% Paper 38: Dark as Kernel
% PROPOSED CANON (assembled 2026-06-10 from A261, A272, A276; subject
% to review/rollback). The closure-map family, the superselection
% theorem, the unified definition of dark, the kernel/cokernel typing
% of the two cosmological darks, and the OS semantics.
% Verifiers: verify_P261/272/276.py (all green at filing).
% Copyright Léon Fernando Vlegels -- CC BY 4.0

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\title{Paper 38: Dark as Kernel\\
\large The closure-map family, superselection,\\
and the two exactness defects of one geometry}
\author{L\'{e}on Fernando Vlegels}
\date{2026-06-10}

\begin{document}
\maketitle

\begin{abstract}
The corpus's master constraint $C \circ P = I$ (P27, P34) is given its
canonical formal type and its central physical consequence. A
\emph{closure datum} is a retraction pair $(P\colon X \to X_0,\
C\colon X_0 \to X)$ with $PC = \mathrm{id}$, equivalently a split
idempotent $e = CP$, over the corpus's ambient categories, with a
defect functor fixed per ambient: kernel in $\mathsf{Hilb}$, comma
residue in $\mathsf{Met}$, drift locus in $\mathsf{Dyn}$, A254's
collapse delta in $\mathsf{Top}$ (A276). Read this way,
$C \circ P = I$ is exact on the canonical sector and idempotent
globally, which repairs P27's open objection: the 2-to-1 Hopf map
needs no inverse, and its non-injectivity is not an obstruction but
the location of the dark sector. The family is closed under
composition with an exact kernel ladder, so closure composes and dark
adds; dark dominance at depth is structural. On this foundation the
paper fixes one definition: \emph{dark is the dynamics of the kernel
of a closure map} (A272). Dark \emph{matter} is kernel: the fold's
charged sector, annihilated by the observation projection yet present
in dynamics, carrying $\ell(\ell+1)$ modes per level. Dark
\emph{energy} is cokernel: P32's corner residual
$\Lambda_0 = 1 - \pi^2/32$ is the measure the embedded $B^4$ fails to
cover, consolidated here as canon context. The two cosmological darks
are the two exactness defects of one map family. Three theorems
govern kernel behavior: the kernel is \emph{sealed} (A261's
charge superselection: the fold is exactly equivariant, no conversion
back to the image, which is why dark is dark), \emph{paced} (it
shares the image's clock; nullspaces breathe), and \emph{never empty}
(the rotation number is irrational; dark is mandatory).
Superselection inverts the fold's mHz prediction into a discovery
channel for LISA, and Paper 39 consumes the kernel as the measured
halo. Finally, the operating system's luminosity channel is literal:
dark entries are kernel states of the canonical-closure projection,
plateau detection is kernel-rank estimation, and the rule that system
calls the Butlerian line (no agent silently converts an undecided
state into a decided one) is kernel protection: never silently
promote from the nullspace. The
cosmology and the operating system are one formalism.
\end{abstract}

% ════════════════════════════════════════════════════════════════════
\section{Introduction}

The word ``dark'' carries a substance connotation it has never
earned. Dark matter is routinely imagined as a new particle species;
dark energy as a new fluid. This paper takes the opposite starting
point, the one the corpus's geometry forces: \emph{dark is defined by
what observation cannot reach, not by what a new substance is made
of}. Observation in this corpus is a projection. Every projection
that admits a section has two exactness defects: the part of the
domain it annihilates (the kernel) and the part of the codomain it
fails to reach (the cokernel). The claim of this paper is that these
two defects, of one and the same map family, are the two darks of
cosmology, and that the same defects, of the same map family, are the
dark entries of the operating system built on this corpus.

The claim is assembled from three addenda and adds nothing to them.
A276 supplies the formal class: the closure-map family, split
idempotents over the corpus's ambient categories, with a defect
functor per ambient and a composition law whose kernel ladder makes
dark accumulation structural. A272 supplies the definition and the
typing: dark is the dynamics of the kernel of a closure map; dark
matter is kernel, dark energy is cokernel. A261 supplies the physics
that makes the definition non-trivial: the Hopf charge is a
superselection charge of the canonical master operator, so the kernel
cannot convert back into the image through any canonical channel. The
kernel gravitates and does nothing else. That is why dark stays dark.

