% Addendum 159: Closure of P158-OP1 — Why E₄(i)/η(i)^8 = J_short
%   P158 left open whether E₄(i)/η(i)^8 = 12 = J_short has a structural
%   explanation independent of j(i) = J_short³. This addendum shows it is a
%   direct algebraic consequence of the j-function definition combined with
%   P158's result. No independent argument is needed.
% L. F. Vlegels, Independent Researcher
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. May 2026.

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\title{Closure of P158-OP1: Why $E_4(i)/\eta(i)^8 = \Js$\\[0.5em]
\large Addendum 159 to the TOE Corpus}

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

\date{May 2026}

\begin{document}
\maketitle

\begin{abstract}
Addendum P158 derived $j(i) = B_{G_2}(H_s,H_s)\times B_{F_4}(H_s,H_s)
= \Js^3 = 12^3 = 1728$ from Killing-form root combinatorics,
and noted as open (P158-OP1) whether $E_4(i)/\eta(i)^8 = 12 = \Js$
has a structural explanation that connects modular forms directly to
Killing-form root-sector sums.
This addendum closes P158-OP1 by observing that the identity is an
immediate algebraic consequence of the $j$-function definition
combined with P158's result.
No independent representation-theoretic argument is required.
All claims are verified numerically by \texttt{verify\_P159.py}.
\end{abstract}

%==========================================================================
\section{Setup: P158-OP1 and the $j$-Function Definition}\label{sec:setup}

Recall from P158 the main result:
\begin{equation}\label{eq:P158}
  j(i) \;=\; B_{G_2}(H_s,H_s)\times B_{F_4}(H_s,H_s) \;=\; \Js^3 \;=\; 12^3 \;=\; 1728.
\end{equation}
P158 also recorded the classical modular identity
$E_4(i)/\eta(i)^8 = 12 = \Js$ and asked (P158-OP1):

\medskip
\noindent\textit{Does there exist a representation-theoretic or geometric argument that
directly identifies $E_4(\tau)$ at a CM point with a Killing-form sector sum,
independent of the route through $j(\tau)=\Js^3$?}

\medskip
\noindent The answer is no: the identity is not independent of that route.
It \emph{is} that route, expressed in different notation.

%==========================================================================
\section{Resolution}\label{sec:resolution}

The $j$-function is defined by
\begin{equation}\label{eq:jdef}
  j(\tau) \;=\; \frac{E_4(\tau)^3}{\Delta(\tau)},
  \qquad
  \Delta(\tau) \;=\; \eta(\tau)^{24}.
\end{equation}

Set $x = E_4(i)/\eta(i)^8$.  Then $E_4(i) = x\cdot\eta(i)^8$, so:
\begin{equation}\label{eq:expand}
  j(i)
  \;=\; \frac{\bigl(x\cdot\eta(i)^8\bigr)^3}{\eta(i)^{24}}
  \;=\; \frac{x^3\cdot\eta(i)^{24}}{\eta(i)^{24}}
  \;=\; x^3.
\end{equation}

\begin{theorem}\label{thm:main}
$E_4(i)/\eta(i)^8 \;=\; j(i)^{1/3}$.
\end{theorem}

\begin{proof}
Equation~\eqref{eq:expand} gives $j(i) = x^3$, so $x = j(i)^{1/3}$.
\end{proof}

\begin{corollary}[Closure of P158-OP1]\label{cor:closure}
$E_4(i)/\eta(i)^8 \;=\; \Js \;=\; 12$.
\end{corollary}

\begin{proof}
By Theorem~\ref{thm:main} and \eqref{eq:P158}:
\[
  \frac{E_4(i)}{\eta(i)^8}
  \;=\; j(i)^{1/3}
  \;=\; \bigl(\Js^3\bigr)^{1/3}
  \;=\; \Js
  \;=\; 12. \qedhere
\]
\end{proof}

%==========================================================================
\section{The Full Chain}\label{sec:chain}

Assembling P158 and Corollary~\ref{cor:closure}:

\begin{equation}\label{eq:chain}
\boxed{
\begin{aligned}
  &\text{$G_2/F_4$ root geometry} \\[4pt]
  &\quad\Longrightarrow\quad
  \Js = 12 \\[4pt]
  &\quad\Longrightarrow\quad
  j(i) \;=\; \Js^3 \;=\; 1728
  \hspace{3em}\text{[P158]}\\[4pt]
  &\quad\Longrightarrow\quad
  \frac{E_4(i)}{\eta(i)^8}
  \;=\; j(i)^{1/3} \;=\; \Js \;=\; 12
  \hspace{1em}\text{[P159]}
\end{aligned}
}
\end{equation}

The 12 in $E_4(i)/\eta(i)^8$ is the same 12 as $\Js$ for a derivable reason:
the $j$-function definition makes $E_4(\tau)/\eta(\tau)^8$ the cube root of
$j(\tau)$ at every $\tau$ where $\Delta(\tau)\neq 0$,
and the Killing-form root geometry places $j(i)$ at a perfect cube.

%==========================================================================
\section{Significance}\label{sec:significance}

\begin{remark}
P158-OP1 asked whether a \emph{direct} representation-theoretic link exists
between $E_4(i)$ and Killing-form sector sums, bypassing $j(\tau)=\Js^3$.
Corollary~\ref{cor:closure} shows no such bypass is needed or available:
the identity $E_4(i)/\eta(i)^8 = \Js$ is already fully explained by the
chain \eqref{eq:chain}.
The $j$-function is the structural bridge between Lie root combinatorics
and modular forms at this CM point.
Searching for an additional, independent identification would be
searching for a second proof of the same equality — possible in principle,
but not required for a closed account.
\end{remark}

\begin{remark}
The route through $j$ is not a coincidence of numbers.
It is a consequence of the definition of $j$ as the cube of
$E_4/\eta^8$, which is the natural weight-0 modular function
built from the two basic modular objects at weight 4 and weight $1/2$.
That the $G_2/F_4$ Killing-form product places $j(i)$ at a perfect cube
is the non-trivial fact; once it is established (P158), the rest follows
from definitions.
\end{remark}

%==========================================================================
\section*{Verification Script Output}

\texttt{Lumen/corpus/addenda/verify/verify\_P159.py} (run 2026-05-15):

\begin{verbatim}
j(i) computed = 1728.000000  (expect 1728)
E4(i)/eta(i)^8 = 12.0000000000  (expect 12)
J_short = 12
j(i)^(1/3) = 12.0000000000  (expect 12)
All assertions PASS

Chain:
  j(tau) = E4(tau)^3 / eta(tau)^24  [definition]
  j(i) = 1728 = J_short^3 = 12^3  [P158]
  E4(i)/eta(i)^8 = j(i)^{1/3} = J_short = 12  [P159, QED]
\end{verbatim}

%==========================================================================
\section*{Summary}

P158-OP1 is closed.
The identity $E_4(i)/\eta(i)^8 = \Js = 12$ is not an independent
coincidence: it is a direct consequence of the $j$-function definition
$j = E_4^3/\eta^{24}$ combined with P158's structural result $j(i) = \Js^3$.
Substituting and cancelling $\eta^{24}$ gives $j(i) = x^3$ where
$x = E_4(i)/\eta(i)^8$, so $x = j(i)^{1/3} = \Js$.
The $j$-function is the bridge; the $12$ appearing in modular forms and
in Lie root combinatorics is the same $12$ for this reason.~$\square$

\bigskip
\noindent\textit{Copyright: L\'eon Fernando Vlegels. License: MIT. May 2026.}

\end{document}
