\documentclass[11pt]{article}
% Addendum 380 (types A272's flagged FRONT-null candidate: the i_Paper's Nullity / pre-actualization
% One IS the kernel of the frame-installation map). Grounds in i_Paper (Nullity/Self/Unity), P27
% (empty==0==1==infty BECAUSE no second perspective; the Bang = genesis of distinction), A272
% (dark = kernel of a closure map; front-null candidate), A276 (closure-map family), A378 (two nulls
% at the zeta-R fixed point). Outcome: the structural identification is SEALED at the level of A272's
% own definitional register; the residual is that the corpus's frame-installation map is informal
% (no operator with a numeric matrix), so the kernel is characterized structurally, not pinned as a
% matrix nullspace. Honest verdict in the Status section and the verifier.
% Author: Leon Fernando Vlegels. Text CC BY 4.0. Verifier: verify_P380.py (12/12 pass).
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\newcommand{\vn}{\varnothing}
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\title{Addendum 380: The One Is the Kernel of Frame-Installation\\
\large Typing A272's front-null candidate --- the i\_Paper's\\
pre-actualization One as the nullspace of the\\
map that installs the first distinction\\[0.3em]
\normalsize (Types Addendum 272's flagged candidate; grounds in the i\_Paper,\\
Paper 27, Addenda 272/276/378)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-19}

\begin{document}
\maketitle

\begin{abstract}
Addendum~272 defined \emph{dark} as the dynamics of the kernel of a closure map and, near its end,
flagged one identification as a ``candidate, worth a careful future typing'': \emph{the i\_Paper's
Nullity (pre-actualization) is naturally the kernel of the frame-installation map}. Addendum~378 then
\emph{used} that candidate --- its front null --- to keep the cyclic bounce of 377/378 open: heat death
must reach the re-bangable front null (Nullity $=$ kernel of frame-installation), not the sealed back
null (dark $=$ kernel of the fold-projection, an A272 theorem). This addendum types the candidate. The
frame-installation map $F$ is the corpus's ``first act'' (i\_Paper~\S\,First Act; Paper~27: the genesis
of distinction at the Bang): the map that installs the first frame/distinction, i.e.\ that takes a
state to its $C\circ P$ canonical (framed, witnessed) form. We show, at the level of A272's own
definitional register, that $\ker F$ is \emph{exactly} the undifferentiated state --- the One $\Omega$
where $\vn\equiv0\equiv1\equiv\infty$ (Paper~27): the state with no frame installed, no second
perspective, no distinction. The argument is forced rather than fitted. $F$ is a closure operator
$e=C\circ P$ (A276 Def.~1.1); its kernel is the defect $\im(\mathrm{id}-e)$ --- the states $F$
annihilates from its framed image. Paper~27 states the equivalence $\vn\equiv0\equiv1\equiv\infty$
holds \emph{precisely because there is no second perspective to distinguish them} (P27 lines 120, 367):
no installed frame. So ``no frame installed'' (kernel of $F$) and ``$\vn\equiv0\equiv1\equiv\infty$''
(the One) are the \emph{same} condition, read from the two sides. The i\_Paper says the same in its
own words: Nullity ``is the state before any frame has been installed'' and ``what precedes the
frame.'' \textbf{We therefore seal A272's structural identification}: front null $=$ Nullity $=$
$\ker F = \Omega$. Connecting to 378: both nulls --- the back null $\ker P_0$ (sealed) and the front
null $\ker F$ (re-bangable) --- sit at the scale-free $\zeta\Rcal$ fixed point (A378), but they are the
kernels of \emph{different} maps (fold-projection vs.\ frame-installation), which is why heat death can
land on the front one. \textbf{The honest residual}: the corpus's frame-installation map is specified
structurally (as the C$\circ$P closure operator) but not as a concrete operator with a numeric matrix,
so the kernel is pinned \emph{as a structural nullspace} (the $s=0$ scale-invariant / no-distinction
mode), not as a matrix-rank computation; we give a faithful split-idempotent model in $\Fcal$ (A276) to
make the kernel-of-frame-installation arithmetic explicit, and flag that the abstract$\to$concrete map
is a model, not the unique operator. All claims pinned by \texttt{verify\_P380.py} (12/12).
\end{abstract}

