% Addendum 167: G1 as a Modular Residual —
%               The Fractional Gap between the TOE Breath Period
%               and One Quarter of j(i)
%   P167 proves: G1 = j(i)/(4*Omega) - 1, where j(i) = 1728 = J_short^3
%   (P158) and Omega = pi*alpha^{-1} is the TOE breath period. G1 is the
%   irreducible fractional gap between the TOE geometric period and the
%   modular value j(i)/4 = 432. The gap is transcendental, unreachable
%   by any CM point of discriminant D=-4, and its near-miss to J_short/2=6
%   is strictly non-exact (deficit 0.04898...).
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. May 2026.

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\newcommand{\Js}{J_{\mathrm{short}}}
\newcommand{\Gone}{G_1}
\newcommand{\ZZ}{\mathbb{Z}}
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\newcommand{\BF}{B_{F_4}}

\title{$G_1$ as a Modular Residual:\\
The Fractional Gap between the TOE Breath Period\\
and One Quarter of $j(i)$\\[0.5em]
\large Addendum 167 to the TOE Corpus}

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

\date{May 2026}

\begin{document}
\maketitle
\tableofcontents
\newpage

%==========================================================================
\begin{abstract}
The $G_1$ gap was introduced as $\Gone = (432 - \Omega)/\Omega$, where
$\Omega = \pi\alpha^{-1}$ is the TOE breath period and $432$ is its
rational approximation.  Addendum~156 (P156) proved $\Gone \ne 0$ via
Lindemann--Weierstrass, and proved that $\Gone$ is unreachable by Haar
averaging over any compact group (Schur obstruction).

The present addendum supplies a new structural identity: since $432 =
j(i)/4 = \Js^3/4 = 1728/4$ (P158), the $G_1$ gap is precisely the
fractional deviation of the breath period $\Omega$ from the modular value
$j(i)/4$ at the CM point $\tau = i$:
\[
  \Gone \;=\; \frac{j(i)}{4\Omega} - 1
  \;=\; \frac{\Js^3}{4\pi\alpha^{-1}} - 1.
\]
We prove: (i)~this is a tautological reformulation of the definition once
P158 is applied, but it carries geometric content: $G_1$ measures exactly
how far $\Omega$ falls from the ``modular slot'' $j(i)/4$; (ii)~the
breath period reconstructs as $\Omega = j(i)/(4(1+\Gone))$, expressing
$\Omega$ as a $\Gone$-corrected quarter of $j(i)$; (iii)~the absolute gap
$j(i) - 4\Omega = 5.9510\ldots$ is close to but strictly less than
$\Js/2 = 6$, proved non-exact by transcendence; (iv)~the polynomial
identity $4\Omega = 16\pi^4 + 4\pi^3 + 4\pi^2 = (4\pi)\alpha^{-1}$
gives $4\Omega$ the geometric interpretation of the surface area of
the unit $S^2$ times $\alpha^{-1}$; (v)~$\Gone$ is transcendental;
and (vi)~no modular form evaluates to $4\Omega$ at any CM point of
discriminant $D = -4$, confirming that $\Gone$ is an irreducible
modular residual.  All numerical claims are verified to 55 significant
figures by \texttt{verify\_G1\_modular.py}.
\end{abstract}

%==========================================================================
\section{Background and motivation}\label{sec:background}
%==========================================================================

\subsection{The $G_1$ gap and its prior status}

The TOE breath period is the positive real number
\begin{equation}\label{eq:Omega}
  \Omega \;=\; \pi\alpha^{-1}, \qquad
  \alpha^{-1} \;=\; 4\pi^3 + \pi^2 + \pi,
\end{equation}
so that $\Omega = 4\pi^4 + \pi^3 + \pi^2$.  Numerically,
$\Omega \approx 430.512$.  The integer $432$ is the nearest integer
to $\Omega$ that arises naturally in the TOE construction (see P\,156
for the complete motivation).  The $G_1$ gap is defined as the fractional
deviation:
\begin{equation}\label{eq:G1-def}
  \Gone \;=\; \frac{432 - \Omega}{\Omega}.
\end{equation}
Addendum~156 established three properties of $\Gone$:
\begin{enumerate}
  \item \emph{Non-zero}: $\Omega$ is transcendental by Lindemann--Weierstrass
    (since $\pi$ is transcendental and $\alpha^{-1}$ is a nonconstant
    polynomial in $\pi$), while $432 \in \QQ$, so $\Omega \ne 432$ and
    $\Gone \ne 0$.
  \item \emph{Irreducible}: The Schur obstruction shows that $\Gone$
    cannot be removed by Haar averaging over any compact group acting on
    the geometry of the breath period.
  \item \emph{Positive}: $\Omega < 432$, so $\Gone > 0$.
\end{enumerate}
What was not addressed in P156 is the structural origin of the integer
$432$.  That question is answered here.

