% 276_Addendum_ClosureMapFamily.tex
% OI-272-1 closed: the C∘P closure-map family is formalized as split
% idempotents over three ambient categories, with a defect functor per
% ambient. A272's Definition 1.1 now quantifies over a defined class;
% the family is closed under composition; the kernel ladder makes dark
% accumulation at depth structural. Verifier: verify_P276.py.
% Copyright Léon Fernando Vlegels -- CC BY 4.0

\documentclass[11pt]{article}
\usepackage{amsmath,amssymb,amsthm}
\usepackage[margin=1in]{geometry}
\usepackage{booktabs}
\usepackage{hyperref}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}
\newcommand{\Tb}{T_{\rm breath}}

\title{Addendum 276: The Closure-Map Family\\
\large Split idempotents, defect functors,\\
and the quantification of ``dark'' (OI-272-1 closed)}
\author{TOE Corpus}
\date{2026-06-10}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case A (formalization; family laws proven; coverage
claim quantified) \quad|\quad \textbf{Closes:} OI-272-1 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P276.py} --- all PASS

\begin{abstract}
A272 defined dark as the dynamics of the kernel of a ``closure map of
the corpus's collapse--projection family,'' quantifying over a list
(fold, chord closure, locking) rather than a class; OI-272-1 asked for
the class. It is supplied here. A \emph{closure datum} is a retraction
pair $(P\colon X \to X_0,\ C\colon X_0 \to X)$ with $P C = \mathrm{id}$,
equivalently a split idempotent $e = C P$ on $X$; this is the formal
content of the corpus's $C \circ P = I$ (P27, P34), read as an identity
on the canonical sector rather than on all of $X$ (on all of $X$ it
would force invertibility --- the precise objection of P27's Open
remark P027\_2 on the 2-to-1 Hopf map, which this typing repairs
without retracting).
\emph{Dark} is the defect of $e$ from $\mathrm{id}_X$, measured by a
defect functor fixed per ambient category: kernel (linear ambient
$\mathsf{Hilb}$), residue distance $x \mapsto d(x, e(x))$ (metric
ambient), divergence locus of the locking projection (dynamical
ambient). The three A272 instances are members
(Proposition~\ref{prop:members}); A254's collapse delta
$\delta_\Omega = \mathrm{rank}\ker(H_0(A) \to H_0(X))$ is the defect
functor of the topological ambient, confirming A272's prototype claim.
\emph{Family laws} (Theorem~\ref{thm:laws}): closure data compose
--- $(P_2 P_1, C_1 C_2)$ is again a closure datum --- and in the linear
ambient the kernel ladder
$0 \to \ker P_1 \to \ker(P_2 P_1) \to \ker P_2$ is exact, so
\emph{dark accumulates along composition}: deeper closure means larger
kernel. A272's consonance remark (dark fraction $\ell/(\ell+1) \to 1$
at depth) is thereby upgraded from consonance to structure: kernel
growth under composed closure is a theorem of the family, not a
property of one instance. The coverage claim is restated over the
defined class and remains falsifiable. Wheel heat (A261's thermalized
defect) is typed as the quantitative defect --- the energy carried by
$\mathrm{id} - e$ --- consistent with A272's watch item; flagged, not
developed.
\end{abstract}

% ────────────────────────────────────────────────────────────────────────────
\section{The Family}

\begin{definition}[Closure datum; the family $\mathcal{F}$]
\label{def:family}
Let $\mathcal{A}$ be one of the corpus's ambient categories:
$\mathsf{Hilb}$ (mode spaces), $\mathsf{Met}$ (metric state spaces with
distinguished lattices), $\mathsf{Dyn}$ (circle/sphere maps). A
\emph{closure datum} in $\mathcal{A}$ is a pair
$(P\colon X \to X_0,\ C\colon X_0 \to X)$ with $P C = \mathrm{id}_{X_0}$.
Its \emph{closure operator} is the idempotent $e = C P\colon X \to X$
($e^2 = C(PC)P = CP = e$); its canonical sector is
$\mathrm{im}\,e \cong X_0$. The family $\mathcal{F}$ is the class of
all closure data over the three ambients. The corpus's $C \circ P = I$
is the statement $e|_{\mathrm{im}\,e} = \mathrm{id}$ --- exact on the
canonical sector, idempotent (not invertible) globally.
\end{definition}

\begin{definition}[Defect functors; dark]
\label{def:defect}
Per ambient, the defect of a closure datum is:
$\mathsf{Hilb}$: $\ker P$ (equivalently $\mathrm{im}(\mathrm{id}-e)$);
$\mathsf{Met}$: the residue function $x \mapsto d(x, e(x))$ (commas);
$\mathsf{Dyn}$: the locus where the locking projection fails to
converge (drift). \emph{Dark} (A272 Def.~1.1) is the dynamics of the
defect. In the topological ambient the defect is A254's collapse delta
$\delta_\Omega(A, X) = \mathrm{rank}\ker(H_0(A) \to H_0(X))$, which is
hereby the defect functor of $\mathsf{Top}$, confirming A272's
prototype identification.
\end{definition}

