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\title{\textbf{Addendum P157}\\[4pt]
       Modular Forms at CM Points:\\
       A Systematic Search for TOE Constant Structure}
\author{L\'eon Fernando Vlegels}
\date{2026-05-15}

\begin{document}
\maketitle

\section{Motivation}

Addendum P153 established
\[
  B_{F_4}(H_s, H_s) \;\times\; B_{G_2}(H_s, H_s)
  \;=\; 36 \times 48 \;=\; 1728 \;=\; 12^3,
\]
where $B_{F_4}$ and $B_{G_2}$ are the Killing forms of $F_4$ and $G_2$
evaluated on the distinguished element $H_s$ of the principal $\mathfrak{sl}_2$.
The observation was not anticipated; 1728 was derived, not sought.

Classical complex multiplication theory gives $j(i) = 1728$ at the CM point
$\tau = i$ (discriminant $D = -4$, ring of integers $\mathbb{Z}[i]$).
These two routes to the same integer are independent: one is Lie-theoretic,
the other number-theoretic.  The question for this addendum is whether this
contact at $D = 4$ is the tip of a systematic structure — TOE constants
appearing broadly in modular forms at CM points — or an isolated instance.

TOE constants under test (tolerance $< 0.5\%$):
\[
  \alpha^{-1} \approx 137.036,\quad
  \Omega \;=\; \pi\alpha^{-1} \approx 430.512,\quad
  \mathcal{P} \;=\; 432,\quad
  \mathcal{C} \;=\; 1728,\quad
  J_s \;=\; 12,\quad
  J_\ell \;=\; 36,\quad
  B_{G_2} \;=\; 48.
\]

\section{Systematic CM Table: $j(\tau_D)$ for $D = 3 \ldots 163$}

Let $\tau_D$ denote the CM point of fundamental discriminant $-D$.  The table
below lists the exact classical value $j(\tau_D)$, verifies it via
$q$-expansion to 60 decimal places (using $j = E_4^3/\Delta$), and records
any near-match ($< 0.5\%$) to a TOE constant.

\begin{center}
\renewcommand{\arraystretch}{1.15}
\begin{tabular}{r r l}
\toprule
$D$ & $j(\tau_D)$ (exact) & TOE hit \\
\midrule
  3 & $0$                          & — \\
  4 & $1728$                       & $|j| = \mathcal{C}$;\; $|j|^{1/3} = J_s$;\; $|j|/4 = \mathcal{P}$ \\
  7 & $-3375$                      & — \\
  8 & $8000$                       & — \\
 11 & $-32768$                     & — \\
 12 & $54000$                      & — \\
 16 & $287496$                     & — \\
 19 & $-884736$                    & — \\
 27 & $-12288000$                  & — \\
 28 & $16581375$                   & — \\
 43 & $-884736000$                 & — \\
 67 & $-147197952000$              & — \\
163 & $-262537412640768000$        & — \\
\bottomrule
\end{tabular}
\end{center}

Verification output (abridged):
\begin{Verbatim}[fontsize=\small]
   D     j(tau_D) exact  q-exp verified  TOE hits
   3              0       OK  —
   4           1728       OK  |j|/4=BREATH; |j|^(1/3)=J_SHORT; |j|/4=PRODUCT; |j|=CUBE
   7          -3375       OK  —
   8           8000       OK  —
  11         -32768       OK  —
  12          54000       OK  —
  16         287496       OK  —
  19        -884736       OK  —
  27      -12288000       OK  —
  28       16581375       OK  —
  43     -884736000       OK  —
  67  -147197952000       OK  —
 163  -2.625e+17          OK  —

  |j(tau_27)|^(1/3) = 230.759931  (not a TOE constant)
  |j(tau_43)|^(1/3) = 960.000000  (= 960; 960/alpha_inv = 7.005)
  |j(tau_163)|^(1/3) = 640320.000000
  Only D=4 yields j(i)=1728=12^3 with 12=J_short
\end{Verbatim}

\noindent \textbf{Finding.} $D = 4$ is the unique discriminant in the full Heegner set
where $|j(\tau_D)|$ or $|j(\tau_D)|^{1/3}$ falls within $0.5\%$ of a TOE
constant.  The cube root $|j(\tau_{43})|^{1/3} = 960$ is exact, but
$960/\alpha^{-1} \approx 7.005$ — not a structurally meaningful ratio.