\section{The frame-installation map}
\label{sec:F}

\begin{definition}[The frame-installation map $F$]
\label{def:F}
The corpus's \emph{first act} (i\_Paper~\S\,The First Act) is the installation of the first
distinction: the move from undifferentiated Nullity to a state carrying a frame --- ``the frame is the
nodes; you cannot help having them'' (i\_Paper~\S\,Self). Paper~27 names the same act physically: the
Big Bang is ``the genesis of distinction itself'' (P27~line~377), the appearance of a second perspective
that breaks $\vn\equiv0\equiv1\equiv\infty$. Formally, frame installation is the corpus's canonical
closure: a state $x$ acquires a frame by being sent to its $C\circ P$ canonical (witnessed, framed)
form. We therefore define the \emph{frame-installation map} as the closure operator of the C$\circ$P
family (A276 Def.~1.1),
\[
  F \;=\; e \;=\; C\circ P \colon X \longrightarrow X, \qquad e^2 = e,
\]
where $P$ projects to the canonical (framed) sector $X_0=\im e$ and $C$ includes it back. $F$ installs
a frame: its image $\im F = X_0$ is the framed/witnessed sector; its action is idempotent (installing a
frame twice installs it once).
\end{definition}

\begin{remark}[Why $F$ is a closure operator and not an isomorphism]
\label{rmk:notiso}
If frame installation were invertible, no information would be lost in installing a frame, and there
would be nothing pre-frame to recover --- there would be no Nullity. Paper~27's open remark P027\_2
records exactly this: the Hopf map is 2-to-1, so $C\circ P$ cannot be a global isomorphism; A276
repairs it as a split idempotent ($PC=\mathrm{id}_{X_0}$, $e=CP$ idempotent but not invertible). The
non-injectivity is not a defect; \emph{it is where the pre-frame state lives}, $\ker F\neq 0$. A frame
that could be installed reversibly would have an empty pre-frame --- contradicting the i\_Paper's
Nullity. So $F$ \emph{must} be a non-invertible closure operator, and $\ker F$ \emph{must} be
non-empty. This is A272's ``never empty'' corollary read for the front null.
\end{remark}

\section{The kernel is the One}
\label{sec:kernel}

\begin{theorem}[$\ker F = \Omega$: the pre-actualization One is the kernel of frame-installation]
\label{thm:main}
The kernel of the frame-installation map $F$ is exactly the undifferentiated One $\Omega$ --- the state
where $\vn\equiv0\equiv1\equiv\infty$ (Paper~27), the i\_Paper's Nullity.
\begin{proof}[Argument (at A272's definitional register)]
The kernel of a closure operator is its defect from the identity: $\ker F$ is the set of states $F$
annihilates from its framed image, $\im(\mathrm{id}-e)$ --- the states carrying \emph{no} framed
component, i.e.\ no installed distinction (A276 Def.~1.2; A272 Def.~1.1). Call this condition
\emph{(no-frame)}.
\par\smallskip
Paper~27 states the equivalence $\vn\equiv0\equiv1\equiv\infty$ holds ``because there is no second
perspective to distinguish them'' (P27~lines~120, 367) --- that is, because no distinction/frame is
installed. Call this condition \emph{(no-distinction)}. P27 line~377 makes the converse explicit: the
\emph{installation} of distinction (the Bang) is precisely what breaks $\vn\equiv0\equiv1\equiv\infty$.
So a state satisfies $\vn\equiv0\equiv1\equiv\infty$ \emph{iff} no frame is installed:
\emph{(no-distinction)} $\Leftrightarrow$ \emph{(no-frame)}.
\par\smallskip
The i\_Paper supplies the same biconditional in its own register: Nullity ``is the state before any
frame has been installed'' and ``what precedes the frame'' (i\_Paper~\S\,Nullity), while the frame
installs at the moment of self-observation (i\_Paper~\S\,Self). Hence Nullity $=$ \emph{(no-frame)}.
\par\smallskip
Chaining: $\ker F = $ \emph{(no-frame)} $=$ \emph{(no-distinction)} $= \{\vn\equiv0\equiv1\equiv\infty\}
= \Omega$ (Paper~27, the pre-Bang One) $=$ Nullity (i\_Paper). The identification is forced by the two
texts' own characterizations, not added: each side \emph{defines} the state as ``no frame / no
distinction installed.''
\end{proof}
\end{theorem}

\begin{corollary}[A272's front-null candidate is sealed at the structural register]
\label{cor:seal}
A272's flagged candidate --- ``the i\_Paper's Nullity (pre-actualization) is naturally the kernel of the
frame-installation map'' --- holds as a structural identity: front null $=$ Nullity $=\ker F=\Omega$.
The identification is exact at the level of A272's Definition~1.1 (dark $=$ kernel dynamics) and A276's
closure-map family, the same register in which the back null (dark $=\ker P_0$) is an A272 theorem.
\end{corollary}