\begin{proposition}[The three instances are members]
\label{prop:members}
(I1) \emph{Fold} ($\mathsf{Hilb}$): $P = $ Hopf pushforward on mode
spaces, $C = $ inclusion of the charge-blind sector; $PC = \mathrm{id}$
because the pushforward reproduces the charge-blind radial density
exactly (A269); $\ker P = $ the charged sector, rank $\ell(\ell+1)$ per
level (A259). (I2) \emph{Chord closure} ($\mathsf{Met}$): $X = \mathbb{R}_{>0}$
(cycle lengths), $X_0 = $ the 3-smooth lattice, $C = $ inclusion,
$P = $ nearest-chord projection; $PC = \mathrm{id}$ (a chord is its own
nearest chord); residue at $\Tb$ is $G_1$ (A266). (I3) \emph{Locking}
($\mathsf{Dyn}$): $X_0 = $ locked orbits (tongues), $P = $ the
phase-locking projection defined on tongue basins, $C = $ inclusion;
defect locus $= $ the drifting class, non-empty since $\rho$ is
irrational (A267/A268).
\end{proposition}

\begin{remark}[P27's Open remark P027\_2, typed without retraction]
P27 (Open remark P027\_2) observes that the Hopf map is 2-to-1, so
$C \circ P = I$ cannot hold by invertibility, and records the gap as
open. Definition~\ref{def:family} is the repair: the identity was never
global --- it is the section law $PC = \mathrm{id}_{X_0}$ plus
idempotency, which a 2-to-1 (indeed any split) projection supports
without contradiction. The non-injectivity is not an obstruction; it is
exactly where dark lives ($\ker P \neq 0$). The remark's demand for
invertibility dissolves; the closure is recorded in the A277 registry.
\end{remark}

% ────────────────────────────────────────────────────────────────────────────
\section{Family Laws}

\begin{theorem}[Composition; the kernel ladder]
\label{thm:laws}
(i) If $(P_1, C_1)$ on $X$ with canonical $X_0$ and $(P_2, C_2)$ on
$X_0$ with canonical $X_{00}$ are closure data, then
$(P_2 P_1,\ C_1 C_2)$ is a closure datum on $X$:
$P_2 P_1 C_1 C_2 = P_2 C_2 = \mathrm{id}$. $\mathcal{F}$ is closed
under composition (and contains identities), hence forms a category
fibred over the ambients. (ii) In $\mathsf{Hilb}$ the sequence
\begin{equation}
0 \;\to\; \ker P_1 \;\to\; \ker(P_2 P_1)
\;\xrightarrow{\;P_1\;}\; \ker P_2 \;\to\; 0
\end{equation}
is exact (the right map is onto because $C_1$ sections it:
$P_1 C_1 = \mathrm{id}$ maps $\ker P_2$ into $\ker(P_2 P_1)$), so
$\dim\ker(P_2 P_1) = \dim\ker P_1 + \dim\ker P_2$: \emph{dark
accumulates additively along composed closure}.
\end{theorem}

\begin{corollary}[Dark dominance at depth is structural]
Iterating closure grows the defect monotonically; no composite of
closure data can shrink a kernel. A272's consonance remark --- per-level
dark fraction $\ell/(\ell+1) \to 1$, ``kernel dominance at depth'' ---
is an instance of the ladder, not a peculiarity of the fold. A universe
reached through many closures is generically dark-dominated.
\end{corollary}

\begin{remark}[Wheel heat as quantitative defect --- flag]
A261's thermalized defect (Wheel heat) is naturally the energy of
$\mathrm{id} - e$: the quantitative size of the defect rather than its
support. This types A272's watch item consistently (heat $=$ kernel
energy after support removal) but is recorded as a flag --- the energy
form on $\mathrm{id} - e$ is not developed here.
\end{remark}

\paragraph{Coverage, restated over the class.} A272's falsifiable claim
now reads: every corpus ``dark'' phenomenon is the defect of a closure
datum in $\mathcal{F}$ (Definition~\ref{def:family}), under one of the
three defect functors (Definition~\ref{def:defect}). A dark phenomenon
admitting no such presentation breaks the claim. The i\_Paper's Nullity
(kernel of frame installation, $\mathsf{Hilb}$-like) and dark energy
(cokernel --- the dual defect, A272 Rem.~2.4) remain consistent;
cokernels enter as the defect of the \emph{opposite} datum
$(C^\dagger, P^\dagger)$ in $\mathsf{Hilb}$, which is again in
$\mathcal{F}$.

\paragraph{Consequences.} OI-272-1 closed; A272's definition quantifies
over a defined class. The kernel ladder gives the corpus a new
structural slogan with theorem backing: closure composes, dark adds.

\paragraph{Verifier.} \texttt{verify\_P276.py}: idempotency and section
laws on explicit Hilb instances (random splittings, 50-digit);
composition law; exactness of the kernel ladder (rank arithmetic on
explicit matrices); I1 rank recomputation; I2 residue $= G_1$ and
idempotency of nearest-chord on a 3-smooth grid; I3 non-empty defect
(irrationality, tongue distance); cokernel-as-opposite-datum check.
All PASS.

\end{document}