\subsection{The $j$-function at $\tau = i$ and the number $432$}

Addendum~158 (P158) derived, from the root geometry of $G_2$ and $F_4$,
the identity
\begin{equation}\label{eq:j-Js}
  j(i) \;=\; \Js^3 \;=\; 12^3 \;=\; 1728,
\end{equation}
where $\Js = 12$ is the $G_2$ short-root sector sum (P151) and $j$ is
the Klein $j$-invariant.  Additively, $j(i)/4 = 1728/4 = 432$.  This is
not a coincidence of notation: $432$ enters the TOE as the integer nearest
to $\Omega$, and it is simultaneously one quarter of the unique CM value
of the $j$-function at the discriminant $D = -4$ point $\tau = i$.

The present addendum makes this identification the foundation of a new
structural reading of $\Gone$.

\subsection{Overview of results}

The main results, in logical order, are the following:
\begin{itemize}
  \item \textbf{Theorem~\ref{thm:G1-j}} (the $j(i)$ reformulation):
    $\Gone = \Js^3/(4\pi\alpha^{-1}) - 1 = j(i)/(4\Omega) - 1$.
  \item \textbf{Proposition~\ref{prop:reconstruction}} (breath period
    reconstruction): $\Omega = j(i)/(4(1+\Gone))$.
  \item \textbf{Proposition~\ref{prop:near-miss}} (near-miss to $\Js/2$):
    $j(i) - 4\Omega = 5.9510\ldots < 6 = \Js/2$, with a strictly
    positive deficit that is transcendental.
  \item \textbf{Proposition~\ref{prop:polynomial}} (polynomial--geometric
    identity): $4\Omega = (4\pi)\alpha^{-1} = 16\pi^4 + 4\pi^3 + 4\pi^2$;
    the factor $4\pi$ is the surface area of the unit sphere $S^2$.
  \item \textbf{Proposition~\ref{prop:modular-residual}} (modular residual):
    $\Gone$ is unreachable by any CM value of $j$ at discriminant $D=-4$.
  \item \textbf{Corollary~\ref{cor:transcendental}} (transcendence): $\Gone$
    is transcendental.
\end{itemize}

%==========================================================================
\section{The $j(i)$ reformulation of $G_1$}\label{sec:reformulation}
%==========================================================================

\begin{theorem}[$j(i)$ reformulation of $\Gone$]\label{thm:G1-j}
With $\Omega = \pi\alpha^{-1}$ as in~\eqref{eq:Omega}, $j(i) = 1728$,
and $\Js = 12$:
\begin{equation}\label{eq:G1-j}
  \Gone \;=\; \frac{j(i)}{4\Omega} - 1 \;=\; \frac{\Js^3}{4\pi\alpha^{-1}} - 1.
\end{equation}
Equivalently:
\begin{equation}\label{eq:Omega-j}
  \Omega \;=\; \frac{j(i)}{4(1 + \Gone)},
\end{equation}
expressing the breath period as a $\Gone$-corrected quarter of $j(i)$.
\end{theorem}

\begin{proof}
By~\eqref{eq:G1-def}, $\Gone = (432 - \Omega)/\Omega$.  Since $j(i) =
1728$ (P158) and $j(i)/4 = 432$, we substitute $432 = j(i)/4$:
\[
  \Gone \;=\; \frac{j(i)/4 - \Omega}{\Omega}
  \;=\; \frac{j(i)}{4\Omega} - 1.
\]
The second form $\Js^3/(4\pi\alpha^{-1}) - 1$ follows from $j(i) = \Js^3$
and $\Omega = \pi\alpha^{-1}$.  Rearranging $\Gone + 1 = j(i)/(4\Omega)$
gives~\eqref{eq:Omega-j}.
\end{proof}

