% Addendum 338: J3(O) -> Monster bridge FEASIBILITY pass under a hard kill-criterion
%   Candidate object (stated FIRST): J3(O) is 27-dim; Str_0(J3(O))=E6, so the
%   concrete number is dim 27 (E6 fundamental). The chain reaches the Monster ONLY
%   via the Leech (A169); the corpus's Monster-adjacent Leech number is 196560
%   (minimal vectors). Single standard target: Monster head-rep dim 196883.
%   MATCH: 196560 != 196883; gap 323 = 17*19 is a bespoke bridge constant -> the
%   A177 failure mode -> DISQUALIFIED. No corpus J3(O)-chain number (27, 78, 1728,
%   196560) equals a Monster head-dim. The only J3(O)->modular function, j^(1/3),
%   is the E8/G2 cube root (A337), not a hauptmodul. VERDICT: NO unforced standard
%   contact; coincidence, left to mainstream moonshine. Stopped at first failure.
%   Outcome: FEASIBILITY. Verifier: verify/verify_P338.py (16 PASS / 0 FAIL).
% TOE Corpus
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. June 2026.

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\newcommand{\Js}{J_{\mathrm{short}}}
\newcommand{\OO}{\mathbb{O}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\Mon}{\mathbb{M}}
\newcommand{\Leech}{\Lambda_{24}}
\newcommand{\JO}{J_3(\OO)}

\begin{document}

\title{Addendum 338 --- $J_3(\OO) \to$ Monster Bridge Feasibility:\\
  A Single Grounding Pass Under a Hard Kill-Criterion\\
  \large (No Unforced Standard Contact; Coincidence Left to Moonshine)}
\author{TOE Corpus\\
  \small Copyright: L\'eon Fernando Vlegels. License: CC BY 4.0.}
\date{June 2026}
\maketitle

\noindent\fbox{\parbox{\textwidth}{\small
\textbf{Outcome:} FEASIBILITY (single pass, no fan-out).\quad
\textbf{Addresses:} whether the corpus's existing $\JO$ (exceptional Jordan
algebra) structure lands on a \emph{standard} monstrous-moonshine object --- a
canonical McKay--Thompson hauptmodul or a genuine Monster head-representation
dimension --- \emph{unforced} (no free parameter, no bespoke function), governed
by a pre-committed kill-criterion.\quad
\textbf{Verifier:} \texttt{Lumen/corpus/addenda/verify/verify\_P338.py}
(16 checks, 16 PASS / 0 FAIL, recomputed from scratch with \texttt{mpmath} at
\texttt{dps}$=40$).\quad
\textbf{Verdict:} \textsc{No unforced standard contact}. The contact stops at the
first honest check; the nearest target requires a bespoke bridge constant
($323 = 17\cdot 19$), the A177 failure mode. Recorded as a coincidence, left to
mainstream moonshine (FLM / Borcherds 1992). \emph{No new TOE bridge is earned.}}}

\begin{abstract}
This addendum runs one grounding-plus-feasibility pass on a deliberately
speculative question: does the exceptional Jordan algebra $\JO$ already present in
the corpus reach a \emph{standard} monstrous-moonshine object without forcing? The
pass is governed by a hard, pre-committed kill-criterion, written precisely because
this is the crank-magnet zone (exceptional algebra $\leftrightarrow$ Monster) where
a connection is always forceable. We first state, before any value-matching, what
$\JO$ concretely produces in the corpus: it is the $27$-dimensional exceptional
Jordan algebra whose reduced structure group is $E_6$ (A166, Prop.~1.1), so the
concrete number is $\dim \JO = 27$, the dimension of the fundamental $\mathbf{27}$
of $E_6$; and the only place the $\JO$ chain touches the Monster is through the
Leech lattice $\Leech$ (A169), whose Monster-adjacent number is the minimal-vector
count $196560$. We then choose, once, the single most plausible standard target:
the smallest faithful Monster head-representation dimension $196883$. The match is
run once: $196560 \neq 196883$, off by $323 = 17\cdot 19$ --- a non-zero bespoke
bridge constant, which is exactly the failure mode A337 retracted (a fitted
quantity standing in for a missing identity). No corpus $\JO$-chain number
($27, 78, 1728, 196560$) equals any Monster head-representation dimension, and the
only $\JO\to$modular function the corpus uses, $j^{1/3} = E_4/\eta^8$, was proved
by A337 to be the $E_8/G_2$ cube-root function, not a Monster hauptmodul. We stop
at the first honest failure, as the kill-criterion requires. Verdict: no unforced
standard contact; the $196560 \sim 196883$ adjacency is the textbook
Leech/Monster near-miss, fully explained by mainstream moonshine, and the corpus
has \emph{not} earned a $\JO$-bridge to it.
\end{abstract}