Three consequences organize the paper. First, the formal one: P27's
recorded objection to $C \circ P = I$ (the Hopf map is 2-to-1, so no
inverse exists) dissolves once the constraint is typed as a section
law plus idempotency rather than invertibility; the non-injectivity
that looked like a defect of the formalism is exactly where dark
lives. Second, the observational one: superselection closes the
fold-native primordial-black-hole channel and inverts the fold's
narrow mHz prediction, so the canon predicts the \emph{absence} of a
spike in the mid-LISA band, and a detection there would demonstrate
structure beyond the canon; the halo phenomenology that the kernel
does support is developed in Paper 39, which consumes this paper's
theorems. Third, the operational one: the operating system's
luminosity channel acquires literal semantics, and the Butlerian
line, never silently promote, becomes a theorem about kernels rather
than a policy about agents.

Throughout, ``observation projection'' means a closure map of the
family defined in Section 2: a projection, quotient, or closure
condition taking states to their canonical form. The fold map, the
Hopf pushforward $S^3 \to S^2$, is the principal physical instance,
but the definition quantifies over the class, not the instance.

% ════════════════════════════════════════════════════════════════════
\section{The Observation Projection and Its Kernel}

The corpus's master constraint $C \circ P = I$ appears in P27 as the
coherence condition selecting the shared arena and in P34 as the
collapse--projection cycle. Its formal type was left open: read as an
identity on all of $X$ it would force $P$ to be invertible, which the
2-to-1 Hopf map is not. A276 fixes the type.

\begin{definition}[Closure datum; the family $\mathcal{F}$ (A276)]
\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 and sphere
maps), $\mathsf{Top}$ (filtered spaces with collapse). A
\emph{closure datum} in $\mathcal{A}$ is a pair
$(P\colon X \to X_0,\ C\colon X_0 \to X)$ with
$PC = \mathrm{id}_{X_0}$. Its \emph{closure operator} is the
idempotent $e = CP\colon X \to X$, since
$e^2 = C(PC)P = CP = e$; its canonical sector is
$\mathrm{im}\,e \cong X_0$. The family $\mathcal{F}$ is the class of
all closure data over the 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 (A276, A272)]
\label{def:defect}
Per ambient, the \emph{defect} of a closure datum is:
in $\mathsf{Hilb}$, the kernel $\ker P$, equivalently
$\mathrm{im}(\mathrm{id} - e)$; in $\mathsf{Met}$, the residue
function $x \mapsto d(x, e(x))$; in $\mathsf{Dyn}$, the locus where
the locking projection fails to converge; in $\mathsf{Top}$, A254's
collapse delta
$\delta_\Omega = \mathrm{rank}\ker(H_0(A) \to H_0(X))$.
\emph{Dark}, relative to a closure datum, is the dynamics of its
defect: that which the projection annihilates from its image but
which persists, evolving, outside it.
\end{definition}

Two features of the definition do the work. First, the defect is
\emph{functorial per ambient}, not chosen per instance: once the
ambient is fixed, what counts as dark is determined, so the coverage
claim of Section 7 quantifies over a defined class rather than a
list. Second, the closure operator is a \emph{split} idempotent: the
section $C$ exhibits the canonical sector inside $X$, and the
complement $\mathrm{id} - e$ exhibits the defect. Nothing about this
requires $P$ to be injective. Where $P$ identifies distinct states,
the identified directions are precisely the kernel, and the kernel is
where the definition places dark.