\begin{remark}
Equation~\eqref{eq:G1-j} is a reformulation, not an independent result:
it follows from the definition of $\Gone$ together with P158's identity
$j(i) = 1728$.  Its content is geometric rather than computational: the
integer $432$ that has appeared in the TOE since the definition of $\Gone$
is now identified as the quarter-$j$ value $j(i)/4$ at the CM point
$\tau = i$.  This locates $\Gone$ precisely in the interface between the
TOE's geometric period $\Omega$ and the modular arithmetic of $\tau = i$.
\end{remark}

\begin{proposition}[Breath period reconstruction]\label{prop:reconstruction}
The breath period $\Omega$ is uniquely determined by $j(i)$ and $\Gone$:
\begin{equation}\label{eq:recon}
  \Omega \;=\; \frac{j(i)}{4(1+\Gone)} \;=\; \frac{1728}{4(1+\Gone)}.
\end{equation}
Conversely, $\Gone$ is uniquely determined by $j(i)$ and $\Omega$.  The
pair $(j(i), \Gone)$ encodes the same information as the pair $(432, \Omega)$.
\end{proposition}

\begin{proof}
Equation~\eqref{eq:recon} is~\eqref{eq:Omega-j} restated.  The converse
is~\eqref{eq:G1-def}.  The bijection between $(j(i), \Gone)$ and
$(432, \Omega)$ follows from $j(i)/4 = 432$ being rational.
\end{proof}

%==========================================================================
\section{The near-miss $j(i) - 4\Omega \approx \Js/2$}\label{sec:near-miss}
%==========================================================================

\subsection{Numerical value and strict non-exactness}

\begin{proposition}[Near-miss to $\Js/2$]\label{prop:near-miss}
The absolute gap satisfies
\begin{equation}\label{eq:abs-gap}
  j(i) - 4\Omega \;=\; 1728 - 4\pi\alpha^{-1}
  \;=\; 5.9510191304042890397\ldots,
\end{equation}
which is close to but strictly less than $\Js/2 = 6$.  More precisely:
\begin{equation}\label{eq:deficit}
  \frac{\Js}{2} - (j(i) - 4\Omega) \;=\; 6 - (1728 - 4\pi\alpha^{-1})
  \;=\; 4\pi\alpha^{-1} - 1722
  \;=\; 0.04898086959571096\ldots
\end{equation}
This deficit is strictly positive and transcendental.
\end{proposition}

\begin{proof}
Compute $j(i) - 4\Omega = 1728 - 4\pi(4\pi^3 + \pi^2 + \pi)
= 1728 - 16\pi^4 - 4\pi^3 - 4\pi^2$.  To 55 significant figures,
using \texttt{verify\_G1\_modular.py} with \texttt{mp.dps = 60}:
\[
  j(i) - 4\Omega \;=\; 5.95101913040428903971145271280803465420267734452966\ldots
\]
The claim $j(i) - 4\Omega < \Js/2 = 6$ is equivalent to
$4\pi\alpha^{-1} > 1722$.  Numerically $4\pi\alpha^{-1} \approx 1722.049$,
confirming the strict inequality.

For non-exactness: the claim $j(i) - 4\Omega = 6$ would require
$4\pi\alpha^{-1} = 1722$, i.e., $\alpha^{-1} = 1722/(4\pi)
\approx 137.028$.  But the TOE constant is $\alpha^{-1} = 4\pi^3 + \pi^2
+ \pi \approx 137.036$; these are unequal because $4\pi^3 + \pi^2 + \pi
\ne 1722/(4\pi)$ is equivalent to $16\pi^4 + 4\pi^3 + 4\pi^2 \ne 1722$,
which holds since the left side is transcendental and $1722 \in \ZZ$.

The deficit $4\pi\alpha^{-1} - 1722 = 16\pi^4 + 4\pi^3 + 4\pi^2 - 1722$
is transcendental because it is a nonconstant polynomial in $\pi$ with
rational coefficients minus a rational, and $\pi$ is transcendental.
\end{proof}

\begin{remark}
The near-miss $j(i) - 4\Omega \approx 6 = \Js/2$ is one more instance of
the pattern of near-integer or near-$\Js$ values that runs through the
TOE addenda corpus.  The deficit $\approx 0.04898$ is approximately
$\pi^2/200 \approx 0.04935$, though no closed form in obvious TOE
quantities appears to be exact.  This near-miss is purely observational;
it is not a theorem.
\end{remark}

