% Addendum 343: Blast-Radius Audit of the A337 T_3B Correction
%   A337 retracted "T_3B = j(tau)^(1/3)" as a Monster 3B McKay-Thompson
%   moonshine statement. This addendum maps everything downstream that
%   ASSERTED or DEPENDED ON the retracted claim, classifies each location
%   SURVIVES / REVISE / REFUTE, applies errata to the affected addenda, and
%   delivers the ToENet luminosity-grounding verdict.
%   Outcome: AUDIT. Addresses: the A337 blast radius. Verifier: verify/verify_P343.py
%   (20 checks, 20 PASS / 0 FAIL). Net: no math retracted (no REFUTE); every
%   affected statement SURVIVES under re-attribution or is a framing REVISE;
%   ToENet's {12^k}-McKay-Thompson luminosity grounding does NOT survive and is
%   flagged for re-derivation.
% TOE Corpus
% Copyright: Léon Fernando Vlegels. License: CC BY 4.0. June 2026.

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\usepackage{mathtools}
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\newcommand{\Js}{J_{\mathrm{short}}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\Tstd}{T_{3B}^{\mathrm{std}}}
\newcommand{\sv}{\textsc{Survives}}
\newcommand{\rv}{\textsc{Revise}}
\newcommand{\rf}{\textsc{Refute}}

\begin{document}

\title{Addendum 343 --- Blast-Radius Audit of the A337\\
  $T_{3B}$ Correction:\\
  Re-grounding the Monster Cluster and the ToENet Luminosity Flag}
\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:} AUDIT.\quad
\textbf{Addresses:} the blast radius of A337's retraction of
``$T_{3B} = j^{1/3}$ as a Monster 3B McKay-Thompson moonshine value.''\quad
\textbf{Verifier:} \texttt{Lumen/corpus/addenda/verify/verify\_P343.py}
(20 checks, 20 PASS / 0 FAIL; A337 facts recomputed from scratch with
\texttt{mpmath} at \texttt{dps}$=40$, plus a machine-checked classification
manifest).\quad
\textbf{Net:} no computed fact is retracted (\emph{no} \rf); every affected
location either \sv{} under re-attribution or is a framing \rv. The ToENet
$\{12^k\}$-McKay-Thompson luminosity grounding does \emph{not} survive and is
flagged for a dedicated re-derivation before the build proceeds.}}

\begin{abstract}
Addendum 337 retracted the identification of $j(\tau)^{1/3}$ with the Monster's
class-3B McKay-Thompson series: $j^{1/3} = E_4/\eta^8$ is the E8/G2 cube-root
function of A160/A161 (value $12$ at $\tau = i$), whereas the canonical 3B
hauptmodul is $\Tstd = (\eta/\eta_3)^{12} + 12$ (value $535.592$ at $\tau = i$).
The retracted claim and its corollaries thread through the Monster/exceptional
cluster (A160--A182) and into one build target (ToENet's canonical/dark
luminosity flag, PRIORITIES \S0a). This addendum maps the full blast radius. We
read every candidate location, classify each \sv{} / \rv{} / \rf, and give a
one-line corrected statement. The audit's central finding is \emph{surgical}:
\textbf{nothing is refuted}. Every E8/G2 root-geometry result (A160, A165--A169,
A182) stands untouched; the perfect-cube CM uniqueness of $\tau = i$ (A172) is a
statement about the function $j^{1/3}$ and survives under re-attribution; the
honest Ramanujan-artefact flag for $T_{2B}(\tau_{-163}) = -24$ (A177/A181)
stands. What requires revision is the \emph{Monster-moonshine framing} layered on
top of the (correct) cube-root number $12$: A161's closing remark, A163, A164,
A170, A172, A177, A181 each attach the number to the wrong provenance and now
carry an erratum. The most consequential framing correction is A170's claim that
the G$_2 \to$ Monster chain \emph{reaches} the Monster: per A338 the corpus
reaches only E8/Leech, and the Ogg/supersingular Monster contact is left to
mainstream moonshine. Finally, the ToENet luminosity criterion --- ``canonical
luminosity $=$ a McKay-Thompson series value landing in $S = \{12^k\}$'' --- had
exactly one example, $T_{3B}(i) = 12$, now re-attributed to E8/G2 root geometry.
No genuine Monster McKay-Thompson value lands in $\{12^k\}$ ($T_{1A}(i) = 984
\notin \{12^k\}$; $\Tstd(i) = 535.6 \notin \{12^k\}$), so the moonshine grounding
of the luminosity flag has no surviving instance. We do not paper over this: the
luminosity flag is flagged for a fresh basis.
\end{abstract}

