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% Addendum 381 (closes 380's residual; extends 378/380, grounds in P18 zeta-R + A276).
% The concrete frame-installation operator: F = the spectral projector onto the scale-bearing modes
% of zeta-R = r d/dr (P18), on the dilation eigenbasis. F^2=F (split idempotent = C-circ-P, A276),
% det F = 0 (non-invertible, P027_2), ker F = the scale-free s=0 mode = Omega = 378's zeta-R fixed
% point. A named, canonical, computed operator whose nullspace IS Omega.
% Author: Leon Fernando Vlegels. Text CC BY 4.0. Verifier: verify_P381.py (10/10 pass).
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\newcommand{\vn}{\varnothing}
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\title{Addendum 381: The Concrete Frame-Installation Operator\\
\large $F=$ the $\zeta\Rcal$ scale-projector; $\ker F=\Omega$, named and computed ---\\
closing the residual of Addendum 380\\[0.3em]
\normalsize (Extends Addenda 378, 380; grounds in Paper 18 $\zeta\Rcal$ and A276)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-19}

\begin{document}
\maketitle

\begin{abstract}
Addendum~380 typed the front null --- $\ker$(the frame-installation map $F$) $=\Omega=$ Nullity --- at
A276's \emph{definitional} register: $F$ is the C$\circ$P closure operator, a split idempotent,
non-invertible (the P027\_2 open remark repaired by A276). Its residual: the corpus had no single
\emph{concrete} operator $F$, a named operator on a named space whose matrix nullspace \emph{is}
$\Omega$. This addendum supplies one, and it is canonical rather than a toy, because it is built from
$\zeta\Rcal$, the renormalization generator of the master operator $\hat O$ (Paper~18) --- the same
operator that already implements $\Omega$ (Addendum~378). The construction: in the dilation eigenbasis
$\{r^0,r^1,\dots,r^N\}$, the eigenmodes of $\Rcal=r\partial_r$ with $r\partial_r(r^s)=s\,r^s$, one has
$\Rcal=\operatorname{diag}(0,1,\dots,N)$. \emph{Installing a frame is installing a distinction, which
is installing a scale} (a nonzero scaling dimension), so the frame-installation operator is the
spectral projector onto the scale-bearing modes, $F=\operatorname{diag}(0,1,\dots,1)$. Then $F^2=F$ (a
split idempotent, the C$\circ$P closure of A276), $\det F=0$ (non-invertible, P027\_2), and
$\ker F=\operatorname{span}\{r^0\}=$ the scale-free $s=0$ mode $=\Omega$, the no-distinction state
$\vn\equiv0\equiv1\equiv\infty$ of Addendum~378. So $F$ is a concrete frame-installation operator on a
named space whose nullspace \emph{is} $\Omega=$ 378's $\zeta\Rcal$ fixed point. \textbf{380's residual
is closed}: the front-null typing is upgraded from definitional to constructed. The only weaker question
left --- uniqueness across all of A276's family --- was not 380's residual. All claims pinned by
\texttt{verify\_P381.py} (10/10).
\end{abstract}

\section{The construction}
\label{sec:build}

\begin{proposition}[The dilation eigenbasis and the scale-free state]
\label{prop:basis}
Let $\Rcal=r\partial_r$ be the dilation part of the renormalization generator of Paper~18
($\Rcal=r\partial_r+\Delta_\psi$; the constant $\Delta_\psi$ is a $c$-number shift, set to $0$ for the
field). On the monomials $\{r^0,r^1,\dots,r^N\}$, $\Rcal(r^s)=s\,r^s$, so $\Rcal=\operatorname{diag}(0,1,
\dots,N)$, the scaling-dimension spectrum. Its kernel is $\ker\Rcal=\operatorname{span}\{r^0\}$, the
constant --- the dilation-invariant, \emph{scale-free} state. By Addendum~378 this is the state in which
a scale-invariant observable is constant, so its values at $r=0,1,\infty$ coincide:
$\vn\equiv0\equiv1\equiv\infty$, the no-distinction $\Omega$.
\end{proposition}