\section{Eta and Eisenstein Series at $\tau = i$}

\subsection{Verification of known identities}

At $\tau = i$, the $q$-expansion computations give (60 d.p.):

\begin{Verbatim}[fontsize=\small]
eta(i) via q-product    = 0.76822542232605661727
eta(i) = Gamma(1/4)/(2 pi^{3/4}) = 0.76822542232605661727
Match (60 dps):           True

E4(i)                   = 1.45576289226870936311
E6(i)                   = 1.03e-48  (-> 0 by symmetry of CM(i))
j(i) = E4^3/Delta       = 1728.000000  (exact: 1728)

E4(i) known 3*Gamma(1/4)^8/(2*pi)^6 = 1.45576289226870936311
Match:  True
\end{Verbatim}

The vanishing of $E_6(i)$ is classical: $\tau = i$ has automorphism of order 4
in $\mathrm{SL}_2(\mathbb{Z})$, forcing $E_6(i) = 0$.

\subsection{Key finding: $E_4(i)/\eta(i)^8 = 12$ exactly}

\begin{Verbatim}[fontsize=\small]
*** E4(i) / eta(i)^8 = 12.0000000000  ***
    This equals J_SHORT = 12 exactly.
    Proof: eta(i)^8 = Gamma(1/4)^8 / (2^8 * pi^6)
           E4(i)    = 3 * Gamma(1/4)^8 / (2*pi)^6
           E4(i)/eta(i)^8 = 3/(2*pi)^6 * (2^8 * pi^6) = 3*4 = 12
\end{Verbatim}

\noindent \textbf{Identity.}
\[
  \frac{E_4(i)}{\eta(i)^8} \;=\; 12 \;=\; J_s.
\]
This is a provable classical identity, derivable from the exact formulae
$\eta(i) = \Gamma(1/4)/(2\pi^{3/4})$ and
$E_4(i) = 3\Gamma(1/4)^8/(2\pi)^6$.  The computation gives $J_s = 12$
by a third independent route: it already appears as $J_{\mathrm{short}}$
(P151) and as $j(i)^{1/3}$ (P153).

No other ratio of $E_4(i)$, $E_8(i) = E_4(i)^2$, or $E_{12}(i)$ to any
power of $\pi$, $\Gamma(1/4)$, or $\alpha^{-1}$ falls within $0.5\%$ of a
TOE constant.

\section{Ramanujan Near-Integer Search}

The Heegner phenomenon: $e^{\pi\sqrt{163}} \approx 640320^3 + 744$
(Ramanujan, proved via $j(\tau_{163}) = -640320^3$).
We search $D \in [1, 300]$ for
\[
  e^{\pi\sqrt{D}} \approx n \cdot \alpha^{-k}, \quad n \in \mathbb{Z},\; k \in \{-3,-2,-1,1,2,3\},
\]
with threshold $|\text{ratio} - n| < 0.01$.

\textbf{Precision note.} For $D \geq 43$ and $k = 3$, the ratio
$e^{\pi\sqrt{D}}/\alpha^{-3}$ exceeds $10^{14}$, exhausting \texttt{float64}
fractional precision.  All $D \geq 43$ hits with error $\approx 0$ are
precision artefacts; only $D \leq 40$ results are numerically meaningful.

\begin{Verbatim}[fontsize=\small]
D <=40 hits (float64-reliable regime):
  D= 2: e^{pi*sqrt(2)} ~= 218789488 x alpha^3,  error=0.009429
  D= 5: e^{pi*sqrt(5)} ~= 21111036 x alpha^2,   error=0.005674
  D=11: e^{pi*sqrt(11)} ~= 4591558 x alpha^1,   error=0.000505
  D=21: e^{pi*sqrt(21)} ~= 245015497 x alpha^1, error=0.002836
  D=29: e^{pi*sqrt(29)} ~= 1185 x alpha^{-2},   error=0.005190
  D=30: e^{pi*sqrt(30)} ~= 216851 x alpha^{-1}, error=0.003012
  D=33: e^{pi*sqrt(33)} ~= 3665 x alpha^{-2},   error=0.001578

Hits in [1,40]: 7  (expected by chance: ~4.8)

D = 43 (Heegner): e^{pi*sqrt(43)} = 884736743.9998
  |j(tau_43)| + 744 = 884736744
  e^{pi*sqrt(43)} / alpha_inv^3 = 343.801016  (not near integer)
\end{Verbatim}

\noindent \textbf{Finding.}  Seven hits in $[1,40]$ against an expectation of
$\sim 4.8$ from equidistribution.  The slight excess is unsurprising at this
sample size.  None of the near-integers $n$ has structural significance in
the TOE.  The Heegner discriminant $D = 43$ does not produce a near-integer
ratio to $\alpha^{-k}$ for any small $k$.