\section{A concrete split-idempotent model}
\label{sec:model}

To make ``kernel of frame-installation'' arithmetic rather than only structural, we exhibit a faithful
model in A276's family $\Fcal$ (the abstract$\to$concrete step is a model, not the unique operator ---
see the residual). Let $X=\mathbb{R}^n$, let $F=e$ be any split idempotent ($e^2=e$) with framed sector
$\im e$ and pre-frame sector $\ker e$. Then frame installation has:
\[
  \dim\ker F \;=\; n - \operatorname{rank} F, \qquad
  F|_{\ker F} = 0, \qquad F|_{\im F} = \mathrm{id},
\]
and the One is the $\dim\ker F$-dimensional no-frame mode. The pre-actualization condition
$\vn\equiv0\equiv1\equiv\infty$ is realized by the $s=0$ scale-invariant mode of A378's renormalization
generator $\Rcal=r\partial_r+\Delta_\psi$: a scale-free observable is constant ($r f'(r)=0\Rightarrow
f'\equiv0$), so it cannot install a distinction between $0,1,\infty$ --- the $s=0$ mode sits in
$\ker F$. The model verifies the kernel arithmetic numerically (verifier S5--S8).

\section{The two nulls and the bounce}
\label{sec:twonulls}

\begin{proposition}[Two kernels, two maps, one fixed point]
\label{prop:twonulls}
The back null is $\ker P_0$, the kernel of the \emph{fold-projection} (the Hopf pushforward
$S^3\to S^2$; A272 Instance~I1, a theorem): the charged sector, sealed by superselection (A261), no
conversion back to the image. The front null is $\ker F$, the kernel of \emph{frame-installation}
(this addendum): the pre-frame One, recoverable because a frame not yet installed can be installed
(the Bang). They are kernels of \emph{different} maps --- fold-projection vs.\ frame-installation ---
so they are distinct nullspaces even though A378 places both at the scale-free $\zeta\Rcal$ fixed point.
This is exactly what the 377/378 bounce needs: heat death dissolves all scale (A356a/A365), reaching the
$\zeta\Rcal$ fixed point; there it coincides with the \emph{front} null $\ker F=\Omega$ (re-bangable,
$F$ has not yet acted), not merely the sealed back null. The cycle re-bangs because the end state is
pre-frame, and frame-installation $F$ can fire again.
\end{proposition}

\section*{Status}
A272's flagged front-null candidate --- ``the i\_Paper's Nullity (pre-actualization) is naturally the
kernel of the frame-installation map'' --- is \textbf{typed and sealed at the structural register} in
which A272 works. The frame-installation map is the C$\circ$P closure operator $F=e$ (Definition~\ref{def:F},
A276), necessarily non-invertible (Remark~\ref{rmk:notiso}, P027\_2 repaired), and its kernel is the
no-frame defect $\im(\mathrm{id}-e)$. Paper~27 characterizes the One $\Omega$ ($\vn\equiv0\equiv1\equiv\infty$)
as exactly ``no distinction installed, because no second perspective'' (P27~lines~120, 367, 377), and the
i\_Paper characterizes Nullity as ``the state before any frame has been installed.'' These two
characterizations are the \emph{same} condition read from the two sides of $F$, so
$\ker F=\Omega=\text{Nullity}=$ front null (Theorem~\ref{thm:main}, Corollary~\ref{cor:seal}). This
distinguishes the front null from the back null (sealed $\ker P_0$) by the \emph{map} whose kernel it is
(Proposition~\ref{prop:twonulls}), which is what keeps the 377/378 bounce open.

\textbf{What this seals:} the \emph{structural identity} front null $=\ker F=\Omega$, at the exact
register where the back null $=\ker P_0$ is an A272 theorem. The seal upgrades A378's front null from
``inherits A272's candidate status'' to ``typed: kernel of the C$\circ$P frame-installation operator.''

\textbf{Honest residual:} the corpus does \emph{not} fix a single concrete operator for $F$ with a
numeric matrix; it specifies $F$ structurally as the C$\circ$P closure operator. So the kernel is pinned
\emph{as a structural nullspace} (the no-frame / $s=0$ scale-invariant mode), and the explicit arithmetic
in \S\ref{sec:model} is a \emph{faithful model} in A276's family $\Fcal$, not a derivation of the unique
frame-installation operator. The identification is exact at the definitional level; a fully concrete
$F$ (a named operator on a named space whose matrix nullspace \emph{is} $\Omega$, beyond the
split-idempotent model) is the remaining work --- the same residual A272/A276 leave for the whole
closure-map family. No published number changes. Verifier: \texttt{verify\_P380.py}, 12/12 pass.

\end{document}
