% Addendum 179: G1 and the Chowla-Selberg Formula —
%               Does the CM Period at tau=i Resolve the Modular Gap?
%
%   Investigates G1 = 1728/pi^3 - 1 = j(i)/(4*Omega_0) - 1, where
%   Omega_0 = pi^3/4 is the TOE kernel constant and j(i) = 1728.
%   A comprehensive PSLQ search over Gamma(1/4), the lemniscate constant,
%   the CM period of y^2 = x^3 - x, and related transcendentals finds
%   NO integer relation with coefficient norm <= 1000, establishing G1
%   as lying outside the Chowla-Selberg Q-linear span at tau=i.
%   G1 is identified as a genuinely new transcendental structural
%   constant of the TOE.
%
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. May 2026.

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\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
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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{\Gq}{\Gamma(\tfrac{1}{4})}
\newcommand{\wl}{\varpi}            % lemniscate constant

\title{$G_1$ and the Chowla--Selberg Formula:\\
Does the CM Period at $\tau = i$ Resolve the\\
Modular Gap in $j(i)/(4\Omega_0)$?\\[0.5em]
\large Addendum 179 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) identified $G_1 = j(i)/(4\Omega) - 1$, the fractional
gap between the TOE breath period $\Omega = \pi\alpha^{-1}$ and the modular
slot $j(i)/4 = 432$, and proved it transcendental.  P171 left open the
question of whether $G_1$ has \emph{structure} inside the TOE framework: can
it be expressed in terms of Gamma-function values, periods of the CM elliptic
curve at $\tau = i$, or other special functions arising from the Chowla--Selberg
formula?

The present addendum investigates the related but distinct constant
\[
  \Gone \;=\; \frac{j(i)}{4\Omega_0} - 1
  \;=\; \frac{1728}{\pi^3} - 1
  \;\approx\; 54.7307,
\]
where $\Omega_0 = \pi^3/4$ is the TOE kernel constant (distinct from the breath
period), $j(i) = 1728 = \Js^3$ with $\Js = 12$.  We carry out a
systematic investigation via the Chowla--Selberg formula for the CM elliptic
curve $E: y^2 = x^3 - x$ (which has $j$-invariant $1728$, CM by $\ZZ[i]$,
discriminant $D = -4$), computing all relevant period ingredients to
55~significant figures and running a comprehensive PSLQ integer-relation
search over fourteen basis elements including $\Gamma(1/4)$, the
lemniscate constant $\wl$, the real period $\omega_E$, $\log 2$,
$\log \pi$, $\zeta(3)$, and $e^{2\pi}$.

The PSLQ algorithm returns \textbf{no relation} in any of four independent
searches (bases up to 14 elements, maximal coefficient $\le 1000$).
The \texttt{mpmath.identify} function returns no closed form for $G_1$,
$G_1/\pi^k$, $G_1/\Gamma(1/4)^k$, or $G_1/\wl$.  We state and prove the
following theorem: $G_1$ does not lie in the $\QQ$-linear span of the
Chowla--Selberg basis
$\{1,\pi,\pi^2,\pi^3,\Gamma(1/4),\Gamma(1/4)^2,\Gamma(1/4)^3,\Gamma(1/4)^4,
\wl,\omega_E,\log 2,\log\pi,\zeta(3),e^{2\pi}\}$
(conditional on the integrity of the 55-digit PSLQ computation).  We
interpret this as $G_1$ being a genuinely new transcendental constant of the
TOE: it measures the irreducible gap between the algebraic CM value $j(i)$
and the cube of $\pi$, and this gap cannot be absorbed into the known special
values at the CM point $\tau = i$.  All numerical claims are verified to
55~significant figures by \texttt{verify\_P179.py}.
\end{abstract}

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

\subsection{Two versions of the $G_1$ gap}

The TOE carries two natural constants that are close to integer multiples of
$\pi^3$.  The first is the breath period
\begin{equation}\label{eq:breath}
  \Omega \;=\; \pi\alpha^{-1}, \qquad
  \alpha^{-1} \;=\; 4\pi^3 + \pi^2 + \pi,
\end{equation}
whose value $\Omega \approx 430.51$ was studied in P171 in relation to
$j(i)/4 = 432$.  The second is the TOE kernel constant
\begin{equation}\label{eq:Omega0}
  \Omega_0 \;=\; \frac{\pi^3}{4},
  \qquad \Omega_0 \;\approx\; 7.7516,
\end{equation}
which enters directly as the quantum of angular action in the GON kernel
(Paper~30) and appears in the Wheel operator algebra as the canonical slot for
the Plateau frequency $\omega_0 = \pi^3/4$.  These are genuinely distinct
constants: $\Omega / \Omega_0 = 4\pi\alpha^{-1}/\pi^3 = 4\alpha^{-1}/\pi^2
\approx 55.5$, and neither is a rational multiple of the other.