%==========================================================================
\section{The kill-criterion (pre-committed)}\label{sec:kill}

This is a feasibility pass, not a result. Before any computation, the spine of the
deliverable is fixed:

\begin{enumerate}
  \item State the candidate object \emph{derived from} $\JO$/corpus structure
        \emph{before} any value-matching --- what graded dimension, character, or
        modular function does $\JO$ concretely produce in the corpus?
  \item It must match a \emph{standard} object with \emph{no} free parameter and
        \emph{no} bespoke function: either (a)~a canonical McKay--Thompson
        hauptmodul $T_g$ (matching $\geq 3$ $q$-expansion coefficients, or its
        value at a CM point against the canonical series), or (b)~a genuine Monster
        head-representation dimension ($1, 196883, 196884, 21296876, \ldots$).
        \textbf{Explicitly disqualified:} $j^{1/3}$ and any non-hauptmodul
        cube-root or bespoke function (the A177 trap); any object requiring a
        fitted constant.
  \item The \emph{first} honest check that fails the standard-object match $\to$
        \textsc{stop}. Record ``no unforced standard contact; coincidence, left to
        mainstream moonshine.'' Iterating past the first failure to find a tuning
        that works is itself the failure mode this pass exists to prevent.
  \item Even a \emph{pass} would be a \emph{consistency} result only (Borcherds
        1992 already explained moonshine) --- a candidate for proper follow-up,
        never ``the bridge earned.''
\end{enumerate}

\paragraph{The A337 lesson, just established.}
A337 retracted A163/A177's claim that $T_{3B} := j^{1/3}$ and that
$T_{3B}(i) = 12$ is a Monster 3B moonshine hit. The function $j^{1/3} =
E_4/\eta^8$ is the $E_8/G_2$ cube-root function (it carries $\dim E_8 = 248$ at
first order; A160/A161), \emph{not} the canonical Monster 3B McKay--Thompson
hauptmodul $(\eta/\eta_3)^{12}+12$ (value $535.59$ at $i$, $q$-expansion
$q^{-1}+0+54q+\cdots$). The leap to the Monster \emph{specifically} was bespoke
and wrong. This pass tests whether \emph{any} unforced standard contact exists,
knowing the prior attempt failed exactly by using a bespoke non-hauptmodul
function.

%==========================================================================
\section{The candidate object (stated first)}\label{sec:candidate}

\begin{proposition}[What $\JO$ concretely produces in the corpus]
\label{prop:candidate}
In the corpus, $\JO$ is the $27$-dimensional exceptional Jordan algebra of
$3\times 3$ Hermitian octonionic matrices, with
\[
  \mathrm{Aut}(\JO) = F_4, \qquad \mathrm{Str}_0(\JO) = E_6
\]
\textup{(A166, \S1.2 and Prop.~1.1)}. The concrete number $\JO$ produces is
$\dim \JO = 27$, identical to the dimension of the fundamental representation
$\mathbf{27}$ of $E_6$. The only place this number touches modular geometry in the
corpus is A166, Thm.~5.1:
\begin{equation}\label{eq:27times64}
  27 \times 64 \;=\; 1728 \;=\; j(i) \;=\; \Js^3 .
\end{equation}
\end{proposition}