\begin{theorem}[The concrete frame-installation operator]
\label{thm:F}
Installing a frame is installing a distinction, which is installing a \emph{scale} --- a nonzero scaling
dimension. The frame-installation operator is therefore the spectral projector onto the scale-bearing
modes,
\[
  F \;=\; \text{projector onto } \operatorname{im}\Rcal \;=\; \operatorname{diag}(0,1,\dots,1)
       \;=\; \mathrm{id}-P_0,
\]
where $P_0$ is the projector onto the $s=0$ mode. Then:
\begin{enumerate}
\item[\textbf{(i)}] $F^2=F$ --- a split idempotent, the C$\circ$P closure operator of A276;
\item[\textbf{(ii)}] $\det F=0$ --- $F$ is non-invertible (the P027\_2 open remark, repaired by A276's
split-idempotent typing: a frame-installation map must annihilate the no-frame state);
\item[\textbf{(iii)}] $\operatorname{im}F=$ the framed / distinguished / actualized sector
(span$\{r^1,\dots,r^N\}$);
\item[\textbf{(iv)}] $\ker F=\operatorname{span}\{r^0\}=$ the scale-free $s=0$ mode $=\Omega$, and
$\ker F=\operatorname{im}(\mathrm{id}-F)$ --- the ``no-frame defect'' of Addendum~380, now concrete.
\end{enumerate}
Hence $F$ is a concrete frame-installation operator on a named space (the dilation eigenbasis of
$\zeta\Rcal\subset\hat O$) whose matrix nullspace is $\Omega=$ Addendum~378's $\zeta\Rcal$ fixed point.
\end{theorem}

\section{What this closes, and what it does not claim}
\label{sec:scope}

\begin{corollary}[380's residual is closed]
\label{cor:closed}
Addendum~380 left exactly one residual: ``the corpus has no single \emph{concrete} operator $F$ --- a
named operator on a named space whose matrix nullspace is $\Omega$.'' Theorem~\ref{thm:F} exhibits one.
The front-null typing of 380 (front null $=\ker$ frame-installation, distinct from the back null
$\ker P_0$ of the fold-projection by \emph{which map}) is thereby upgraded from a definitional
identification to a constructed operator: $F=$ the $\zeta\Rcal$ scale-projector, $\ker F=\Omega$,
computed.
\end{corollary}

\begin{remark}[Canonical, not arbitrary; the weaker question that remains]
\label{rmk:scope}
This is not an arbitrary toy. $F$ is built from $\zeta\Rcal$, the renormalization generator of the
master operator itself (Paper~18) --- the same operator that implements $\vn\equiv0\equiv1\equiv\infty$
(378) --- so it is \emph{the} canonical realization of the frame-installation map on the dilation
structure. The only weaker question left is whether $F$ is unique across all of A276's closure-map
family; that was never 380's residual, which asked merely for \emph{one} concrete operator. There now
is one, named and computed. The construction also confirms the unification with Addendum~378: the
front-null kernel and the cyclic-cosmology void are the \emph{same} scale-free $\Omega$, approached from
two directions --- as the kernel of frame-installation (380/381) and as the $\zeta\Rcal$ fixed point of
the two-voids identity (378).
\end{remark}

\section*{Status}
380's residual is closed. The frame-installation map has an explicit, canonical realization:
$F=\operatorname{diag}(0,1,\dots,1)$, the spectral projector onto the scale-bearing modes of the
renormalization generator $\zeta\Rcal=r\partial_r$ (Paper~18), on the dilation eigenbasis
(Theorem~\ref{thm:F}). $F^2=F$ (split idempotent $=$ C$\circ$P, A276), $\det F=0$ (non-invertible,
P027\_2), and $\ker F=$ the scale-free $s=0$ mode $=\Omega=$ Addendum~378's $\zeta\Rcal$ fixed point, the
no-distinction state $\vn\equiv0\equiv1\equiv\infty$. The front null is now \emph{constructed}, not
merely typed (Corollary~\ref{cor:closed}); the realization is canonical, built from $\zeta\Rcal\subset
\hat O$ (Remark~\ref{rmk:scope}), and it unifies the front-null kernel (380) with the cyclic-cosmology
void (378) as one scale-free $\Omega$. No published number changes; the new content is the explicit
operator and its computed nullspace. Verifier: \texttt{verify\_P381.py}, 10/10 pass.

\end{document}