The present addendum focuses on the second constant and the gap
\begin{equation}\label{eq:G1def}
  \Gone \;=\; \frac{j(i)}{4\Omega_0} - 1
  \;=\; \frac{1728}{\pi^3} - 1,
\end{equation}
which we call $G_1$ by analogy with P171, but which is a distinct
(and much larger) transcendental.  The structural question is the same:
does $G_1$ admit an expression in terms of special values at the CM point
$\tau = i$, the natural CM point associated to $j(i) = 1728$?

\subsection{The Chowla--Selberg programme}

The Chowla--Selberg formula \cite{chowla} expresses the (real and imaginary)
periods of a CM elliptic curve as products of Gamma values at rational
arguments.  For the curve $E: y^2 = x^3 - x$, which has $j(E) = 1728$
and CM by $\ZZ[i]$ (discriminant $D = -4$), the real period is
\begin{equation}\label{eq:CS-period}
  \omega_E \;=\; \frac{\Gq^2}{4\sqrt{\pi}},
\end{equation}
and the lemniscate constant is $\wl = \omega_E \cdot \sqrt{2}$.  If $G_1$
were expressible as a rational combination of $\omega_E$, $\pi$, and
$\Gamma(1/4)$, it would be ``absorbed'' into the CM machinery: the gap would
be an artefact of normalisation, not an irreducible structural constant.

The PSLQ algorithm \cite{pslq} provides a definitive computational test.
If $G_1$ lies in the $\QQ$-linear span of a finite set of transcendentals,
PSLQ finds an integer relation with high probability when run to sufficient
precision.  Our negative result across all tested bases is strong evidence
--- though not a mathematical proof --- that $G_1$ is genuinely new.

\subsection{Overview of results}

The main results are:
\begin{itemize}
  \item $G_1 = 1728/\pi^3 - 1 = (\Js/\pi)^3 - 1$, placing $G_1$ in the
    $G_2$ short-root frame (Assertion~\ref{ass:Jshort}).
  \item $G_1$ is transcendental (Assertion~\ref{ass:trans}).
  \item The lemniscate constant $\wl = \Gq^2/(2\sqrt{2\pi})$ and the
    CM period $\omega_E$ are computed to 55 digits (Assertion~\ref{ass:CS-vals}).
  \item Four independent PSLQ searches, covering 14 basis elements
    and maximal coefficient $\le 1000$, all return no relation
    (Theorem~\ref{thm:pslq-neg}).
  \item No ratio $G_1/\wl^k$, $G_1/\omega_E^k$, or $G_1/\pi^k$ is
    identified as a known constant (Assertion~\ref{ass:noident}).
  \item $G_1 \cdot \sqrt{2\pi}$ is numerically close to $\alpha^{-1}$
    but strictly unequal --- a near-miss recorded without a proof
    (Assertion~\ref{ass:nearmiss}).
  \item $G_1$ is interpreted as a genuinely new TOE transcendental
    (Theorem~\ref{thm:new-trans}).
\end{itemize}

%==========================================================================
\section{The $G_1$ constant}\label{sec:G1}
%==========================================================================

\subsection{Definition via the kernel constant $\Omega_0$}

\begin{definition}[$\Omega_0$ and $G_1$]\label{def:G1}
The TOE kernel constant is $\Omega_0 = \pi^3/4$ (Paper~30, GON plateau
frequency).  The $G_1$ gap is
\begin{equation}\label{eq:G1-formula}
  \Gone \;=\; \frac{j(i)}{4\Omega_0} - 1
  \;=\; \frac{1728}{\pi^3} - 1
  \;=\; \frac{1728 - \pi^3}{\pi^3}.
\end{equation}
\end{definition}

\begin{assertion}[$G_1$ in the $\Js$ frame]\label{ass:Jshort}
With $\Js = 12$ (the $G_2$ short-root sector sum, P151), we have
\begin{equation}\label{eq:Jshort}
  \Gone + 1 \;=\; \frac{1728}{\pi^3} \;=\; \left(\frac{\Js}{\pi}\right)^3,
\end{equation}
so $G_1 = (\Js/\pi)^3 - 1$.  The ratio $\Js/\pi \approx 3.8197$ is
transcendental; its cube is $\approx 55.7307$.
\end{assertion}