\subsection{Exact 55-digit value}

For the record, to 55 significant figures (verified by
\texttt{verify\_G1\_modular.py}):

\begin{align}
  \Omega &= 430.5122452173989277400721368217979913364493306638675851\ldots
  \label{eq:Omega-digits}\\
  \Gone &= 0.003455778085591479075380623365743477970730119293270778\ldots
  \label{eq:G1-digits}\\
  j(i) - 4\Omega &= 5.951019130404289039711452712808034654202677344529660\ldots
  \label{eq:gap-digits}
\end{align}

%==========================================================================
\section{The polynomial identity and $S^2$ geometry}\label{sec:polynomial}
%==========================================================================

\begin{proposition}[Polynomial--geometric identity]\label{prop:polynomial}
The following identities hold:
\begin{equation}\label{eq:poly-id}
  4\Omega \;=\; 4\pi\alpha^{-1} \;=\; 16\pi^4 + 4\pi^3 + 4\pi^2,
\end{equation}
and
\begin{equation}\label{eq:S2-id}
  4\Omega \;=\; \mathrm{vol}(S^2) \cdot \alpha^{-1},
\end{equation}
where $\mathrm{vol}(S^2) = 4\pi$ is the surface area of the unit sphere
in $\RR^3$.
\end{proposition}

\begin{proof}
Substituting $\alpha^{-1} = 4\pi^3 + \pi^2 + \pi$ into $4\pi\alpha^{-1}$:
\[
  4\pi(4\pi^3 + \pi^2 + \pi) \;=\; 16\pi^4 + 4\pi^3 + 4\pi^2.
\]
The identity $\mathrm{vol}(S^2) = 4\pi$ is the standard formula for the
surface area of the unit 2-sphere.  Hence $4\pi\alpha^{-1} =
\mathrm{vol}(S^2) \cdot \alpha^{-1}$.
\end{proof}

\begin{remark}
The geometric reading of~\eqref{eq:S2-id} is as follows.  The TOE is
formulated on $S^3$, the 3-sphere.  The quantity $\alpha^{-1}$ is a
dimensionless coupling derived from $S^3$ geometry.  The breath period
$\Omega = \pi\alpha^{-1}$ has $\pi$ as the ratio of the circumference to
the diameter; $4\pi = \mathrm{vol}(S^2)$ is the area of the equatorial
2-sphere inside $S^3$.  The identity $4\Omega = \mathrm{vol}(S^2) \cdot
\alpha^{-1}$ therefore says that the $4\Omega$ slot — the modular
reference value $j(i)$ displaced by $G_1$ from $4\Omega$ — is the product
of the $S^2$ volume and the fine-structure denominator.  Whether this has
a direct interpretation in the $S^3$ Hopf fibration (with $S^2$ as the
base and $S^1$ as the fibre) is left as an open direction.
\end{remark}

%==========================================================================
\section{$G_1$ as a modular residual}\label{sec:modular}
%==========================================================================

\subsection{The CM point $\tau = i$ and its $j$-value}

Recall the basic facts about the CM point $\tau = i$, established in
P156--P159.  The discriminant of $\QQ(i)/\QQ$ is $D = -4$.  The
$j$-function at this point takes the value
\begin{equation}\label{eq:j-i}
  j(i) \;=\; 1728 \;\in\; \ZZ,
\end{equation}
an integer — this is the defining property of CM points: $j$ is an
algebraic integer at every CM point.  The value $j(i) = 1728$ is the
unique value associated to the lattice $\ZZ[i]$ (the Gaussian integers),
and it is the smallest CM value in absolute terms (for imaginary quadratic
fields of small discriminant).

\subsection{The modular residual interpretation}

\begin{proposition}[$\Gone$ as modular residual at $\tau = i$]
\label{prop:modular-residual}
The $G_1$ gap is the fractional deviation of the TOE breath period from
the CM modular slot $j(i)/4 = 432$:
\begin{equation}\label{eq:modular-residual}
  \Gone \;=\; \frac{j(i)/4 - \Omega}{\Omega}
  \;=\; \frac{j(i)}{4\Omega} - 1.
\end{equation}
No modular form $f$ for $\mathrm{SL}(2,\ZZ)$ evaluates to $4\Omega$ at
any CM point $\tau_0$ of discriminant $D = -4$ (i.e., at $\tau_0 = i$).
Equivalently, $4\Omega$ does not lie in the ring of special values
$\{j(\tau_0)^{k/n} : k, n \in \ZZ_{>0}\}$ computed at $\tau_0 = i$.
\end{proposition}

\begin{proof}
The first statement is Theorem~\ref{thm:G1-j}.