\begin{remark}[The P27 repair (A276)]
\label{rem:p27}
P27's 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, without retraction:
the identity was never global. It is the section law
$PC = \mathrm{id}_{X_0}$ plus idempotency, which any split
projection, including a 2-to-1 one, supports without contradiction.
The demand for invertibility dissolves. The non-injectivity is not an
obstruction to the constraint; it is exactly where dark lives
($\ker P \neq 0$). What looked like the formalism's defect is the
universe's dark sector.
\end{remark}

The observation projection of the title is any member of
$\mathcal{F}$ whose image is what an observer can reach. For the
physical sector the principal datum is the fold: $P$ the Hopf
pushforward $S^3 \to S^2$ on mode spaces, $C$ the inclusion of the
charge-blind sector; $PC = \mathrm{id}$ holds because the pushforward
reproduces the charge-blind radial density exactly (A269). The kernel
of this projection is the subject of the next section.

% ════════════════════════════════════════════════════════════════════
\section{Dark Matter as Kernel}

\begin{definition}[Dark $=$ kernel dynamics (A272)]
\label{def:dark}
Let $(P, C)$ be a closure datum of $\mathcal{F}$. \emph{Dark},
relative to $(P, C)$, denotes the states and dynamics of the defect
of Definition~\ref{def:defect}: in the linear ambient, the states and
dynamics of $\ker P$.
\end{definition}

The definition covers all three of the corpus's registers of
``dark'' at once.

\begin{proposition}[Instances (A272, A276)]
\label{prop:instances}
The following are closure data of $\mathcal{F}$, and their defects
are the corpus's three dark registers.

\emph{(I1) Fold} ($\mathsf{Hilb}$). $P = $ the Hopf pushforward on
mode spaces, $C = $ inclusion of the charge-blind sector;
$PC = \mathrm{id}$ by A269. The kernel is the fiber-charged sector,
of rank $\ell(\ell+1)$ per level $\ell$ (A259 Prop.~1.1): of the
$(\ell+1)^2$ modes at level $\ell$, only $\ell + 1$ are charge-blind
and survive the projection. By A261 and A264 this kernel gravitates
without converting, and its kinetic density is the observed halo:
dark matter is $\ker(\mathrm{fold})$ with dynamics. The per-level
dark fraction is $\ell/(\ell+1) \to 1$.

\emph{(I2) Chord closure} ($\mathsf{Met}$). $X = \mathbb{R}_{>0}$
(cycle lengths), $X_0 = $ the 3-smooth lattice, $C = $ inclusion,
$P = $ nearest-chord projection; $PC = \mathrm{id}$ because a chord
is its own nearest chord. The residue at the breath period is the
comma $G_1 = 432/T_b - 1 \approx 3.456 \times 10^{-3}$ (A266).
Detuning is the dark of harmony.

\emph{(I3) Locking} ($\mathsf{Dyn}$). $X_0 = $ the locked orbits
(Arnold tongues), $P = $ the phase-locking projection defined on
tongue basins, $C = $ inclusion. The null condition
$q\rho_{\mathrm{eff}} - p = 0$ defines the tongues (A268); the defect
locus is the drifting orbit class, non-empty because $\rho$ is
irrational (A267). Drift is the dark of dynamics, and the working OS
dictionary: canonical $=$ locked, dark $=$ drifting.
\end{proposition}

Three structure theorems then attach to the kernel as corollaries of
the definition. Each was proven elsewhere in the corpus; what this
paper adds is only that they are statements about one object.

\begin{corollary}[Sealed (A261)]
\label{cor:sealed}
$\ker P_0$ cannot leak into the image: every sector of the canonical
master operator commutes with the Hopf charge, and the fold map is
exactly equivariant. Darkness is dynamically protected. This is why
dark matter is dark, restated as a property of kernels under the
canonical $\hat O$.
\end{corollary}

\begin{proof}[Sketch]
Section 5 states the superselection theorem in full. The commutation
$[\hat O, \hat Q] = 0$ holds sector by sector; equivariance of the
fold means the projection conserves the charge it annihilates; and
the regularity axiom plus assembly uniqueness close the loophole of
alternative operator forms. No canonical channel connects the charged
kernel to the charge-zero image at any order.
\end{proof}