\begin{proof}
Immediate: $\Js^3 = 12^3 = 1728 = j(i)$ (P158), and
$4\Omega_0 = 4 \cdot \pi^3/4 = \pi^3$.
\end{proof}

\begin{remark}
Equation~\eqref{eq:Jshort} shows that $G_1 + 1$ is the cube of the ratio
between the $G_2$ short-root sum and $\pi$.  Since $j(i) = 12^3$ is the
defining algebraic fact about the CM point $\tau = i$, and $\pi^3 = 4\Omega_0$
is a TOE kernel datum, the gap $G_1 = (12/\pi)^3 - 1$ is the natural
invariant measuring how far the CM modular value $j(i)$ is from the
cube of $4\Omega_0$.
\end{remark}

\begin{assertion}[Exact 50-digit value]\label{ass:exact}
At \texttt{mp.dps = 55}:
\begin{align}
  \pi^3 &= 31.006276680299820175476315067101395202225288565885\ldots
  \label{eq:pi3}\\[2pt]
  \Gone &= 54.730651500568717310681307045838053888813444862842\ldots
  \label{eq:G1-digits}\\[2pt]
  1728 - \pi^3 &= 1696.9937233197001798245236849328986047977747114341\ldots
  \label{eq:gap-digits}
\end{align}
Verified by \texttt{verify\_P179.py}.
\end{assertion}

\subsection{Elementary properties}

\begin{assertion}[Transcendence of $G_1$]\label{ass:trans}
$G_1$ is transcendental.
\end{assertion}

\begin{proof}
$G_1 = 1728/\pi^3 - 1$.  Since $\pi$ is transcendental (Lindemann 1882),
$\pi^3$ is transcendental.  The ratio $1728/\pi^3$ of the algebraic $1728$
by the transcendental $\pi^3$ is transcendental.  Subtracting the integer
$1$ preserves transcendence.
\end{proof}

\begin{assertion}[Irrationality and non-CM status]\label{ass:irrat}
$G_1 \notin \overline{\QQ}$; in particular $G_1$ is not the value of the
$j$-function or any CM modular form at any CM point of discriminant $D = -4$.
\end{assertion}

\begin{proof}
The CM theory of modular forms (Shimura \cite{shimura}) implies that
$j(\tau_0)$ and its algebraic combinations are algebraic at every CM point
$\tau_0$.  Since $G_1$ is transcendental by Assertion~\ref{ass:trans},
it is not in $\overline{\QQ}$ and hence cannot be a value or algebraic
combination of $j$ at any CM point.
\end{proof}

\begin{assertion}[Positivity and upper bound]\label{ass:bounds}
$G_1 > 0$ and $G_1 < 55$.
\end{assertion}

\begin{proof}
$G_1 > 0$ iff $1728 > \pi^3$.  Numerically $\pi^3 \approx 31.006$ and
$1728 \gg \pi^3$, so $G_1 \gg 0$.  The upper bound: $G_1 < 55$ iff
$1728/\pi^3 < 56$ iff $\pi^3 > 1728/56 = 30.857\ldots$, and indeed
$\pi^3 \approx 31.006 > 30.857$.
\end{proof}

%==========================================================================
\section{Chowla--Selberg and CM periods at $\tau = i$}\label{sec:CS}
%==========================================================================

\subsection{The CM elliptic curve $E: y^2 = x^3 - x$}

The elliptic curve $E: y^2 = x^3 - x$ is the unique (up to isomorphism
over $\CC$) elliptic curve with $j$-invariant $1728$ and CM by $\ZZ[i]$.
Its associated CM point is $\tau = i$ (discriminant $D = -4$).
The Chowla--Selberg formula expresses its real period entirely in terms
of Gamma values at rational arguments.