For the second statement: any CM value $f(i)$ of a modular form with
algebraic Fourier coefficients is an algebraic number (by the CM theory
of modular forms, see Shimura~\cite{shimura} or Silverman~\cite{silverman}).
In particular, $j(i)^{k/n} = 1728^{k/n}$ is algebraic for all
$k, n \in \ZZ_{>0}$.  However, $4\Omega = 4\pi\alpha^{-1} =
16\pi^4 + 4\pi^3 + 4\pi^2$ is transcendental: it is a nonconstant
polynomial in $\pi$ with rational coefficients, and $\pi$ is
transcendental by Lindemann--Weierstrass.  A transcendental number
cannot equal an algebraic number.  Hence $4\Omega$ lies outside the
field of CM values at $\tau = i$, and $\Gone = j(i)/(4\Omega) - 1$
is the residual measuring this gap.
\end{proof}

\begin{remark}
This is the precise sense in which $\Gone$ is a \emph{modular residual}:
it is not the value of any modular invariant at the relevant CM point,
and it cannot be made zero by any algebraic adjustment to the lattice
$\ZZ[i]$.  The gap between $\Omega$ (a transcendental geometric period)
and $j(i)/4$ (an integer CM value) is irreducible in the same sense
as a period of an elliptic curve is transcendental while the $j$-invariant
of that curve is algebraic.  The present addendum establishes that $\Gone$
occupies precisely this interface.
\end{remark}

%==========================================================================
\section{Transcendence of $G_1$}\label{sec:transcendence}
%==========================================================================

\begin{corollary}[$\Gone$ is transcendental]\label{cor:transcendental}
$\Gone$ is a transcendental number.
\end{corollary}

\begin{proof}
By~\eqref{eq:G1-j}, $\Gone = j(i)/(4\Omega) - 1 = 1728/(4\Omega) - 1
= 432/\Omega - 1$.  Since $432 \in \QQ$ and $\Omega$ is transcendental
(Lindemann--Weierstrass; $\Omega = \pi\alpha^{-1}$, a product of the
transcendental $\pi$ and the nonzero algebraic $\alpha^{-1}$... but wait:
$\alpha^{-1} = 4\pi^3 + \pi^2 + \pi$ is itself transcendental, being
a nonconstant polynomial in $\pi$), $\Omega$ is transcendental.  The
ratio $432/\Omega$ of a nonzero rational by a transcendental is
transcendental.  Subtracting the integer $1$ preserves transcendence.
Hence $\Gone = 432/\Omega - 1$ is transcendental.
\end{proof}

\begin{remark}
The transcendence of $\Gone$ is not surprising given that $\Omega$ is
transcendental, but it has a conceptual consequence: $\Gone$ will never
appear as a root of a polynomial with rational (or even algebraic)
coefficients.  In particular, it cannot be a rational number, cannot be
$1/n$ for any integer $n$, and cannot be expressed in radicals or as
a CM value.  The TOE framework treats $\Gone$ as a structural constant
of the same transcendental type as $\pi$: it is not a fitting parameter
but a definite mathematical number whose value is forced by the TOE geometry.
\end{remark}

%==========================================================================
\section{Connection to Ramanujan tau (P165)}\label{sec:tau}
%==========================================================================

\begin{remark}[Ramanujan tau and $432$]\label{rem:tau}
Addendum~165 (P165) observed that $\tau(3) = 252 = 7 \times 36 = 7 \times
\BF$, where $\tau(n)$ denotes the Ramanujan tau function and $\BF = B_{F_4}
= 36$ is the $F_4$ branching factor of P165.  In the present context,
$252 = 432 - 180 = 432 - 4 \times 45$, and separately
\begin{equation}\label{eq:tau3-Js}
  \frac{\tau(3)}{j(i)/4} \;=\; \frac{252}{432} \;=\; \frac{7}{12}
  \;=\; \frac{7}{\Js}.
\end{equation}
The ratio $\tau(3)/432 = 7/12$ expresses the third Ramanujan coefficient
as a rational multiple of $\Js^{-1}$ of $432 = j(i)/4$.  We record
this as an observation: whether the factor $7$ admits a root-system or
moonshine interpretation complementary to $\Js = 12$ in this ratio is
open.  No proof is claimed here beyond the numerical verification.
\end{remark}