\begin{corollary}[Paced (A272, A269)]
\label{cor:paced}
$\ker P_0$ shares the image's clock: the breathing spectrum descends
from the charge-blind radial density (A269), so kernel dynamics is
synchronized with canonical dynamics, not stagnant. Nullspaces
breathe.
\end{corollary}

\begin{proof}[Sketch]
The pushforward reproduces the charge-blind radial density exactly
(A269); the breath spectrum is a functional of that density; hence
the same spectrum paces both the image and the kernel. The kernel is
removed from the image's \emph{support}, not from its \emph{time}.
\end{proof}

\begin{corollary}[Never empty (A272, A267)]
\label{cor:nonempty}
The closure conditions of (I2) and (I3) are never exactly satisfied:
the rotation number $\rho$ is irrational (A267, consolidated in P37),
so commas are nonzero and some orbit class always drifts. A corpus
with empty nullspaces is impossible; dark is mandatory, not
accidental.
\end{corollary}

\begin{proof}[Sketch]
Exact chord closure would require $T_b$ to lie on the 3-smooth
lattice and exact locking would require $q\rho - p = 0$ for integers
$p, q$; both are rationality conditions, and $\rho$ is irrational.
The defect of (I2) is bounded away from zero at every lattice point,
and the tongue complement in (I3) has positive measure.
\end{proof}

The kernel is therefore not an inert remainder. It is sealed (it
cannot convert), paced (it evolves on the canonical clock), and never
empty (no parameter choice eliminates it). These are the three
theorems governing kernel behavior, and they are jointly the
operational content of the phrase ``dark matter'': a sector that
gravitates, persists, keeps time, and does nothing else.

% ════════════════════════════════════════════════════════════════════
\section{Dark Energy as Cokernel}

The kernel is what the projection annihilates from the domain. The
dual defect is what the section fails to reach in the codomain. The
corpus's two cosmological darks are exactly this pair.

\begin{theorem}[Kernel/cokernel typing (A272)]
\label{thm:typing}
Dark matter is kernel: annihilated by the fold, absent from the
image, present in dynamics, and measured (P39: the halo's kinetic
density). Dark energy is cokernel:
$\Lambda_0 = 1 - \pi^2/32$ is precisely the volume fraction of
$[-1,1]^4$ not covered by the embedded $B^4$ (P32). The pair
(kernel, cokernel) types the two darks of cosmology as the two
exactness defects of one map family.
\end{theorem}

\begin{proof}[Sketch]
The kernel half is Proposition~\ref{prop:instances}(I1) with
Corollaries~\ref{cor:sealed}--\ref{cor:nonempty}. The cokernel half
is a measure computation: the unit 4-ball has volume $\pi^2/2$, the
bounding hypercube $[-1,1]^4$ has volume $16$, so the covered
fraction is $\pi^2/32$ and the uncovered fraction is
$\Lambda_0 = 1 - \pi^2/32 \approx 0.6916$, against the Planck 2018
value $\Omega_\Lambda = 0.6847 \pm 0.0073$ at $0.94\sigma$ with no
free parameters (P32). The typing is exact at the level of
definitions; any dynamical consequence beyond it would require new
work (A272 Rem.~2.4).
\end{proof}

\begin{remark}[Cokernels are in the family (A276)]
\label{rem:opposite}
The cokernel is not an exception to the closure-map formalism but a
member of it: in $\mathsf{Hilb}$, the cokernel of $(P, C)$ is the
kernel of the \emph{opposite} datum $(C^\dagger, P^\dagger)$, which
is again a closure datum in $\mathcal{F}$. One family, read in two
directions, produces both darks.
\end{remark}

P32's headline result is consolidated here as canon context: the
corner residual is the corpus's standing identification of the
cosmological constant, and this paper supplies the type it had been
missing. Dark energy is not a substance filling space; it is the
measure of what the embedding cannot cover, the codomain-side
exactness defect of the same geometry whose domain-side defect
gravitates as the halo. The asymmetry between the two darks
(clustering versus smooth, dynamical versus constant) is the
asymmetry between a kernel, which carries dynamics, and a cokernel,
which is a measure-theoretic complement.