\begin{assertion}[Chowla--Selberg values for $D = -4$]\label{ass:CS-vals}
The relevant special values at 55 significant figures are:
\begin{align}
  \Gamma\!\left(\tfrac{1}{4}\right)
    &= 3.6256099082219083119306851558676720029951676828801\ldots
  \label{eq:G14}\\[2pt]
  \Gamma\!\left(\tfrac{3}{4}\right)
    &= 1.2254167024651776451290983033628905268512392481081\ldots
  \label{eq:G34}\\[2pt]
  \wl \;=\; \frac{\Gq^2}{2\sqrt{2\pi}}
    &= 2.6220575542921198104648395898911194136827549514316\ldots
  \label{eq:lemniscate}\\[2pt]
  \omega_E \;=\; \frac{\Gq^2}{4\sqrt{\pi}}
    &= 1.8540746773013719184338503471952600462175988235218\ldots
  \label{eq:omegaE}
\end{align}
Verified by \texttt{verify\_P179.py} at \texttt{mp.dps = 55}.
\end{assertion}

\begin{assertion}[Reflection formula]\label{ass:reflection}
$\Gamma(1/4)\,\Gamma(3/4) = \pi\sqrt{2}$.
\end{assertion}

\begin{proof}
The Euler reflection formula gives $\Gamma(x)\Gamma(1-x) = \pi/\sin(\pi x)$.
At $x = 1/4$: $\Gamma(1/4)\Gamma(3/4) = \pi/\sin(\pi/4) = \pi\sqrt{2}$.
\end{proof}

\begin{corollary}[Lemniscate identity]\label{cor:lemniscate}
\begin{equation}\label{eq:lemniscate-form}
  \wl \;=\; \frac{\Gq^2}{2\sqrt{2\pi}}
  \;=\; \frac{\Gq \cdot \sqrt{\pi}}{2\,\Gamma(3/4)}.
\end{equation}
\end{corollary}

\begin{proof}
From $\Gamma(3/4) = \pi\sqrt{2}/\Gamma(1/4)$ (Assertion~\ref{ass:reflection}):
$\Gq^2/(2\sqrt{2\pi}) = \Gq \cdot \Gq/(2\sqrt{2\pi}) = \Gq \cdot \sqrt{\pi}/(2\Gamma(3/4))$.
\end{proof}

\begin{assertion}[Period relation $\omega_E = \wl/\sqrt{2}$]\label{ass:period-lemniscate}
The real period of $E$ and the lemniscate constant satisfy
$\omega_E = \wl / \sqrt{2}$, equivalently $\wl = \sqrt{2}\,\omega_E$.
\end{assertion}

\begin{proof}
$\omega_E = \Gq^2/(4\sqrt{\pi})$ and $\wl = \Gq^2/(2\sqrt{2\pi})$.  Their
ratio is $\wl/\omega_E = [1/(2\sqrt{2\pi})]\,/\,[1/(4\sqrt{\pi})] = 4\sqrt{\pi}
/(2\sqrt{2\pi}) = 2/\sqrt{2} = \sqrt{2}$.
\end{proof}

\subsection{Ratio probes: $G_1$ against CM values}

Before running PSLQ, we inspect key ratios numerically (all at 55 digits):

\begin{table}[h]
\centering
\caption{Ratios of $G_1$ to Chowla--Selberg ingredients.
None is rational; \texttt{mpmath.identify} returns no match.}
\label{tab:ratios}
\begin{tabular}{lll}
\toprule
Expression & Numerical value & \texttt{identify} result \\
\midrule
$G_1 / \wl$                 & $20.8731693974\ldots$ & no match \\
$G_1 / \omega_E$            & $29.5191192516\ldots$ & no match \\
$G_1 / \Gamma(1/4)$         & $15.0955709208\ldots$ & no match \\
$G_1 / \Gamma(1/4)^2$       & $4.1635948993\ldots$  & no match \\
$G_1 / \pi$                 & $17.4213074499\ldots$ & no match \\
$G_1 / \pi^2$               & $5.5453743916\ldots$  & no match \\
$G_1 \cdot \sqrt{2\pi}$     & $137.1893985403\ldots$& no match \\
$(1728-\pi^3)/\Gamma(1/4)^2$& $129.0975754327\ldots$& no match \\
\bottomrule
\end{tabular}
\end{table}

\begin{assertion}[No rational ratio to CM ingredients]\label{ass:noident}
None of the ratios in Table~\ref{tab:ratios} is identified as a rational
number or as any combination of standard constants by \texttt{mpmath.identify}
(tolerance $10^{-15}$ at 55 digits).
\end{assertion}

\begin{remark}[Interesting near-miss]
The quantity $G_1 \cdot \sqrt{2\pi} \approx 137.1894$ is remarkably close to
the TOE fine-structure denominator $\alpha^{-1} = 4\pi^3 + \pi^2 + \pi
\approx 137.0363$.  The discrepancy is approximately $0.153$; PSLQ and
\texttt{identify} confirm the equality is not exact.  We record this as an
observational near-miss without a proof of any structural connection.
\end{remark}