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

\subsection{Main results}

The central identity of this addendum is:

\begin{equation}\label{eq:main}
\boxed{
  \Gone \;=\; \frac{j(i)}{4\Omega} - 1
  \;=\; \frac{\Js^3}{4\pi\alpha^{-1}} - 1,
  \qquad
  \Omega \;=\; \frac{j(i)}{4(1+\Gone)}.
}
\end{equation}

The supporting results are:
\begin{align}
  j(i) - 4\Omega &= 5.951019\ldots < 6 = \tfrac{\Js}{2}
  \quad\text{(near-miss, strictly non-exact),}
  \label{eq:summary-near-miss}\\
  4\Omega &= (4\pi)\alpha^{-1} = \mathrm{vol}(S^2)\cdot\alpha^{-1}
  \quad\text{(sphere-volume identity),}
  \label{eq:summary-sphere}\\
  \Gone &\notin \overline{\QQ}
  \quad\text{(transcendental).}
  \label{eq:summary-trans}
\end{align}

\noindent Together these establish that $\Gone$ is the irreducible modular
residual at $\tau = i$: it measures the gap between the TOE's geometric
period $\Omega$ and the CM modular slot $j(i)/4$, and this gap is
transcendental and unreachable by any modular or algebraic construction.

\subsection{Open directions}

\begin{enumerate}
  \item \emph{The near-miss deficit $\approx 0.04898$.}  The deficit
    $6 - (j(i) - 4\Omega) = 4\pi\alpha^{-1} - 1722 \approx 0.04898$
    has no obvious closed form in TOE quantities.  Whether it is a
    rational linear combination of powers of $\pi$ with a combinatorial
    interpretation (analogous to the way $\Omega$ itself is $\pi\alpha^{-1}$
    with $\alpha^{-1}$ a cubic in $\pi$) is an open question.

  \item \emph{$S^3$ Hopf fibration interpretation of $4\Omega = \mathrm{vol}(S^2)
    \cdot \alpha^{-1}$.}  The $S^3$ geometry of the TOE carries a Hopf
    fibration with $S^2$ base and $S^1$ fibre.  The identity $4\Omega
    = \mathrm{vol}(S^2)\cdot\alpha^{-1}$ places $4\Omega$ on the base sphere.
    Whether the Hopf lapse function $m = \sqrt{1 - u^2}$ (GON, Paper~30)
    integrates to $4\Omega$ over the $S^2$ base in any natural sense is
    an open geometric question.

  \item \emph{Higher CM points.}  The present addendum focuses on $\tau = i$
    (discriminant $D = -4$).  The CM points of other discriminants give
    different algebraic values of $j$.  For example, $j(e^{2\pi i/3})
    = 0$ (discriminant $D = -3$), and $j((1+i\sqrt{7})/2) = -3375$
    (discriminant $D = -7$).  Whether the breath period $\Omega$ stands in
    a similarly structured near-miss relationship to quarter-$j$ values at
    other CM points is not known.

  \item \emph{Tau-function connection.}  Remark~\ref{rem:tau} records that
    $\tau(3)/432 = 7/\Js$.  This ratio connects the Ramanujan tau function
    at prime $3$ to the modular reference value $432 = j(i)/4$.  A structural
    explanation --- possibly via the McKay--Thompson series for the Monster
    group evaluated at $\tau = i$ (P163) --- is an open direction.

  \item \emph{Period integrals.}  In classical CM theory, the period of an
    elliptic curve $E$ with CM by $\ZZ[i]$ is $\omega = \Gamma(1/4)^2/(4\sqrt{\pi})$,
    which is transcendental and unrelated to $\Omega$ by any known algebraic
    identity.  Whether the ratio $\omega/\Omega$ carries a number-theoretic
    meaning is open.
\end{enumerate}

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\bibitem{silverman}
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\bibitem{serre}
J.-P.\ Serre, \emph{A Course in Arithmetic}, Springer, 1973.

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

\bibitem{borwein}
J.~M.\ Borwein and P.~B.\ Borwein, \emph{Pi and the AGM}, Wiley, 1987.

\bibitem{gelfond}
A.~O.\ Gel'fond, \emph{Transcendental and Algebraic Numbers},
Dover, 1960.
\end{thebibliography}

\end{document}
