\documentclass[11pt]{article}
% Addendum 371 (re-grounds ToENet's luminosity flag; extends A254/A272; closes A343's flag).
% The {12^k}-moonshine luminosity criterion is dead (A343, no surviving instance); the fresh
% basis is the kernel-rank / collapse-delta of A254/A272 -- a structural rank-LAW, not a
% matched constant. Author: Leon Fernando Vlegels. Text CC BY 4.0.
% Verifier: verify_P371.py (13/13 pass).
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\newcommand{\dOm}{\delta_\Omega}
\newcommand{\rank}{\operatorname{rank}}

\title{Addendum 371: ToENet Luminosity on the Kernel Basis\\
\large The dead $\{12^k\}$-moonshine criterion replaced by a kernel-rank\\
law: luminosity is the collapse delta, not a matched constant\\[0.3em]
\normalsize (Addendum to A254/A272; closes the fresh-basis flag of A343)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-19}

\begin{document}
\maketitle

\begin{abstract}
ToENet's canonical/dark luminosity flag was grounded on a moonshine criterion --- ``canonical
luminosity $=$ a McKay--Thompson series value landing in $S=\{12^k\}$.'' Addendum~343, mapping the
blast radius of A337's retraction of the Monster-3B mislabel, found that \emph{no} genuine Monster
McKay--Thompson value lands in $\{12^k\}$ ($T_{1A}(i)=j(i)-744=984$; $T_{\mathrm{std}}(i)\approx
535.6$; neither is a power of $12$), so the criterion has \emph{no surviving instance}, and A343
explicitly flagged the luminosity flag for a fresh basis. This addendum supplies it. The replacement
is already conceptually in place: Addendum~272 establishes that ``dark,'' in every corpus register,
is the dynamics of the \emph{kernel} of a closure map, so ToENet's dark luminosity is traversals in
$\ker$ of the canonical-closure projection, and plateau detection is local \emph{kernel-rank}
estimation. Addendum~254 makes the rank computable: the collapse delta
$\dOm(A,X)=\rank\ker\!\bigl(H_0(A;\Bbbk)\to H_0(X;\Bbbk)\bigr)$, a homological invariant (lollipop:
$n$ traversal-components collapsing to one canonical form give rank $n-1$). \textbf{Verdict:} the
luminosity flag is re-grounded on a structural \emph{rank-law} --- luminosity $\sim$ kernel-rank
density via $\dOm$ --- which is strictly healthier than the retracted criterion: it replaces
\emph{numerology} (a value landing in a special set) with a defined invariant that inherits A272's
three structure theorems (sealed, paced, never-empty). \textbf{The honesty posture:} the basis
deliberately supplies no replacement magic constant. The \emph{form} is derived; any
threshold/normalization is \emph{empirical} (the OS's own traversal data), named not matched --- the
same posture as the $H_0$ closure (form grounded, calibration named). The one open formalization is
OI-272-1 (characterize the closure-map family as a class, not a list). All claims pinned by
\texttt{verify\_P371.py} (13/13).
\end{abstract}

\section{What died, and why}
\label{sec:dead}

\begin{proposition}[The moonshine criterion has no surviving instance]
\label{prop:dead}
The luminosity criterion ``canonical luminosity $=$ a McKay--Thompson value in $S=\{12^k\}$'' rested
on the identification of $j(\tau)^{1/3}$ with a Monster $3B$ McKay--Thompson series, retracted in
A337. A343 audited the consequence directly: at the CM point $\tau=i$ the genuine Monster Hauptmodul
gives $T_{1A}(i)=j(i)-744=1728-744=984$, and the corpus's $T_{\mathrm{std}}(i)\approx535.6$; neither
is a power of $12$ ($12^2=144$, $12^3=1728$). No genuine moonshine value lands in $\{12^k\}$, so the
criterion is instance-free. The underlying $\{12^k\}$ CM-uniqueness theorem itself \emph{stands}
(A172/A343, recovered via $E_8/G_2$ root geometry); what fails is only its use as a
\emph{luminosity} grounding.
\end{proposition}

