% 272_Addendum_NullspaceBehavior.tex
% Synthesis addendum: "dark," in every corpus register, denotes the dynamics
% of the kernel of a closure map. Definition; three instances with their
% already-proven structure theorems as corollaries; kernel/cokernel typing of
% dark matter vs dark energy; literal kernel semantics for ToENet's
% luminosity channel. No new computation; one vocabulary.
% Verifier: verify_P272.py.
% Copyright Léon Fernando Vlegels -- CC BY 4.0

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\title{Addendum 272: Nullspace Behavior\\
\large Dark as kernel dynamics: one definition,\\
three instances, three inherited theorems}
\author{TOE Corpus}
\date{2026-06-10}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Synthesis (definitions and corollaries only; no new
computation; falsifiable typing) \quad|\quad
\textbf{Verifier:} \texttt{verify\_P272.py} --- all PASS

\begin{abstract}
The corpus uses ``dark'' in three registers: dark matter (the
gravitating, non-converting sector, A259--A264), dark luminosity
(ToENet's failed-traversal channel), and the detuned/drifting side of
the closure dichotomies introduced in A266--A268 (commas; quasiperiodic
orbits outside Arnold tongues). This addendum fixes one definition
covering all three: \emph{dark denotes the dynamics of the kernel of a
closure map}. Each register is an instance: (I1) the fold map $P_0$
(Hopf pushforward) has $\ker P_0 = $ the charged sector --- the dark
halo is literally the fold's nullspace, with dynamics; (I2) the
just-lattice closure homomorphism has commas as its non-closure residue;
(I3) the mode-locking condition $q\rho - p = 0$ is a null condition
whose complement is the drifting (``dark'') orbit class. Three structure
theorems already proven this cycle then attach as corollaries of the
definition: the kernel is \textbf{sealed} (A261 superselection: no
conversion back to the image), \textbf{paced} (A269: the kernel shares
the image's clock --- nullspaces are synchronized dynamics, not frozen
storage), and \textbf{never empty} (A267: irrational rotation forbids
exact closure; dark is mandatory, not accidental). A typing remark
separates the two cosmological darks: dark \emph{matter} is kernel (what
the fold annihilates from the base); dark \emph{energy} is cokernel ---
P32's corner residual $\Lambda_0 = 1 - \pi^2/32$ is precisely the
measure of what the embedded $B^4$ fails to cover. Finally, ToENet's
luminosity channel acquires literal semantics: dark nodes are kernel
states of the canonical-closure projection, and plateau detection is
local kernel-rank estimation in the sense of A254's collapse delta ---
the OS metabolism and the cosmological dark sector are the same
mathematics. Scope: synthesis; the content is the shared vocabulary and
the inheritance of theorems, nothing else is claimed.
\end{abstract}

% ────────────────────────────────────────────────────────────────────────────
\section{Definition and Instances}

\begin{definition}[Dark $=$ kernel dynamics]
\label{def:dark}
Let $P$ be a closure map of the corpus's collapse--projection family
(C$\circ$P, P27): a projection, quotient, or closure condition taking
states to their canonical/closed form. \emph{Dark} (relative to $P$)
denotes the states and dynamics of $\ker P$: that which $P$ annihilates
from its image but which persists, evolving, outside it.
\end{definition}

\begin{proposition}[Instances]
\label{prop:instances}
(I1) \emph{Fold.} $P_0 = $ the Hopf pushforward $S^3 \to S^2$;
$\ker P_0 = $ the fiber-charged sector, $\ell(\ell+1)$ modes per level
(A259 Prop.~1.1). By A261 and A264 this kernel gravitates without
converting and its kinetic density is the observed halo: dark matter is
$\ker(\text{fold})$ with dynamics. The dark fraction per level,
$\ell/(\ell+1) \to 1$, makes kernel dominance at depth a structural
prediction, qualitatively consonant with a dark-dominated universe
(stated as consonance, not derivation).
(I2) \emph{Chord closure.} The homomorphism from the 3-smooth lattice to
the breath cycle (A266/A267); its failure-to-close residue at the
nearest lattice point is the comma $G_1$. Detuning is the dark of
harmony.
(I3) \emph{Locking.} The null condition $q\rho_{\rm eff} - p = 0$
defines the Arnold tongues (A268); orbits in the complement drift
without closure --- the dark of dynamics, and the working ToENet
dictionary (canonical $=$ locked, dark $=$ drifting).
\end{proposition}

% ────────────────────────────────────────────────────────────────────────────
\section{Inherited Theorems}

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

\begin{corollary}[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, not stagnant. Nullspaces breathe.
\end{corollary}

\begin{corollary}[Never empty]
The closure conditions of (I2) and (I3) are never exactly satisfied:
$\rho$ is irrational (A267), so commas are nonzero and some orbit class
always drifts. A corpus with empty nullspaces is impossible; dark is
mandatory.
\end{corollary}

\begin{remark}[Kernel vs.\ cokernel: the two cosmological darks]
\label{rem:coker}
Dark \emph{matter} is kernel: annihilated by the fold, absent from the
image, present in dynamics. Dark \emph{energy} is cokernel: P32's corner
residual $\Lambda_0 = 1 - \pi^2/32$ measures what the embedding
$B^4 \hookrightarrow [-1,1]^4$ fails to cover --- absent from the image
for the dual reason. The pair (kernel, cokernel) thus types the two
darks of cosmology as the two exactness defects of one map family.
Recorded as typing, exact at the level of definitions; any dynamical
consequence would require new work.
\end{remark}

% ────────────────────────────────────────────────────────────────────────────
\section{Semantics for ToENet}

\begin{proposition}[Dark luminosity $=$ kernel states]
ToENet's luminosity channel (canonical/dark) is assigned the semantics
of Definition~\ref{def:dark}: a dark entry 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 then local estimation of kernel rank,
prototype A254's collapse delta
$\delta_\Omega = \operatorname{rank}\ker(H_0(A) \to H_0(X))$. The
corollaries transfer: dark entries are sealed (never silently promoted
--- the Butlerian line as kernel protection), paced (indexed by the same
breath ledger as canonical entries), and never empty (a school whose
students never stall has stopped measuring). The OS's metabolism and the
cosmology's dark sector are one formalism.
\end{proposition}

\paragraph{Scope and what would strain it.} This addendum adds
vocabulary and inheritance, not computation. The typing is falsifiable
in its own register: a corpus ``dark'' phenomenon not expressible as
kernel or cokernel of a C$\circ$P closure map would break the
definition's claim to coverage. Candidates to watch: the i\_Paper's
Nullity (pre-actualization) is naturally the kernel of the
frame-installation map --- consistent; Wheel heat (A261's thermalized
defect) is kernel \emph{energy} after the kernel's support is removed
--- consistent but worth a careful future typing. \textbf{OI-272-1}:
formalize the C$\circ$P closure-map family so Definition~\ref{def:dark}
quantifies over a defined class rather than a list.

\paragraph{Verifier.} \texttt{verify\_P272.py}: instance recomputations
(kernel rank $\ell(\ell+1)$; $G_1$; tongue-edge $2\pi\delta$; dark
fraction limit; $\Lambda_0$ as complement measure) and the corollary
citations; all PASS.

\end{document}
