% Staged Draft for Addendum 254: Omega as Coequalizer in the Observer Graph
%
% Status: Codex/Read staged draft, not promoted to corpus canon.
% TASK_019 verdict: PARTIAL.
%
% Core correction:
%   H_1(K_n^Omega, Omega; k) has rank C(n,2), not n-1.
%   The n-1 invariant is the collapse delta
%      rank ker(H_0({O_i}) -> H_0(X)) = n-1.
%
% L. F. Vlegels, Independent Researcher
% Copyright: Leon Fernando Vlegels. License: CC BY 4.0. May 2026.

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\title{Addendum 254: Omega as Coequalizer\\
in the Observer Graph\\[0.4em]
\large Collapse Delta, Lollipop Compactification,\\
and the Corrected Status of the P27 Multi-Observer Rank}

\author{L.\ F.\ Vlegels\\
Independent Researcher\\
\texttt{axiomofchoicepuzzle@gmail.com}}

\date{May 2026}

\begin{document}
\maketitle

\begin{abstract}
P27 defines $\Omega$ as the pre-disagreement state with one observer and,
later, as the consensus limit where all observers' projections become
consistent.  TASK~018 attempted to extend P27's two-observer
Time-from-Disagreement theorem to $n$ observers by the graph identity
\[
  \binom{n}{2} - \beta_1(K_n) = n-1,
\]
but that synthesis was quarantined because it promoted a provisional
calculation to Case~A$'$ closure and because the coefficient-module question
for the P34 associativity defect remained open.  TASK~014 independently found
an $n-1$ region-count jump when $n$ bounded lollipop stem-tips are identified
to the single point at infinity in the one-point compactification.

This addendum gives the first-pass correction.  The shared invariant is not
\[
  H_1(K_n^\Omega,\Omega;\mathbb{k}),
\]
whose rank is $\binom{n}{2}$, but the \emph{collapse delta}
\[
  \delta_\Omega(A,X)
  :=
  \rank \Ker\bigl(H_0(A;\mathbb{k}) \to H_0(X;\mathbb{k})\bigr),
\]
where $A=\{O_1,\ldots,O_n\}$ is the set of free observer endpoints or
lollipop stem-tips in a connected ambient graph~$X$.  Since
$H_0(A;\mathbb{k})\cong \mathbb{k}^n$ and
$H_0(X;\mathbb{k})\cong \mathbb{k}$, the component map has matrix
$[1\;1\;\cdots\;1]$ and kernel rank $n-1$.  This proves the lollipop
$n-1$ jump and supplies a cleaner structural interpretation of the P27
rank arithmetic.

The result is partial.  The OEIS/TASK~018 quantity
$D(n)=\beta_1(K_n)-1$ is not ordinary graph torsion: graph homology over
$\ZZ$ is free, so the $\ZZ$ and $\RR$ ranks agree.  If $D(n)$ has a TOE-native
homological explanation, it requires the non-trivial P34/Kleisli
coefficient module left open in TASK~018.  Status: \textbf{Case B+} for
TASK~018; rank-level value confirmed, torsion/coefficient level open.
\end{abstract}

\tableofcontents
\newpage

%==========================================================================
\section{Context}
\label{sec:context}

\subsection{P27: Omega is a state, not an ordinary peer vertex}

P27's relevant claims are twofold.  First, before disagreement there is one
observer:
\begin{quote}
At $\Omega$---the pre-Bang state---there is one observer. Self-coherence is
trivial.
\end{quote}
Second, in the Great Attractor passage, P27 identifies $\Omega$ as the
consensus state:
\begin{quote}
The attractor $\Omega$ is not a place but a state: the state where all
observers' projections become consistent.
\end{quote}
These two statements make $\Omega$ unlike an ordinary additional observer.
Adding $\Omega$ as a peer vertex would form $K_{n+1}$, but the lollipop
analogy points instead to a coequalizer or collapse operation: many free
endpoints become one shared reference.

\subsection{The three target appearances}

The first-pass research problem was to compare three similar-looking counts:
\begin{enumerate}[label=(\roman*)]
  \item TASK~014: identifying $n$ bounded lollipop stem-tips to one point at
        infinity increases the face count by $n-1$ in connected arrangements.
  \item TASK~018 N4: the rank arithmetic
        $\binom{n}{2}-\beta_1(K_n)=n-1$ matches P27's $n=2$ base case.
  \item TASK~018 N5: the OEIS/lollipop surplus candidate
        $D(n)=\beta_1(K_n)-1$ appears at $n=4$ as $D(4)=2$.
\end{enumerate}
The bridge-first discipline requires these to be related by explicit maps,
not identified by resemblance.

%==========================================================================
\section{The Candidate Omega Graphs}
\label{sec:graphs}

\begin{definition}[Reference observer graph]
For $n\geq 2$, let $K_n$ be the complete graph on observer vertices
$O_1,\ldots,O_n$.  It has
\[
  |V(K_n)|=n,\qquad |E(K_n)|=\binom{n}{2},\qquad
  \beta_1(K_n)=\binom{n}{2}-n+1.
\]
\end{definition}

\begin{definition}[Graph cone 1-skeleton]
Let $\Cone(K_n)$ be the graph obtained by adding one vertex $\Omega$ and one
spoke edge $\Omega O_i$ for each observer vertex.  This is only the
1-skeleton of a cone, not the full topological cone with filled 2-cells.
\end{definition}