%==========================================================================
\section{Method}\label{sec:method}

A337 corrects an \emph{attribution}, not a computation. The audit therefore
classifies each downstream location by whether the underlying number/lemma is
true under its \emph{correct} provenance:
\begin{itemize}
  \item \sv{} (re-attribute): the statement is true under the corrected
    attribution (the $12$ as E8/G2 root geometry; $T_{1A}(i) = 984$; the E8
    dimension $248$ as $[q^1]\,j^{1/3}$). No retraction of the math; note the
    re-attribution.
  \item \rv: the statement is partly right but its Monster-moonshine framing
    must change (e.g.\ ``unique $\{12^k\}$ McKay-Thompson hit'' $\to$
    ``$j^{1/3}(i) = 12$, an E8/G2 root-geometry value, not a Monster series
    value'').
  \item \rf: the statement depends essentially on the false identification and
    does not survive.
\end{itemize}
The verifier \texttt{verify\_P343.py} recomputes the A337 facts from scratch and
encodes the classification as a machine-checked manifest (checks 11--20). The
audit's strongest single result is that the \rf{} column is \emph{empty}: the
math is never wrong; only the Monster-moonshine attribution is.

%==========================================================================
\section{The classification table}\label{sec:table}

Locations are listed by addendum (or build file). ``Asserts'' marks whether the
location states one of: (i) ``$T_{3B} = j^{1/3}$''; (ii) ``$T_{3B}(i) = 12$ is a
Monster McKay-Thompson value''; (iii) ``the unique $S \cap \{12^k\}$
McKay-Thompson hit''; (iv) ``canonical luminosity $=$ a McKay-Thompson value in
$\{12^k\}$''; (v) ``the corpus chain reaches the Monster.''

\renewcommand{\arraystretch}{1.25}
\begin{longtable}{@{}p{0.20\textwidth} c p{0.56\textwidth}@{}}
\toprule
\textbf{Location} & \textbf{Verdict} & \textbf{Corrected one-line statement} \\
\midrule
\endfirsthead
\toprule
\textbf{Location} & \textbf{Verdict} & \textbf{Corrected one-line statement} \\
\midrule
\endhead
\bottomrule
\endfoot

A160 Monodromy\-Weyl$A_2$ & \sv & Pure cube-root geometry: monodromy of
$j^{1/3}$ is $\ZZ/3\ZZ = \mathrm{Weyl}(A_2)$; $12 = \Js$ at $\tau = i$. No
Monster claim; untouched. \\

A161 E8Monster\-Bridge & \rv & Theorems A--C (the $j^{1/3} = E_4/\eta^8 =
q^{-1/3}(1 + 248q + \cdots)$ q-expansion, McKay's $248 = \dim E_8$, the
$G_2\!\leftrightarrow\!E_8$ chain) \sv{} as E8/G2 root geometry; the
\emph{closing remark} ``$j^{1/3}$ is the McKay-Thompson series for class 3B,
placing $G_2$ in contact with the Monster'' is the retracted conflation ---
\rv{} to ``$j^{1/3}$ is the E8/G2 cube-root function, not the Monster 3B series
(A337).'' \\

A163 Monster\-McKay\-Thompson & \rv & Already errata'd by A337. The
$j^{1/3}$/$T_{1A}/T_{2A}/T_{2B}/T_{3A}$ computations stand; the ``$T_{3B}(i) =
12$ is a Monster 3B value'' framing is retracted. \\

A164 Stable\-Null\-Grid & \rv & $\Delta(i) = \eta(i)^{24}$ and $j(i) = \Js^3 =
1728$ \sv{} (independently derived in A158 from $G_2/F_4$ root geometry); the
provenance line ``$j(i) = \Js^3$ was \emph{proved using} the McKay-Thompson
identification $T_{3B}(i) = \Js$ (A163)'' is \rv{} to ``$j(i) = \Js^3$ follows
from A158's root geometry; the $j^{1/3}(i) = \Js$ contact is E8/G2, not Monster
3B.'' \\

A165 BF4\_36, A166 E6\-Jordan, A168 E8\_OM, A169 Leech, A182 Root\-Complement &
\sv & Every use of $12 = \Js$ is E8/G2/$F_4$/$E_6$ root geometry (Coxeter
numbers, root counts, $E_4(i) = \Js\,\eta(i)^8$, Golay $[24,12,8]$ dimension).
No Monster McKay-Thompson dependence; untouched. A169's Leech$\to$Monster
remark is mainstream moonshine, not a corpus bridge (consistent with A338). \\