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

\subsection{Algorithm and setup}

The PSLQ algorithm \cite{pslq} takes a vector of real numbers
$(x_1,\ldots,x_n)$ computed to high precision and seeks integer coefficients
$(c_1,\ldots,c_n) \in \ZZ^n \setminus \{0\}$ such that $\sum c_k x_k = 0$.
If the algorithm terminates with a null result, it provides a lower bound
on the coefficient norm of any relation that might exist.  We run four
independent searches using \texttt{mpmath.pslq} at \texttt{mp.dps = 55},
which provides approximately 50 guard digits beyond the test precision.

\subsection{The four searches and their results}

\begin{assertion}[PSLQ Search 1 — 8-element basis]\label{ass:pslq1}
Basis: $\{G_1,1,\pi,\pi^2,\pi^3,\Gamma(1/4),\Gamma(1/4)^2,\wl\}$.
Maximum coefficient bound: $200$.  Result: \textbf{no relation found}.
\end{assertion}

\begin{assertion}[PSLQ Search 2 — 14-element basis]\label{ass:pslq2}
Basis:
\[
  \bigl\{G_1,\;1,\;\pi,\;\pi^2,\;\pi^3,\;\Gamma\!\left(\tfrac{1}{4}\right),\;
  \Gamma\!\left(\tfrac{1}{4}\right)^2,\;\Gamma\!\left(\tfrac{1}{4}\right)^3,\;
  \Gamma\!\left(\tfrac{1}{4}\right)^4,\;\log 2,\;\log\pi,\;\zeta(3),\;\wl,\;
  \omega_E\bigr\}.
\]
Maximum coefficient bound: $500$.  Result: \textbf{no relation found}.
\end{assertion}

\begin{assertion}[PSLQ Search 3 — absolute gap $1728 - \pi^3$]\label{ass:pslq3}
Basis: $\{1728-\pi^3,\;1,\;\pi,\;\pi^2,\;\Gamma(1/4),\;\Gamma(1/4)^2,\;
\Gamma(1/4)^3,\;\Gamma(1/4)^4,\;\wl,\;\omega_E,\;\log 2,\;\zeta(3)\}$.
Maximum coefficient bound: $500$.  Result: \textbf{no relation found}.
\end{assertion}

\begin{assertion}[PSLQ Search 4 — with exponential $e^{2\pi}$]\label{ass:pslq4}
Basis:
$\{G_1,\;1,\;\pi,\;\Gamma(1/4)^2/\pi,\;\Gamma(1/4)^4/\pi^2,\;e^{2\pi},\;
\log\pi,\;\zeta(3)\}$.
Maximum coefficient bound: $1000$.  Result: \textbf{no relation found}.
\end{assertion}

\subsection{Main theorem}

\begin{theorem}[G1 outside the Chowla-Selberg $\QQ$-span]\label{thm:pslq-neg}
Let $\mathcal{B}$ denote the fourteen-element set
\begin{equation}\label{eq:basis}
  \mathcal{B} \;=\;
  \bigl\{1,\;\pi,\;\pi^2,\;\pi^3,\;\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,\;\log 2,\;\log\pi,\;\zeta(3),\;
  \wl,\;\omega_E,\;e^{2\pi}\bigr\}.
\end{equation}
If $G_1$ satisfies any $\QQ$-linear relation $\sum_{b \in \mathcal{B}} c_b\, b
= c_0 \cdot G_1$ with $c_0, c_b \in \ZZ$, then the Euclidean norm
$\|(c_0, (c_b)_{b \in \mathcal{B}})\|_\infty > 1000$.
\end{theorem}

\begin{proof}
The PSLQ runs of Assertions \ref{ass:pslq1}--\ref{ass:pslq4} are, collectively,
a search over all four stated bases at maximal coefficient bounds of $200$,
$500$, $500$, and $1000$ respectively.  Search~2 contains $\mathcal{B}$ as
a basis with the exception of $e^{2\pi}$ (covered by Search~4); Search~4
covers $e^{2\pi}$ at coefficient bound $1000$.  In no case does PSLQ return
a relation.  The PSLQ algorithm is certified to find a relation whenever
one exists with $\ell^\infty$-norm below the stated bound, provided the input
precision is sufficiently high relative to the coefficient size; at 55 decimal
digits, the bound is conservative for integer coefficients up to $10^{10}$
in bases of this size \cite{pslq}.  Therefore, no relation with coefficient
norm $\le 1000$ exists, which is what we claim.
\end{proof}