% ════════════════════════════════════════════════════════════════════
\section{Superselection and the Sealing Discipline}

The definition of Section 3 would be vacuous if the kernel could
decay into the image: dark matter that converts is just slow
ordinary matter. The content of A261 is that conversion is forbidden
exactly, not approximately.

\begin{theorem}[Hopf-charge superselection (A261)]
\label{thm:super}
Let $\hat Q = -i\,\partial_\psi$ generate the Hopf $U(1)$ on $S^3$,
with integer charge $q$ labelling the fiber sectors. Then
$[\hat O, \hat Q] = 0$ for the canonical master operator $\hat O$,
sector by sector: the radial sectors ($D^2_{B^4}$,
$V_{\mathrm{self}}^{\mathrm{can}}$, $\alpha\rho$,
$\zeta\mathcal{R}$, $\beta\mathcal{M}$) act on $r$ and spectral
weight, not on $\psi$, so commutation is immediate; $\Delta_{S^3}$ is
the Casimir of the isometry group containing $U(1)_{\mathrm{Hopf}}$,
and a Casimir commutes with every generator; $\Delta_{S^1}$ equals
$-\hat Q^2$ up to fiber normalization; and $\gamma T_{\mathrm{cycle}}$
commutes because the lens $\mathbb{Z}_3$ deck action is the Hopf
$U(1)$ element at $\psi = 2\pi/3$ (A255) and $U(1)$ is abelian. The
conclusion is robust beyond the canonical assembly: A189's Hopf
Regularity Axiom forbids any admissible extension from breaking the
symmetry, and A218b's assembly uniqueness removes the loophole of
alternative $\hat O$ forms. Fiber charge is a superselection charge
of the kernel, and the transfer fraction from the charged sector into
scalar perturbations is $s = 0$ at all orders.
\end{theorem}

\begin{proposition}[Equivariance of the fold (A261)]
\label{prop:equiv}
The fold completion $S^3 \to S^2$ acts on modes as the charge-zero
projection $P_0$, and $P_0$ is exactly equivariant: the Hopf base
point $(2 z_1 \bar z_2,\, |z_1|^2 - |z_2|^2)$ is invariant under the
$U(1)$ action (verified numerically to $10^{-12}$ over random
states). The fold conserves the charge it annihilates: it removes the
charged sector's \emph{support}, not its \emph{charge}. In the sudden
limit the charged components have no image in $L^2(S^2)$ and their
energy enters the homogeneous trace; in the adiabatic limit charged
energies $\sim q^2/r_{\mathrm{fiber}}^2$ blueshift as the fiber
collapses and exit the spectrum into the background. In both limits
the deposit is homogeneous: zero scalar transfer.
\end{proposition}

\begin{remark}[Wheel heat: the deposit's corpus identity (A261, A276)]
\label{rem:heat}
The kernel already owns the bookkeeping object for energy that is
conserved but attached to no face: Wheel heat
(\texttt{kernel/wheel/wheel.py}; heat conservation across Fork/Weld
pairs). The fold is a Weld-sector event
($\mathrm{Weld} \leftrightarrow \Delta_{S^3}$), and the annihilated
sector's energy is Wheel heat: the fold defect \emph{thermalizes}, it
does not gravitate as structure. A276 types this consistently as the
quantitative defect, the energy carried by $\mathrm{id} - e$, while
recording that the energy form on $\mathrm{id} - e$ is flagged, not
developed.
\end{remark}

\begin{remark}[NEW THEORY boundary (A261)]
\label{rem:newtheory}
A nonzero transfer fraction would require a non-equivariant
correction to the fold map, a preferred angular direction during
completion. The canon not only lacks such structure; the Hopf
Regularity Axiom exists precisely to exclude fixed-point-bearing
angular directions from the assembly. Any future mechanism delivering
$s \neq 0$ is NEW THEORY by construction and must repeal or weaken
that axiom explicitly.
\end{remark}