A170 Monster\_\-Modular\-Boundary & \rv & The Moonshine exposition (Conway--Norton,
Borcherds, $V^\natural$) \sv{} as mainstream mathematics; the two load-bearing
\emph{corpus} claims --- ``$T_{3B}(i) = j(i)^{1/3} = \Js = 12$ is the unique
McKay-Thompson series returning an exact TOE constant'' and ``the chain from
$G_2$ \emph{reaches} the Monster'' --- are \rv. Corrected: the $12$ is the E8/G2
cube-root value, not a McKay-Thompson value; the corpus chain reaches E8/Leech,
and the Ogg/supersingular Monster contact is real but unearned, left to
mainstream moonshine (A337 \S5, A338). \\

A172 CM\_Uniqueness & \rv & The result ``$\tau = i$ is the unique class-number-one
CM point with $j(\tau_D) \in \{12^k\}$'' \sv{} --- it is a statement about the
\emph{function} $j$ (perfect-cube CM theory), independent of moonshine. The
framing line ``recovers $\Js = j(i)^{1/3}$ via the $3B$ McKay-Thompson series''
is \rv{} to ``via the E8/G2 cube-root function $j^{1/3} = E_4/\eta^8$ (A337).'' \\

A177 McKay\-Thompson\_CM\_Scan & \rv & Already errata'd by A337. The 45-evaluation
scan stands as arithmetic; $T_{1A}(i) = 984$ and the $T_{2B}(\tau_{-163}) = -24$
Ramanujan-artefact flag survive; the ``$(T_{3B}, \tau_i) = 12$ is the unique
$S \cap \{12^k\}$ Monster McKay-Thompson hit'' (Prop.~5.1) is retracted \emph{as
a moonshine statement}. \\

A181 T2B\_Ramanujan & \rv & The result ``$T_{2B}(\tau_{-163}) \approx -24$ is a
Ramanujan eta-expansion artefact, $-24 = -2\Js$'' \sv. The provenance ``$\Js$
established in P163 ($T_{3B}(i) = 12$)'' is \rv{} to ``$\Js = 12$ established in
P151/P158 as $G_2$ root geometry; $j^{1/3}(i) = 12$ is the E8/G2 contact, not a
Monster 3B value.'' \\

A185 G1P171\_Char\-acterisation & \sv & ``Moonshine'' appears only as the name of
one PSLQ basis (a null result ruling out moonshine-type formulae for $G_1$); no
dependence on the $T_{3B}$ identification. Untouched. \\

PRIORITIES \S0a (ToENet) & \rv & The G$_2\to$Monster-chain claim is \rv{} (the
chain reaches E8/Leech, not the Monster --- A338); the luminosity grounding
``canonical $=$ a McKay-Thompson value in $\{12^k\}$'' is \rv{} to
\emph{flagged-for-re-derivation} (\S\ref{sec:toenet}). \\

\texttt{prime\_survey\_toe\_map.md} & \sv & Already updated by A337 (RESOLVED
2026-06-14 note); consistent with this audit. No change. \\

\end{longtable}

\noindent\textbf{Verdict tally:} \sv{} $\times$ 6 location-groups, \rv{} $\times$
8, \rf{} $\times$ 0. \emph{No computed fact is retracted.}

%==========================================================================
\section{The ToENet luminosity verdict}\label{sec:toenet}

PRIORITIES \S0a grounds ToENet's canonical/dark luminosity flag thus:
\begin{quote}\small
``The G$_2\to$Monster chain (P164--P170) $+$ P177 (CM scan, $T_{3B}(i)=12$
unique $\{12^k\}$ hit) $+$ P181 \dots\ establish what `canonical luminosity'
means: a McKay-Thompson series value landing in $S = \{12^k\}$. Dark nodes are
everything else.''
\end{quote}

The \emph{sole example} of that criterion was $T_{3B}(i) = 12$. Under A337 that
$12$ is not a McKay-Thompson value --- it is $j^{1/3}(i) = E_4(i)/\eta(i)^8$,
E8/G2 root geometry. We ask honestly whether the criterion can be recast.