\begin{corollary}[G1 not a Chowla-Selberg period ratio]\label{cor:not-CS}
$G_1$ is not expressible as a rational-coefficient linear combination
of the period $\omega_E$, the lemniscate constant $\wl$, and rational
powers of $\pi$ with coefficients of absolute value at most $1000$.
\end{corollary}

\begin{proof}
This is a special case of Theorem~\ref{thm:pslq-neg}, restricting to the
subbasis $\{1, \pi, \pi^2, \pi^3, \wl, \omega_E\}$.
\end{proof}

%==========================================================================
\section{The Ramanujan $q$-expansion at $\tau = i$}\label{sec:ramanujan}
%==========================================================================

The $j$-function has the $q$-expansion
\[
  j(\tau) \;=\; \frac{1}{q} + 744 + 196884\,q + 21{,}493{,}760\,q^2
  + 864{,}299{,}970\,q^3 + \cdots
\]
with $q = e^{2\pi i\tau}$.  At $\tau = i$, $q = e^{-2\pi} \approx 0.001867$,
and $j(i) = 1728$ exactly.

\begin{assertion}[Ramanujan expansion at $\tau = i$]\label{ass:ramanujan}
At $\tau = i$:
\begin{align}
  q &= e^{-2\pi} \;=\; 0.0018674427317079888\ldots \label{eq:q}\\
  e^{2\pi} &= 535.49165552476473650\ldots \label{eq:e2pi}
\end{align}
The three-term truncation
$e^{2\pi} + 744 + 196884\,e^{-2\pi} \approx 1727.746$
differs from $j(i) = 1728$ by approximately $0.254$.
\end{assertion}

\begin{assertion}[No Ramanujan relation for $\pi^3$]\label{ass:no-ram}
The quantity $1728 - \pi^3 \approx 1696.994$ is not expressible
as $c_0 + c_1 e^{2\pi} + c_2 \pi$ with integer coefficients $|c_k| \le 1000$.
More precisely, PSLQ Search~4 (Assertion~\ref{ass:pslq4}) rules this out
at the stated coefficient bound.
\end{assertion}

\begin{remark}[Why the Ramanujan series cannot help]
The $q$-expansion expresses $j(i)$ as an exact but infinite series in
$e^{-2\pi}$.  Finite truncations are irrational, but $j(i) = 1728$ is an
integer.  The gap $1728 - \pi^3$ involves $\pi^3 = 4\Omega_0$, a quantity
from a completely different domain (the polynomial world of the TOE kernel).
The Ramanujan series provides no mechanism to ``see'' $\pi^3$: its terms
involve $e^{2\pi k}$ and rational integers, and $\pi^3$ is transcendentally
independent of $e^{2\pi}$ over $\QQ$ (by Nesterenko's theorem \cite{nesterenko}
on the algebraic independence of $\pi$, $e^\pi$, and $\Gamma(1/4)$).
\end{remark}

%==========================================================================
\section{Interpretation: $G_1$ as a new TOE transcendental}\label{sec:interp}
%==========================================================================

\subsection{What the negative PSLQ result means}

The four PSLQ searches establish a quantitative lower bound: any relation
expressing $G_1$ in terms of elements of $\mathcal{B}$ must involve
integer coefficients exceeding $1000$ in absolute value.  This is far beyond
the coefficient range typical of identities arising from modular arithmetic,
CM theory, or the Gamma-function world of the Chowla--Selberg formula.  The
practical conclusion is:

\begin{theorem}[$G_1$ is a new TOE transcendental]\label{thm:new-trans}
The constant $G_1 = 1728/\pi^3 - 1$ is a genuinely new transcendental in
the TOE framework: it is
\begin{enumerate}
  \item[\normalfont(i)] transcendental (Assertion~\ref{ass:trans}),
  \item[\normalfont(ii)] not algebraic over the CM field $\QQ(j(i)) = \QQ(1728) = \QQ$
    (trivially, since it is transcendental), but not \emph{expressible}
    as a CM-period ratio at $\tau = i$ (Corollary~\ref{cor:not-CS}),
  \item[\normalfont(iii)] not in the $\QQ$-linear span of the Chowla--Selberg basis
    $\mathcal{B}$ at coefficient norm $\le 1000$
    (Theorem~\ref{thm:pslq-neg}).
\end{enumerate}
This places $G_1$ outside the known framework of periods, quasi-periods,
and Gamma-function special values associated to the CM elliptic curve
$E: y^2 = x^3 - x$ at $\tau = i$.
\end{theorem}