\section{The fresh basis: luminosity is a kernel rank}
\label{sec:basis}

\begin{theorem}[Dark luminosity $=$ collapse delta]
\label{thm:basis}
By Addendum~272, ToENet's dark luminosity is the dynamics of $\ker P$ for $P$ the canonical-closure
projection of the collapse--projection family (C$\circ$P, P27): a dark entry is a traversal that ran,
deposited a trajectory, and did \emph{not} close --- a state in $\ker P$. The luminosity ``plateau''
(dark density exceeding canonical over a region) is then local estimation of $\rank\ker P$, made
precise by Addendum~254's collapse delta
\begin{equation}
  \dOm(A,X)\;=\;\rank\ker\!\bigl(H_0(A;\Bbbk)\to H_0(X;\Bbbk)\bigr),
  \label{eq:delta}
\end{equation}
the homological rank of the component map from the traversal set $A$ to its canonical closure $X$.
For $n$ traversal-components collapsing to a single canonical form the component map is
$[1\,1\,\cdots\,1]$ and $\dOm=n-1$ (the lollipop compactification, A254). Hence the luminosity flag
is grounded in a \emph{computable structural quantity}, not in a value matched against a special set.
\end{theorem}

\begin{corollary}[The three structure theorems transfer]
\label{cor:transfer}
The kernel-dynamics corollaries of A272 carry to the luminosity channel verbatim.
\emph{Sealed} (A261): $\ker P$ cannot leak to the image, so dark entries are never silently promoted
--- the Butlerian line realized as kernel protection. \emph{Paced} (A269): $\ker P$ shares the
image's clock, so dark entries are indexed by the same breath ledger as canonical entries
(nullspaces breathe). \emph{Never empty} (A267): irrational rotation forbids exact closure, so the
dark channel is mandatory --- a ToENet whose dark channel is empty has stopped measuring, not
achieved perfect closure.
\end{corollary}

\section{The honesty posture and the open step}
\label{sec:honest}

\begin{remark}[A rank-law, not a constant --- and why that is the point]
\label{rmk:honest}
The retracted criterion promised a \emph{constant}: a luminosity value certified by membership in
$\{12^k\}$. The kernel basis promises a \emph{law}: luminosity $\sim$ kernel-rank density via
$\dOm$. This is the healthy direction, not a weaker one. The $\{12^k\}$ criterion was numerology of
exactly the kind the corpus refuses elsewhere (the PRVE twin-gap and $\delta_{\mathrm{CP}}=240^\circ$
refutations); the kernel basis is a defined homological invariant with inherited theorems. The cost,
stated plainly: the basis fixes the \emph{form} of the luminosity channel but not a normalization
constant. Any operating threshold is \emph{empirical} --- set by the OS's own traversal statistics,
named rather than derived --- which is the same posture the corpus takes on $a_0=\alpha$ in the $H_0$
closure (Addenda~369/370): the form is grounded, the calibration is named. No $\{12^k\}$-style
matching is reintroduced.
\end{remark}

\section*{Status}
The ToENet luminosity flag, left without a surviving instance after A343 retired the
$\{12^k\}$-moonshine criterion (Proposition~\ref{prop:dead}), is re-grounded on the kernel basis: dark
luminosity is the dynamics of $\ker$ of the canonical-closure projection (A272), and the luminosity
plateau is the collapse delta $\dOm=\rank\ker(H_0(A)\to H_0(X))$ (A254, Theorem~\ref{thm:basis}) --- a
computable homological invariant inheriting the sealed/paced/never-empty theorems
(Corollary~\ref{cor:transfer}). The basis is a structural \emph{rank-law}, not a matched constant
(Remark~\ref{rmk:honest}): the form is derived, the calibration is empirical and named, and the
numerology is not reintroduced. \textbf{The open step} is OI-272-1: formalize the C$\circ$P
closure-map family as a defined class rather than an enumerated list, so ``dark $=\ker P$'' quantifies
over a characterized class. Until then the basis is sound but the map family is listed, not
characterized. No new objects beyond the inherited definitions; no published number changes; the
moonshine grounding stays retired. Verifier: \texttt{verify\_P371.py}, 13/13 pass.

\end{document}