\begin{enumerate}
  \item \textbf{Recast onto genuine McKay-Thompson values in $\{12^k\}$?} The
    genuine Monster McKay-Thompson values the corpus actually computed at $\tau =
    i$ are $T_{1A}(i) = 984$ and $\Tstd(i) = 535.592$. Neither is a power of
    $12$ ($984 = 2^3\cdot 3\cdot 41$; $535.6$ is not even an integer). The
    verifier confirms the set of genuine McKay-Thompson values in $\{12^k\}$ is
    \emph{empty} (check 18). The $\{12^k\}$ criterion has no surviving Monster
    instance. \emph{Does not survive as a McKay-Thompson criterion.}
  \item \textbf{Recast as an E8/G2 root-geometry criterion?} The number $12 =
    \Js$ does survive --- as the $G_2$ short-root sector sum / $E_6$--$F_4$
    Coxeter number / Golay-code dimension. One \emph{could} define ``canonical
    luminosity $=$ contact with an E8/G2 root-geometry constant.'' But this is a
    \emph{different and weaker} criterion: it has no $\tau = i$ uniqueness, no
    genus-zero modular boundary, and no dark/canonical \emph{dichotomy} (root
    constants are everywhere in the corpus). It is a candidate, not a
    drop-in replacement; asserting it now would be inventing a grounding.
  \item \textbf{Recast onto $T_{1A}(i) = 984$?} $984 \in S$ is a genuine
    surviving McKay-Thompson value, but it is a single point, not a
    $\{12^k\}$-membership rule, and membership in the hand-assembled set $S$ is
    weak evidence (A337 \S3.2 already noted this). Not a basis for a
    canonical/dark \emph{dichotomy}.
\end{enumerate}

\paragraph{Verdict: NEEDS-FRESH-BASIS.} The ToENet luminosity flag's moonshine
grounding does \emph{not} survive A337. Its only example was the re-attributed
$12$; no genuine Monster McKay-Thompson value lands in $\{12^k\}$; and the
nearest survivors (E8/G2 root constants; $T_{1A} = 984$) do not reproduce the
canonical/dark dichotomy the criterion was supposed to supply. We do \emph{not}
manufacture a replacement here. The luminosity flag requires a dedicated
re-derivation probe before the ToENet build proceeds. A live candidate worth
that probe --- and notably \emph{independent} of moonshine --- is the corpus's
own kernel-nullspace semantics (A272: dark $=$ dynamics of the kernel of a
closure map; canonical $=$ locked, dark $=$ drifting, A268), which already
carries the dark/canonical dichotomy the $\{12^k\}$ criterion was reaching for.

%==========================================================================
\section{Errata applied}\label{sec:errata}

A337 placed errata on A163 and A177. This audit (A343) extends the same surgical
erratum --- a header comment and a visible \textbf{Erratum} paragraph after
\texttt{\textbackslash maketitle}, bodies preserved as audit trail --- to the
remaining addenda that assert the retracted Monster framing and did not yet carry
one:
\begin{center}\small
A161 (closing-remark conflation), A164 (``proved using $T_{3B}$'' provenance),
A170 (the Monster-reaching chain), A172 (``$3B$ McKay-Thompson'' provenance),
A181 ($\Js$-via-P163 provenance).
\end{center}
The build file \texttt{Documents/Specs/PRIORITIES.md} \S0a is corrected in place:
the G$_2\to$Monster-chain claim now reads ``reaches E8/Leech, not the Monster
(A338)'' and the luminosity grounding now reads ``needs a fresh basis (A343).''
Addenda A160, A165--A169, A182, A185 and
\texttt{Science/Primes/prime\_survey\_toe\_map.md} require no change.

%==========================================================================
\section{What I could not fully resolve}\label{sec:open}

Two honest residuals.
\begin{itemize}
  \item \textbf{A replacement luminosity grounding is not derived here.} This
    audit establishes that the moonshine grounding fails and names a candidate
    (A272 kernel-nullspace semantics); it does not derive a criterion. That is a
    separate probe, correctly gated behind this one.
  \item \textbf{The classification reads each addendum's \emph{stated} claims,}
    not every transitive citation. The manifest is exhaustive over the
    grep-identified cluster (A160--A185 plus the two build files); a citation
    that uses $T_{3B}(i) = 12$ only through an intermediate result already
    classified \rv{} inherits that verdict without separate listing.
\end{itemize}

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

The A337 retraction is surgical and its blast radius is contained. No computed
fact across the Monster/exceptional cluster is wrong; every affected statement
either survives under its correct E8/G2 root-geometry provenance or needs only a
framing revision away from Monster moonshine. The one genuine \emph{loss} is at
the build layer: ToENet's canonical/dark luminosity flag was grounded in a single
$\{12^k\}$-McKay-Thompson example that A337 re-attributed, and no genuine Monster
McKay-Thompson value replaces it. That grounding is flagged for re-derivation,
not patched over. Errata are recorded on A161/A164/A170/A172/A181 (joining the
A337 errata on A163/A177); PRIORITIES \S0a is corrected; the verifier
\texttt{verify\_P343.py} confirms the A337 facts and the classification at 20/20.

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

\end{document}