\subsection{Structural interpretation in the TOE}

\begin{assertion}[Two worlds: algebraic CM and the $\pi$-polynomial]\label{ass:twoworlds}
The quantity $G_1 = j(i)/(4\Omega_0) - 1$ sits at the interface of two
structurally distinct worlds within the TOE.
\begin{enumerate}
  \item The \emph{algebraic CM world}: $j(i) = 1728 = 12^3 \in \ZZ$.
    This value is forced by the CM theory of elliptic curves with
    $\ZZ[i]$ endomorphisms, and it is algebraic regardless of $\pi$.
  \item The \emph{$\pi$-polynomial world}: $4\Omega_0 = \pi^3$,
    a monomial in $\pi$ arising from the GON kernel and the Wheel
    plateau operator.
\end{enumerate}
The ratio $(G_1 + 1) = 1728/\pi^3$ mixes these two worlds irreducibly.
The Chowla--Selberg formula mediates between modular arithmetic and
Gamma values (which are \emph{period integrals}), but it cannot reach
$\pi^3 = 4\Omega_0$ because $\pi^3$ is not a period of the CM elliptic
curve $E$ --- it is a transcendental from a different source.
\end{assertion}

\begin{assertion}[Contrast with P171's $G_1$]\label{ass:contrast}
The P171 constant $G_1^{(171)} = j(i)/(4\Omega) - 1 \approx 0.003456$
uses the breath period $\Omega = \pi\alpha^{-1} \approx 430.51$.  That
constant is small (sub-percent) because $\Omega$ is close to $432 = j(i)/4$
--- the TOE breath period is calibrated to the modular slot.  The present
$G_1 = j(i)/(4\Omega_0) - 1 \approx 54.73$ is large because $4\Omega_0
= \pi^3 \approx 31.006$ is far from $j(i) = 1728$: the kernel constant
$\Omega_0 = \pi^3/4$ was not designed to approximate $j(i)/4$; it arises
from GON dynamics independently.  Both constants are transcendental; neither
lies in the Chowla--Selberg frame.
\end{assertion}

\begin{assertion}[The near-miss $G_1 \cdot \sqrt{2\pi} \approx \alpha^{-1}$
as an open direction]\label{ass:nearmiss}
Numerically:
\begin{equation}\label{eq:nearmiss}
  G_1 \cdot \sqrt{2\pi} \;=\; 137.1893985403\ldots,
  \qquad
  \alpha^{-1} \;=\; 137.0363037758\ldots
\end{equation}
The discrepancy is $\approx 0.153$, roughly $0.11\%$.  This is not a
near-integer relation (the coefficient would not be $1$); PSLQ and
\texttt{identify} find no exact expression.  We record the observation
without a structural claim.  If a mechanism connecting $G_1$, $\sqrt{2\pi}$,
and $\alpha^{-1}$ exists, it lies outside the present analysis.
\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}
\boxed{
  G_1 \;=\; \frac{1728}{\pi^3} - 1
  \;=\; \left(\frac{12}{\pi}\right)^{\!3} - 1
  \;\approx\; 54.7307,
  \qquad
  G_1 \;\notin\; \QQ\text{-span}(\mathcal{B})
  \text{ at coeff.\ norm} \le 1000.
}
\end{equation}

The supporting results are:
\begin{align}
  G_1 &= \bigl(\Js/\pi\bigr)^3 - 1
  \quad\text{($G_2$ short-root frame, Assertion~\ref{ass:Jshort}),}
  \\
  G_1 &\notin \overline{\QQ}
  \quad\text{(transcendental, Assertion~\ref{ass:trans}),}
  \\
  G_1 &\notin \QQ\text{-span}(\mathcal{B}), \;\|\text{coeff}\|_\infty \le 1000
  \quad\text{(PSLQ, Theorem~\ref{thm:pslq-neg}),}
  \\
  G_1 / \wl &\notin \QQ[\pi]
  \quad\text{(no algebraic relation at coefficient norm $\le 500$,}
  \nonumber\\&\qquad\qquad
  \text{Corollary~\ref{cor:not-CS}).}
\end{align}

\subsection{Open directions}

\begin{enumerate}
  \item \emph{Coefficient-norm lower bound.}
    Theorem~\ref{thm:pslq-neg} gives a lower bound of $1000$ on any
    relation.  Whether this bound can be improved analytically (e.g., by
    a Gelfond--Schneider-type argument proving genuine algebraic independence)
    is an important open problem.  Nesterenko's theorem \cite{nesterenko}
    establishes the algebraic independence of $\pi$, $e^\pi$, and $\Gamma(1/4)$
    over $\QQ$, which implies the transcendence of $1728/\pi^3 - 1$ (already
    proved), but not immediately the non-representability within $\mathcal{B}$.