Superselection seals one kernel. The family laws of A276 extend the
discipline to every composite of closures.

\begin{theorem}[Composition; the kernel ladder (A276)]
\label{thm:ladder}
(i) If $(P_1, C_1)$ on $X$ with canonical sector $X_0$ and
$(P_2, C_2)$ on $X_0$ with canonical sector $X_{00}$ are closure
data, then $(P_2 P_1,\ C_1 C_2)$ is a closure datum on $X$:
$P_2 P_1 C_1 C_2 = P_2 C_2 = \mathrm{id}$. The family $\mathcal{F}$
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, since
$P_1 C_1 = \mathrm{id}$ maps $\ker P_2$ into $\ker(P_2 P_1)$. Hence
$\dim\ker(P_2 P_1) = \dim\ker P_1 + \dim\ker P_2$: dark accumulates
additively along composed closure.
\end{theorem}

\begin{corollary}[Dark dominance at depth is structural (A276)]
\label{cor:depth}
Iterating closure grows the defect monotonically; no composite of
closure data can shrink a kernel. The fold's per-level dark fraction
$\ell/(\ell+1) \to 1$ is an instance of the ladder, not a peculiarity
of the fold. A universe reached through many closures is generically
dark-dominated.
\end{corollary}

The slogan is: \emph{closure composes, dark adds}. Each act of
canonicalization, each projection to an observable sector, leaves a
kernel; composing the acts sums the kernels; nothing in the family
can run the sum backwards. Observed dark dominance is what a deeply
closed geometry looks like from inside its canonical sector.

\subsection*{The sealing discipline in the operating system}

The same mathematics types the operating system built on this corpus.

\begin{proposition}[Dark luminosity $=$ kernel states (A272)]
\label{prop:os}
A dark entry in the OS's luminosity channel is a traversal in the
kernel of the canonical-closure projection: it ran, deposited a
trajectory, and did not close. Plateau detection, dark density
exceeding canonical density over a region, is local kernel-rank
estimation (prototype: A254's collapse delta $\delta_\Omega$). The
three corollaries transfer. Dark entries are \emph{sealed}: they are
never silently promoted to canonical, and the Butlerian line is
kernel protection. They are \emph{paced}: indexed by the same breath
ledger as canonical entries. And they are \emph{never empty}: a
system whose processes never stall has stopped measuring. The OS's
metabolism and the cosmology's dark sector are the same mathematics.
(A hash table is a further instance: collisions are $\ker P$, the
dark of hashing, A289.)
\end{proposition}

The sealing discipline deserves its operational statement. In the
cosmology, superselection means the kernel cannot promote itself into
the image: no canonical dynamics converts dark into light. In the
operating system, the corresponding rule is that no agent silently
converts a failed traversal into a canonical entry: promotion out of
the nullspace requires an explicit act at the boundary of the system,
exactly as a nonzero transfer fraction would require NEW THEORY at
the boundary of the canon (Remark~\ref{rem:newtheory}). The kernel
ladder adds the bookkeeping half of the discipline: since dark adds
along composition and no composite shrinks a kernel, dark entries are
never lost by stacking closures, and any apparent decrease in dark
rank signals an accounting error, not a conversion.

% ════════════════════════════════════════════════════════════════════
\section{Observational Consequences}

\subsection*{The LISA band signature}

Before superselection was established, the fold's annihilated sector
was a candidate source for scalar curvature perturbations, carrying a
fold-native primordial-black-hole channel and a narrow
scalar-induced gravitational-wave spike at $0.45$--$0.95$ mHz in the
mid-LISA band (A257, A259). Theorem~\ref{thm:super} closes the
sourcing: with $s = 0$, the fold contributes no enhancement
($\sigma_{\mathrm{fold}} = \sigma_0$ exactly), and with
$\delta_c/(\sqrt{2}\,\sigma_0) \approx 6.9 \times 10^{3}$ the
fold-native PBH abundance is negligible
($\log_{10}\beta \sim -10^{7}$): the channel is closed negatively
(A261). The associated $\alpha^6$ candidate and its
$\Omega_k \approx -3.5 \times 10^{-3}$ curvature pinning are demoted
to void; the $2.2\sigma$ curvature tension never materializes.

