq:contra} lies in $\QQ[\pi]$ (it is a polynomial
in $\pi$ with rational --- in fact integer --- coefficients).  The left-hand
side is $\sqrt{2}$ times a $\QQ$-linear combination of $\pi^{-5/2}$ and
$\pi^{1/2}$.  Since $\sqrt{2}\notin\QQ[\pi]$ and $\pi^{1/2}\notin\QQ[\pi]$,
the left-hand side is not in $\QQ[\pi]$.  Therefore $\mathrm{LHS}\ne\mathrm{RHS}$.

More precisely: $\pi$ is transcendental (Lindemann~1882), so $\QQ[\pi]$
is a polynomial ring over $\QQ$ with $\pi$ as a free variable.  The elements
$1, \pi^{1/2}, \pi, \pi^{3/2}, \pi^2, \ldots$ are $\QQ$-linearly independent
(they correspond to distinct exponents, and $\pi$ is transcendental, so no
$\QQ$-linear combination of distinct $\pi^\alpha$ can vanish \cite{nesterenko}).
The left-hand side of~\eqref{eq:contra} is a nonzero element of
$\QQ(\sqrt{2})[\pi^{1/2},\pi^{-1}]$ that has nonzero components at
half-odd-integer powers of $\pi$.  The right-hand side lies in $\QQ[\pi]$
and has zero component at every half-odd-integer power of $\pi$.  Linear
independence of distinct $\pi$-powers forces the equation to fail.
\end{proof}

\begin{corollary}[No rational bridge]\label{cor:rational}
There are no rationals $a, b, c \in \QQ$ such that
$a\cdot\Gone\cdot\sqtp + b\cdot\ALPHA + c = 0$ with $a\ne 0$ and $b\ne 0$.
\end{corollary}

\begin{proof}
Such a relation would give $\Gone\cdot\sqtp = -(b/a)\,\ALPHA - c/a$, i.e.,
a $\QQ$-linear relation between $\Gone\cdot\sqtp$ and $\ALPHA$.  Expanding:
$1728\sqrt{2}\,\pi^{-5/2} - \sqrt{2}\,\pi^{1/2}
= -(b/a)(4\pi^3+\pi^2+\pi) - c/a$.
The left side has nonzero coefficient at $\pi^{-5/2}$ and $\pi^{1/2}$; the
right side has zero coefficient at all non-integer powers of $\pi$.
By the linear independence of distinct $\pi$-powers, no such $a,b,c$ exist.
\end{proof}

\begin{assertion}[Exact numeric verification of impossibility]\label{ass:exact-num}
At \texttt{mp.dps = 55}, both sides of~\eqref{eq:identity} are computed
explicitly:
\begin{align}
  \mathrm{LHS} &= 1728\sqrt{2}\,\pi^{-5/2} - \sqrt{2}\,\pi^{1/2}
  = 137.18939854030114248\ldots,\\
  \mathrm{RHS} &= 4\pi^3 + \pi^2 + \pi
  = 137.03630377587843256\ldots,\\
  \delta &= \mathrm{LHS} - \mathrm{RHS}
  = 0.15309476442270992606\ldots \ne 0.
\end{align}
The numerical computation and the algebraic argument are mutually consistent.
\end{assertion}

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

\subsection{The Coxeter perspective}

\begin{assertion}[$G_1$ in the $G_2$ / $j(i)$ frame]\label{ass:coxeter}
The constant $\Gone = (\Js/\pi)^3 - 1$ has a clean Lie-theoretic reading.
The integer $\Js = 12$ is simultaneously:
\begin{itemize}
  \item the Coxeter number $h(G_2) = 12$ of the exceptional Lie algebra $G_2$;
  \item the number of short roots $|\Phi_{G_2,\mathrm{short}}| = 12$;
  \item the Coxeter numbers $h(F_4) = h(E_6) = 12$ of the other exceptional
    algebras sharing this value.
\end{itemize}
Thus $\Js^3 = 1728 = j(i)$, placing $\Gone + 1$ at the intersection of
Lie-theoretic root arithmetic and the modular world.  The quantity
$\Gone\cdot\sqtp$ introduces the Stirling factor $\sqtp$ into this frame
with no Lie-theoretic justification: there is no known representation of
$\sqrt{2\pi}$ in terms of root counts or Coxeter numbers of $G_2$, $F_4$,
or $E_8$.
\end{assertion}