\noindent Equation~\eqref{eq:27times64} is recomputed from scratch by the verifier
(checks 3--4): $j(i) = 1728$ and $27\cdot 64 = 1728$. So the corpus's
$\JO$-to-modular contact is the number $27$ (and its cube-partner $j(i)=1728$),
not any character of the Monster.

\begin{proposition}[The chain to the Monster runs only through the Leech]
\label{prop:leech}
The corpus reaches the Monster sporadic group $\Mon$ from $\JO$ along a single
documented path \textup{(A169)}:
\[
  \JO \;\longrightarrow\; E_6 \;\longrightarrow\; \cdots \;\longrightarrow\; E_8
  \;\longrightarrow\; \Leech \;\longrightarrow\; V^\natural \;(\text{FLM})
  \;\longrightarrow\; \Mon .
\]
The corpus's Leech-level number adjacent to the Monster is the minimal-vector
count $|\mathrm{minvec}(\Leech)| = 196560$, the $q^2$ coefficient of the Leech
theta series $\Theta_{\Leech} = E_4^3 - 720\,\Delta$ \textup{(A169)}.
\end{proposition}

\noindent The verifier recomputes $196560$ from the theta series (check~5) and its
factorisation $196560 = 2^4\cdot 3^3\cdot 5\cdot 7\cdot 13$ (check~6). Crucially,
the Monster's head-representation dimensions come from the \emph{graded structure}
of $V^\natural$ (Borcherds 1992), not from $\JO$; the $\JO$ chain supplies the
Leech only as a lattice \emph{input} to the FLM construction. The candidate object
for a $\JO\to$Monster contact is therefore the Leech number $196560$.

%==========================================================================
\section{The single standard target}\label{sec:target}

We choose, once, the single most plausible standard-moonshine target: the smallest
faithful Monster head-representation dimension,
\begin{equation}\label{eq:target}
  196883 ,
\end{equation}
together with McKay's identity $[q^1]\,j = 196884 = 1 + 196883$ (the $q^1$
coefficient of the $V^\natural$ partition function $j$; verifier checks 7--8). This
is the most plausible candidate for an \emph{unforced} contact for three reasons:
(i)~$196560$ is a genuine corpus-derived $\JO$/Leech number, not a bespoke
function; (ii)~$196883$ is a genuine Monster head-rep dimension, the kill-criterion's
allowed target; (iii)~the two differ by only $\sim 0.16\%$, the textbook near-miss
that A169 \S6.2 already flags ($196883 - 196560 = 323 = 17\cdot 19$). If the corpus
contact were unforced, $196560$ would simply \emph{equal} a Monster head-dimension
under canonical normalization.

%==========================================================================
\section{The single match, and the kill-criterion verdict}\label{sec:match}

\begin{proposition}[The standard-object match, run once]\label{prop:match}
The corpus $\JO$/Leech number $196560$ is \emph{not} equal to any Monster
head-representation dimension:
\begin{equation}\label{eq:nohit}
  196560 \;\notin\; \{1,\,196883,\,21296876,\,842609326,\,\ldots\}.
\end{equation}
The nearest target, $196883$, is reached only by adding a non-zero bespoke bridge
constant:
\begin{equation}\label{eq:gap}
  196883 - 196560 \;=\; 323 \;=\; 17 \times 19 \;\neq\; 0 .
\end{equation}
More generally, \emph{no} corpus $\JO$-chain number
$\{27,\,78,\,1728,\,196560\}$ (the $\mathbf{27}$ dimension, $\dim E_6 = 78$,
$j(i)$, and the Leech minimal vectors) equals any Monster head-representation
dimension.
\end{proposition}

\begin{proof}
Direct comparison against the canonical Monster head-dimension list, recomputed
where modular (verifier S3, checks 9--11). $196560 \neq 196883$; the gap is
$323 = 17\cdot 19$, a non-zero integer. The set
$\{27, 78, 1728, 196560\}$ has empty intersection with the head-dimension list.
\end{proof}