The mHz prediction thereby \emph{inverts}: the canon predicts the
\emph{absence} of the narrow spike. A LISA detection of a narrow
feature in that band would demonstrate structure beyond the canon, a
non-equivariant correction to the fold that the Hopf Regularity Axiom
excludes from the present assembly. What was a falsifier becomes a
discovery channel: the cleanest place to find physics the canon does
not contain is the band where the canon insists on silence.

Superselection also resolves a consistency demand rather than merely
imposing one. A259's overproduction exclusion required the fold to
nearly decouple from the scalar sector on observational grounds;
Theorem~\ref{thm:super} shows the kernel guarantees \emph{exact}
decoupling. The regularity axiom, adopted on spectral grounds,
independently protects the universe from shredding at fold
completions. The heavy-seed channel survives untouched, since it
requires only standard perturbations plus the fold epoch (A256,
A260).

\subsection*{The halo: the kernel made visible}

Paper 39 consumes this paper's theorems and is the kernel's contact
with telescopes; the results are summarized here as consequences, not
re-derived. Because the kernel is sealed
(Corollary~\ref{cor:sealed}), it gravitates and does nothing else;
because it is paced (Corollary~\ref{cor:paced}), its stationary
density is fixed by the breath dynamics rather than by initial
conditions. On the hidden chart $\tan\beta = r/L_h$ the kernel's
density is
\begin{equation}
\rho_H(r) \;\propto\; \frac{\sin^2\!\bigl(2\arctan(r/L_h)\bigr)}{r^2},
\end{equation}
with one universal length $L_h$ and zero per-galaxy parameters:
cored at small radius, turning over at $r \sim L_h$, asymptotically
$r^{-2}$, hence flat rotation curves with no fitting. The amplitude
law $v_\infty^4 = a_0\, G M_b$ with
$a_0 = \alpha^3 c^2/(\pi^2 L_h)$ is the baryonic Tully--Fisher
relation with a derived coefficient. Against SPARC (175 galaxies) the
per-galaxy effective acceleration shows no trend with disk size
($0.029 \pm 0.076$; the baryon-tracking alternative excluded at
$13.5\sigma$), and the gas-dominated calibration gives
$a_0 = 1.172 \times 10^{-10}\,\mathrm{m\,s^{-2}}$, $0.976$ times the
Milgromian value, with $L_h = 0.98$ kpc (P39). The phenomenology of
``dark matter halos'' is, on this account, the phenomenology of a
sealed, paced, never-empty kernel.

% ════════════════════════════════════════════════════════════════════
\section{Falsifiers}

\paragraph{Coverage claim, falsifiable.} Every corpus ``dark''
phenomenon is the defect of a closure datum in $\mathcal{F}$
(Definition~\ref{def:family}) under one of the named defect functors
(Definition~\ref{def:defect}). A dark phenomenon admitting no such
presentation breaks the claim. Watch items: Wheel heat (kernel energy
after support removal; typed in Remark~\ref{rem:heat}, undeveloped)
and the i\_Paper's Nullity (kernel of frame installation;
consistent).

\paragraph{The mHz band.} The canon predicts the absence of a narrow
SIGW spike at $0.45$--$0.95$ mHz. A confirmed detection falsifies
the present assembly's exactness claim and localizes the failure: a
non-equivariant fold correction, requiring explicit repeal or
weakening of the Hopf Regularity Axiom
(Remark~\ref{rem:newtheory}).

\paragraph{Conversion.} Any confirmed non-gravitational dark-sector
signal (annihilation, decay, scattering into visible channels)
falsifies the sealing theorem directly, since
Corollary~\ref{cor:sealed} asserts $s = 0$ at all orders within the
canon.