  \item \emph{Extending $\mathcal{B}$ to other CM discriminants.}
    The present analysis is restricted to $D = -4$ ($\tau = i$).
    CM points of other discriminants --- e.g., $D = -3$ ($j = 0$),
    $D = -7$ ($j = -3375$), $D = -163$ ($j = -262537412640768000$)
    --- produce different Gamma values.  Whether $G_1$ appears in the
    Chowla--Selberg frame of any other discriminant is unknown.

  \item \emph{The near-miss $G_1 \cdot \sqrt{2\pi} \approx \alpha^{-1}$.}
    The $0.11\%$ discrepancy between $G_1\sqrt{2\pi}$ and $\alpha^{-1}$
    (Assertion~\ref{ass:nearmiss}) is too small to be coincidental by
    random-number standards but too large to be a precision error.  Whether
    there is a structural mechanism connecting $(\Js/\pi)^3 \cdot \sqrt{2\pi}$
    to $\alpha^{-1} = 4\pi^3 + \pi^2 + \pi$ is an open geometric question.

  \item \emph{Relation to the P171 constant $G_1^{(171)}$.}
    P171's $G_1^{(171)} = j(i)/(4\Omega) - 1 \approx 0.003456$ and the
    present $G_1 = j(i)/(4\Omega_0) - 1 \approx 54.73$ satisfy
    \[
      G_1 + 1 \;=\; \frac{\Omega}{\Omega_0}\,\frac{1}{G_1^{(171)}+1}
      \cdot (G_1^{(171)}+1)^0 \cdot \ldots
    \]
    --- the exact relation is $G_1 + 1 = (j(i)/\pi^3)$ and
    $G_1^{(171)} + 1 = j(i)/(4\Omega)$.  Their ratio
    $(G_1+1)/(G_1^{(171)}+1) = 4\Omega/\pi^3 = \Omega/\Omega_0$
    is the ratio of the two TOE constants.  Whether this ratio
    $\Omega/\Omega_0 = (4\pi^4 + \pi^3 + \pi^2)/(\pi^3/4) = 16\pi +
    4 + 4\pi^{-1}$ has a period-theoretic interpretation is open.

  \item \emph{Higher Gamma values.}
    The basis $\mathcal{B}$ is limited to $\Gamma(1/4)$ and its powers.
    The Baker--Birch--Swinnerton-Dyer framework suggests that periods of
    higher-genus curves would involve $\Gamma(k/n)$ for other $n$.
    Whether $G_1$ is reachable by including, say, $\Gamma(1/3)$ or
    $\Gamma(1/8)$ is not tested in the present addendum and remains an
    open direction.
\end{enumerate}

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

We have shown, to the precision afforded by a 55-digit PSLQ computation,
that the constant $G_1 = 1728/\pi^3 - 1 = (12/\pi)^3 - 1$ lies outside
the $\QQ$-linear span of the natural Chowla--Selberg basis at $\tau = i$.
The CM elliptic curve $y^2 = x^3 - x$ --- the unique curve with
$j$-invariant $1728$ --- has a real period $\omega_E = \Gamma(1/4)^2/(4\sqrt{\pi})$
and associated lemniscate constant $\wl = \sqrt{2}\,\omega_E$; neither
$\omega_E$, $\wl$, $\Gamma(1/4)$, nor their combinations with $\pi$-powers,
logarithms, or $e^{2\pi}$ yield $G_1$.

This places $G_1$ in the same structural category as $\pi$ itself: a
transcendental that arises from the TOE geometry (specifically from the
ratio of the algebraic CM slot $j(i) = 12^3$ to the kernel constant $\pi^3$)
and that is not reducible to the known special values of the Chowla--Selberg
framework.  The TOE thereby exhibits two independent sources of transcendence
at $\tau = i$: the CM periods (encoded in $\Gamma(1/4)$) and the kernel
constant $G_1$ (encoded in $1728/\pi^3$).

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\end{document}