\section{Depth of the 1728 Connection}

\begin{Verbatim}[fontsize=\small]
Exact integer arithmetic:
  1728 = 12^3                    True
  1728 = B_F4 x B_G2 = 36x48    True
  1728 = j(i)                    classical CM theorem
  1728 / 4 = 432 = PRODUCT       True
  1728^(1/3) = 12 = J_SHORT      True
  12 appears: J_SHORT, cbrt(j(i)), E4(i)/eta(i)^8 -- three independent routes

BREATH vs j(i)/4:
  j(i)/4 = 432.000000 (exact integer = J_short x J_long)
  BREATH = pi * alpha_inv = 430.512245217399
  j(i)/4 - BREATH = G1_abs = 1.487754782601  (P156, the known unknown)
  relative gap     = G1_rel = 0.003455778086  (0.3456%)
\end{Verbatim}

The structural picture at $\tau = i$ is:
\begin{align*}
  j(i) &= 1728 = 12^3 = B_{F_4} \times B_{G_2} \\
  j(i)^{1/3} &= 12 = J_s \\
  j(i)/4 &= 432 = \mathcal{P} = J_s \times J_\ell \\
  E_4(i)/\eta(i)^8 &= 12 = J_s \quad\text{(provable identity)} \\
  E_6(i) &= 0 \quad\text{(CM symmetry at $\tau = i$)} \\
  \Omega = \pi\alpha^{-1} &= 430.512\ldots \neq 432\;\; (\text{gap} = G_1)
\end{align*}

All roads through $\tau = i$ lead back to $12 = J_s$.  The integer 432 is
$j(i)/4$; the breath period $\Omega$ misses it by $G_1$ (the known unknown
of P156, $\approx 0.35\%$).  The chain is $j(i) \to 12 \to 432$ via exact
arithmetic; $\Omega$ is not part of it.

\section{Verdict}

\paragraph{What was found.}
\begin{enumerate}
  \item $j(i) = 1728 = B_{F_4} \times B_{G_2}$: a genuine, exact coincidence.
  Both sides equal 1728 by independent computation.

  \item $E_4(i)/\eta(i)^8 = 12 = J_s$: a classical, provable identity.
  The same $12$ that is $J_{\mathrm{short}}$ in P151 and $j(i)^{1/3}$ in P153.
  One underlying arithmetic fact ($j(i) = 12^3$), three appearances of~12.

  \item $j(i)/4 = 432 = \mathcal{P} = J_s \times J_\ell$: exact integers,
  no transcendental input.  The breath period $\Omega$ differs by $G_1$.
\end{enumerate}

\paragraph{What was not found.}
\begin{itemize}
  \item No CM discriminant other than $D = 4$ yields a $j$-value within
  $0.5\%$ of any TOE constant.
  \item No Eisenstein or eta combination at $\tau = i$ produces $\alpha^{-1}$,
  $\Omega$, $\phi$, or $\sin^2\theta_W$ within $0.5\%$.
  \item The Ramanujan scan ($D \leq 40$, reliable precision) produces hits
  statistically consistent with equidistribution; no structurally motivated
  near-integer emerges.
\end{itemize}

\paragraph{Conclusion.}
$j(i) = 1728$ is an isolated structural contact between CM theory and
TOE root-system geometry.  The contact is real and exact; it is not
coincidence in the pejorative sense — both computations are forced.  But it
does not extend.  The evidence does not support a hypothesis of systematic TOE
constant structure in modular forms.  The observation belongs with P153 as a
boundary condition: something the correct unified theory must eventually
account for, but not yet a derivation.

\vspace{2em}
\noindent\texttt{[ged-traversal: query="modular forms CM points j-invariant alpha TOE constants"
nodes="j-invariant CM point; Killing form root system; G1 known unknown; Eisenstein series"
score=0.41]}

\end{document}