\paragraph{First honest check $\to$ stop.}
Check~9 (the $196560 \notin$ head-dims test) is the \emph{first} standard-object
match in the pass, and it fails: there is no exact, unforced hit. Per the
kill-criterion we stop here. We do \emph{not} iterate to a variant --- we do not
try a different CM point, a different Leech shell, a rescaling, or a constructed
function that would close the $323$. The non-zero gap $323$ is precisely the A177
failure mode: a fitted quantity standing in for a missing identity. The
kill-criterion disqualifies it (check~10).

\begin{remark}[The A177 trap is closed, not re-entered]
The only $\JO\to$modular function the corpus actually uses is $j^{1/3} =
E_4/\eta^8$ (A161), which A337 proved is the $E_8/G_2$ cube-root function
($q^{-1/3}(1+248q+\cdots)$, carrying $\dim E_8 = 248$), not a Monster hauptmodul.
Its value $j^{1/3}(i) = 12$ is the $G_2$ short-root sum $\Js$, and $12$ is not a
Monster head-representation dimension either (verifier checks 12--13). The trap
that produced the A177 error is explicitly avoided, not re-entered under a new name.
\end{remark}

\begin{remark}[What \emph{is} true is mainstream, not a $\JO$ result]
The partition function of $V^\natural$ is $j(\tau)$, so $j(i) = 1728$ at the corpus
CM point (verifier check~14). This is FLM/Borcherds mainstream moonshine, true
regardless of $\JO$. The adjacency $196560 \sim 196883$ is real --- both numbers
live inside the degree-one graded piece $\dim V^\natural_1 = 196884 = 1 + 196883$,
and $196884 - 196560 = 324$ (check~15) --- but it is the standard Leech/Monster
near-miss explained by the moonshine VOA, not an unforced $\JO$ identity. The
corpus supplies the Leech as a lattice input; it does not supply the Monster
head-dimensions.
\end{remark}

\begin{corollary}[Kill-criterion verdict]\label{cor:verdict}
There is no unforced standard contact between the corpus's $\JO$ structure and a
canonical monstrous-moonshine object. The candidate object ($\JO \to \dim 27$;
chain-to-Monster number $= $ Leech $196560$) does not match the single standard
target ($196883$) with canonical normalization and no free parameter; the contact
requires a bespoke bridge constant $323 = 17\cdot 19$, which the kill-criterion
disqualifies. The result is a coincidence, left to mainstream moonshine
\textup{(FLM; Borcherds 1992)}. No new TOE/$\JO$ bridge to the Monster is earned.
\end{corollary}

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

Asked whether the corpus's existing $\JO$ structure lands on a standard
monstrous-moonshine object unforced, the honest answer is no. The concrete
$\JO$ number in the corpus is $27$ (the $E_6$ fundamental, via $\mathrm{Str}_0(\JO)
= E_6$), and the only number the $\JO$ chain carries toward the Monster is the
Leech minimal-vector count $196560$ (A169). The single most plausible standard
target, the smallest faithful Monster head-representation dimension $196883$, is a
near-miss, not a hit: it differs from $196560$ by the bespoke constant
$323 = 17\cdot 19$. No corpus $\JO$-chain number equals a Monster head-dimension,
and the corpus's only $\JO\to$modular function is the $E_8/G_2$ cube root, not a
hauptmodul. We stopped at the first honest failure, as the kill-criterion requires,
and did not iterate. This mirrors A337's posture: the corpus legitimately reaches
$E_6$, $E_8$, and the Leech, but the leap to the Monster \emph{specifically} is not
earned --- it is a real coincidence, fully explained by mainstream moonshine.

\medskip
\noindent\textit{Copyright: L\'eon Fernando Vlegels. License: CC BY 4.0. June 2026.}

\end{document}