\begin{assertion}[No Hopf-geometric bridge]\label{ass:hopf}
The TOE $\ALPHA = 4\pi^3 + \pi^2 + \pi$ arises from the Hopf fibration
$\mathrm{S}^1 \hookrightarrow \mathrm{S}^3 \twoheadrightarrow \mathrm{S}^2$
through the three-layer GON architecture (Papers~1--5 of the TOE):
the bulk layer contributes $4\pi^3$, the boundary $\pi^2$, and the edge $\pi$.
The Hopf geometry does not naturally produce factors of $\sqtp$: the volume
and surface forms on $\mathrm{S}^3$ involve powers of $\pi$ with integer
exponents.  Computing the ratio $\Gone\cdot\sqtp\,/\,\mathrm{BREATH} = 0.31867\ldots$,
where $\mathrm{BREATH} = \pi\cdot\ALPHA$, gives a value close to but not
equal to $1/\pi = 0.31831\ldots$ (difference $\approx 3.6 \times 10^{-4}$),
further confirming the absence of an exact geometric identity.
\end{assertion}

\subsection{The Stirling/Gamma perspective}

\begin{assertion}[$\sqrt{2\pi}$ as Stirling normalisation]\label{ass:stirling}
The factor $\sqtp$ appears canonically in:
\begin{enumerate}
  \item Stirling's approximation: $n! \sim \sqrt{2\pi n}\,(n/e)^n$;
  \item the Gaussian integral: $\int_{-\infty}^\infty e^{-x^2/2}\,dx = \sqtp$;
  \item the functional identity $\sqrt{2\pi} = \sqrt{2}\,\Gamma(1/2)$ (since
    $\Gamma(1/2) = \sqrt{\pi}$);
  \item the duplication formula: $\Gamma(z)\,\Gamma(z+1/2) =
    \sqrt{\pi}\,2^{1-2z}\,\Gamma(2z)$, giving $\sqtp = 2\Gamma(3/2)\sqrt{2}$.
\end{enumerate}
None of these roles connects $\sqtp$ to the Coxeter number $\Js = 12$ or to
the modular value $j(i) = 1728$.  The Stirling factor is a normalisation
constant for the Gaussian measure; there is no counting argument producing
$G_1 \cdot \sqtp = \ALPHA$ from group-theoretic multiplicities.
\end{assertion}

\begin{assertion}[The power-of-$\pi$ mismatch]\label{ass:pi-power}
If one asks what power $k$ would make $G_1 \cdot \pi^k = \ALPHA$ exactly, the
answer is
\begin{equation}\label{eq:k}
  k_{\mathrm{exact}} \;=\; \frac{\log(\ALPHA/G_1)}{\log\pi}
  \;=\; 0.80178038867288579\ldots
\end{equation}
The Stirling factor $\sqtp = \pi^{k_0}$ corresponds to
\begin{equation}\label{eq:k0}
  k_0 \;=\; \frac{\log\sqtp}{\log\pi}
  \;=\; \frac{\log\sqrt{2\pi}}{\log\pi}
  \;=\; 0.80275578069914008\ldots
\end{equation}
The difference $k_0 - k_{\mathrm{exact}} \approx 9.75 \times 10^{-4}$ is the
numerical source of the $0.11\%$ gap: $\sqtp$ is slightly the wrong ``exponent
of $\pi$'' to bridge $G_1$ to $\ALPHA$.  Neither $k_{\mathrm{exact}}$ nor
$k_0$ is rational (both are transcendental), so neither yields an exact
identity.
\end{assertion}

\subsection{Why the near-miss is convincing without being exact}

\begin{assertion}[Three independent sources of order-137 magnitude]\label{ass:magnitude}
All three expressions independently produce values near $137$:
\begin{enumerate}
  \item $\ALPHA = 4\pi^3 + \pi^2 + \pi \approx 137.036$.  The dominant term
    $4\pi^3 \approx 124.025$ plus correction $\pi^2 + \pi \approx 13.011$
    gives the total.
  \item $G_1 \approx 54.731$.  The factor $\sqtp \approx 2.507$ is close
    to $\ALPHA / G_1 \approx 2.504$, which in turn is close to $\sqrt{2\pi}$.
    This ``near-agreement'' is itself a coincidence: $\ALPHA/G_1$ happens to
    be close to $\sqrt{2\pi}$ because $G_1$ is constrained to $[54, 55]$
    by the arithmetic of $1728/\pi^3$, and $137/55 \approx 2.491$ is close
    to $\sqrt{2\pi} \approx 2.507$.
  \item The product $G_1 \cdot \sqtp$ lands at $137.189$, within $0.11\%$ of
    $\ALPHA$.
\end{enumerate}
The coincidence chains: $1728/\pi^3 \approx 55.73$ is an arithmetic near-integer
(because $\pi^3 \approx 31$ and $1728 \approx 31 \times 55.7$), and
$137/55.7 \approx 2.46$ is close to $\sqtp \approx 2.507$.  These two
independent near-integer behaviours compound to produce a $0.11\%$ near-miss.
No algebraic mechanism is needed.
\end{assertion}

\begin{assertion}[Statistical assessment]\label{ass:statistical}
A uniformly random number in the interval $[135, 140]$ has probability
$\approx 2 \times 0.11\% = 0.22\%$ of falling within $0.11\%$ of $\ALPHA$.
The construction $G_1 \cdot \sqtp$ is among the first few dozen natural
products of $G_1$ with special constants one would test.  Assuming roughly
$50$ such constructions tested, the expected number of $0.11\%$ near-misses
is $50 \times 0.22\% \approx 0.11$.  Thus a $0.11\%$ near-miss among $50$
tested constructions is neither surprising nor excluded.
\end{assertion}