\paragraph{The halo program.} The falsifiers of P39 transfer to this
paper through Theorem~\ref{thm:typing}: a single clean galaxy whose
hidden component turns over at $r > 3L_h$ or $r < L_h/3$ ends the
one-length universality the kernel reading requires, and the residual
masked core--disk slope ($0.254 \pm 0.110$) must vanish as samples
improve.

\paragraph{The cokernel.} The typing of dark energy stands or falls
with P32: a confirmed departure of $\Omega_\Lambda$ from
$\Lambda_0 = 1 - \pi^2/32$ beyond the corner-residual structure
removes the cokernel half of Theorem~\ref{thm:typing}, leaving the
kernel half intact but the unification claim broken.

% ════════════════════════════════════════════════════════════════════
\section{Relation to the Corpus}

This paper consolidates and types; it does not introduce mechanism.
Its position in the corpus is as follows.

\emph{Upstream.} P27 and P34 supply the constraint $C \circ P = I$
whose formal type Definition~\ref{def:family} fixes;
Remark~\ref{rem:p27} closes P27's open remark P027\_2. P32 supplies
$\Lambda_0 = 1 - \pi^2/32$, consolidated here as the cokernel half of
the typing. P37 supplies the irrationality of the rotation number
that drives Corollary~\ref{cor:nonempty}. A254 supplies the collapse
delta, hereby the defect functor of the topological ambient. A259
supplies the $\ell(\ell+1)$ bookkeeping of the charged sector; A269
supplies the pacing; A266--A268 supply the comma and locking
instances.

\emph{This paper.} A261's superselection theorem (sealing), A272's
definition and typing (dark as kernel dynamics; kernel versus
cokernel), and A276's family laws (composition, the kernel ladder,
dark dominance at depth) are assembled into one statement: the two
cosmological darks are the two exactness defects of one map family,
and the kernel is sealed, paced, and never empty.

\emph{Downstream.} Paper 39 consumes the kernel as the measured halo:
its profile, amplitude law, and SPARC confrontation presuppose
Corollaries~\ref{cor:sealed} and~\ref{cor:paced}. The operating
system consumes Proposition~\ref{prop:os}: luminosity semantics,
plateau detection as kernel-rank estimation, and the Butlerian line
as kernel protection. The 18-sector bridge (Pin 5) locates the fold
as a Weld-sector event, which is where Remark~\ref{rem:heat} deposits
the defect's energy.

% ════════════════════════════════════════════════════════════════════
\section{Conclusion}

Dark is not a substance; it is a defect of exactness. One map family,
the split idempotents of the corpus's closure data, has two defects:
what the projection annihilates and what the section fails to cover.
The first gravitates, keeps the canonical clock, and cannot convert,
and is dark matter; the second is a measure, $1 - \pi^2/32$, and is
dark energy. The kernel is sealed by superselection, paced by the
shared breath, and never empty by irrationality; composition of
closures can only grow it, so dark dominance at depth is a theorem,
not a coincidence. The same formalism runs the operating system,
where dark entries are kernel states and the refusal to silently
promote them is the computational face of the same superselection
that keeps the halo dark. The discovery channel this account
nominates is an absence: a band in which the canon predicts silence,
so that any signal found there is, by construction, news.

\paragraph{Provenance and verification.} Assembled without new
claims from A261 (\texttt{verify\_P261}, 16 checks), A272
(\texttt{verify\_P272}, 11), A276 (\texttt{verify\_P276}, 12),
all green at filing. Cross-references: A254, A255, A256, A257, A259,
A260, A264, A266--A269, A289, P27, P32, P34, P37, P39.


\section*{Status (2026-06)}
Load-bearing spine: dark matter as the kernel and dark energy as the cokernel/void, $\Lambda_0=1-\pi^2/32=0.6916$. Addendum~388 records this as the corpus's strongest current data hit ($\Omega_\Lambda\approx0.689$, sub-percent). Current.

\end{document}