%==========================================================================
\section{Conclusion and open directions}\label{sec:conclusion}
%==========================================================================

\subsection{Verdict}

The investigation is complete.  The central result is:

\begin{theorem}[Verdict: COINCIDENCE]\label{thm:verdict}
The near-miss $\Gone\cdot\sqtp \approx \ALPHA$ is a numerical coincidence, not an algebraic identity.  Specifically:
\begin{enumerate}
  \item[\normalfont(i)]
    \emph{Provably false:} The identity $\Gone\cdot\sqtp = \ALPHA$ is
    impossible over $\QQ$ by the power-structure argument of
    Theorem~\ref{thm:impossible}.
  \item[\normalfont(ii)]
    \emph{PSLQ:} No integer relation for $\delta = \Gone\cdot\sqtp - \ALPHA$
    with coefficient norm $\le 500$ exists in any tested basis up to 16 elements,
    including all natural $\pi$-powers, $\sqtp$, $\Gamma(1/4)$, the lemniscate
    constant, $\log\pi$, $\zeta(3)$, and $e^{2\pi}$.
  \item[\normalfont(iii)]
    \emph{No correction:} No natural TOE constant $T$ and small integer $c$
    satisfy $\delta = c \cdot T$ exactly.
  \item[\normalfont(iv)]
    \emph{Structural explanation:} The near-miss arises from two compounding
    arithmetic accidents: $1728/\pi^3 \approx 55.73$ is close to an integer,
    and $137/55.7 \approx 2.46$ is close to $\sqtp \approx 2.507$.
    These are independent accidents from genuinely different mathematical
    domains with no algebraic entanglement.
\end{enumerate}
\end{theorem}

\begin{corollary}[P179 direction resolved]\label{cor:resolved}
The open direction recorded as item~3 of Section~6.2 of P179 is resolved:
the $0.11\%$ discrepancy between $\Gone\cdot\sqtp$ and $\ALPHA$ has no
algebraic structural mechanism, and the near-miss is not evidence of an
undiscovered identity.
\end{corollary}

\subsection{Open directions}

\begin{enumerate}
  \item \emph{Sharper coefficient bound.}
    Theorem~\ref{thm:pslq} certifies no relation with norm $\le 500$ in the
    tested bases.  A Gelfond--Schneider-type analytic argument proving genuine
    algebraic independence of $\Gone\cdot\sqtp$ and $\ALPHA$ over $\QQ$ ---
    beyond the power-structure argument, which proves impossibility of a specific
    functional form --- remains open.

  \item \emph{Other $G_1$-variants.}
    P171 introduced $G_1^{(171)} = j(i)/(4\Omega) - 1 \approx 0.003456$, with
    $\Omega = \pi\ALPHA$ the breath period.  The product
    $G_1^{(171)}\cdot\sqtp \approx 0.00867$ is not close to any known constant.
    Whether $G_1^{(171)}$ has a Stirling-bridge near-miss of comparable quality
    to the present one is not investigated here.

  \item \emph{Other normalisation factors.}
    Replacing $\sqtp$ by other Gamma-function normalisations --- $\Gamma(1/4)$,
    $\sqrt{\pi}$, $\Gamma(3/4)$ --- in the product with $G_1$ produces values
    far from $\ALPHA$ (checked numerically).  The $\sqtp$ near-miss appears
    to be isolated.

  \item \emph{The deeper question: why is $1728/\pi^3$ close to an integer?}
    The near-miss traces ultimately to $1728/\pi^3 \approx 55.73$, i.e., to
    the fact that $\pi^3 \approx 1728/56 = 30.857$, close to $\pi^3 = 31.006$.
    This is equivalent to asking why $12^3 / (4\Omega_0) \approx 56$, or
    equivalently $\Omega_0 \approx j(i)/224$.  Whether the GON kernel constant
    $\Omega_0 = \pi^3/4$ has any modular-arithmetic relation to $j(i)/224$ is
    an open and likely intractable question.
\end{enumerate}

%==========================================================================
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\end{thebibliography}

\end{document}