\begin{definition}[Omega quotient graph]
Let $K_n^\Omega$ be the quotient graph obtained from $K_n$ by identifying
all observer vertices:
\[
  O_1=\cdots=O_n=\Omega.
\]
The original edges of $K_n$ become loops based at $\Omega$.
\end{definition}

\begin{proposition}[First Betti numbers of the two Omega graph operations]
\label{prop:beta}
For $n\geq 2$,
\[
  \beta_1(\Cone(K_n))=\binom{n}{2}
  \qquad\text{and}\qquad
  \beta_1(K_n^\Omega)=\binom{n}{2}.
\]
The two graph operations are distinct, but they do not differ in first Betti
number.
\end{proposition}

\begin{proof}
For $\Cone(K_n)$,
\[
  |V|=n+1,\qquad |E|=\binom{n}{2}+n,\qquad C=1.
\]
Thus
\[
  \beta_1=|E|-|V|+C
  =\binom{n}{2}+n-(n+1)+1
  =\binom{n}{2}.
\]
For $K_n^\Omega$,
\[
  |V|=1,\qquad |E|=\binom{n}{2},\qquad C=1,
\]
where loop edges count as edges.  Hence
\[
  \beta_1=\binom{n}{2}-1+1=\binom{n}{2}.
\]
\end{proof}

\begin{table}[h]
\centering
\begin{tabular}{r r r r r}
\toprule
$n$ & $\beta_1(K_n)$ & $\binom{n}{2}-\beta_1(K_n)$
& $\beta_1(\Cone(K_n))$ & $\beta_1(K_n^\Omega)$ \\
\midrule
2 & 0 & 1 & 1 & 1 \\
3 & 1 & 2 & 3 & 3 \\
4 & 3 & 3 & 6 & 6 \\
5 & 6 & 4 & 10 & 10 \\
\bottomrule
\end{tabular}
\caption{TASK~019 W1 exact enumeration.}
\label{tab:w1}
\end{table}

\begin{remark}[Negative result]
Neither $\Cone(K_n)$ nor $K_n^\Omega$ produces $n-1$ through
$\binom{n}{2}-\beta_1$.  Both produce zero by that expression.  The value
$n-1$ is therefore not a first-Betti invariant of the Omega graph operation.
\end{remark}

%==========================================================================
\section{Collapse Delta}
\label{sec:collapse}

\begin{definition}[Collapse delta]
\label{def:collapse}
Let $X$ be a connected graph and let
$A=\{O_1,\ldots,O_n\}$ be a set of $n$ distinguished vertices.  Over a field
$\mathbb{k}$, define the Omega collapse delta by
\[
  \delta_\Omega(A,X)
  :=
  \rank \Ker\bigl(i_*:H_0(A;\mathbb{k})\to H_0(X;\mathbb{k})\bigr),
\]
where $i:A\hookrightarrow X$ is inclusion.
\end{definition}

\begin{theorem}[Omega collapse delta]
\label{thm:collapse}
If $X$ is connected and $A$ has $n$ vertices, then
\[
  \delta_\Omega(A,X)=n-1.
\]
\end{theorem}

\begin{proof}
Since $A$ is a discrete set of $n$ points,
\[
  H_0(A;\mathbb{k})\cong \mathbb{k}^n.
\]
Since $X$ is connected,
\[
  H_0(X;\mathbb{k})\cong \mathbb{k}.
\]
The inclusion-induced map sends every endpoint component to the unique
component of $X$.  In the natural bases, its matrix is
\[
  i_*=\begin{bmatrix}1&1&\cdots&1\end{bmatrix}.
\]
This map has rank~$1$.  By rank-nullity,
\[
  \rank \Ker(i_*)=n-1.
\]
\end{proof}

\begin{corollary}[Explicit small cases]
For $n=2,3,4,5$, the collapse matrices and kernel ranks are:
\[
\begin{array}{c|c|c}
n & i_* & \rank\Ker(i_*)\\
\hline
2 & [1\;1] & 1\\
3 & [1\;1\;1] & 2\\
4 & [1\;1\;1\;1] & 3\\
5 & [1\;1\;1\;1\;1] & 4
\end{array}
\]
\end{corollary}

%==========================================================================
\section{Lollipop Compactification}
\label{sec:lollipop}

\begin{proposition}[Tip-identification face jump]
\label{prop:lollipop}
Let $G$ be a connected bounded lollipop arrangement graph in the plane with
$n$ free stem-tip vertices.  Let $G^\Omega$ be obtained by identifying those
$n$ tip vertices to a single point at infinity.  If no edges are removed or
added, then the number of faces changes by
\[
  F(G^\Omega)-F(G)=n-1.
\]
\end{proposition}

\begin{proof}
Euler's formula for a planar graph with $C$ connected components is
\[
  F=E-V+1+C.
\]
For connected lollipop witnesses, identifying the $n$ tips to one point
changes
\[
  \Delta V=-(n-1),\qquad \Delta E=0,\qquad \Delta C=0.
\]
Therefore
\[
  \Delta F=\Delta E-\Delta V+\Delta C=n-1.
\]
\end{proof}

\begin{table}[h]
\centering
\begin{tabular}{r r r r r r r r r}
\toprule
$n$ & $V$ & $E$ & $C$ & $F$ & $V^\Omega$ & $E^\Omega$ & $C^\Omega$ & $F^\Omega-F$\\
\midrule
2 & 11 & 18 & 1 & 9  & 10 & 18 & 1 & 1\\
3 & 18 & 30 & 1 & 14 & 16 & 30 & 1 & 2\\
4 & 31 & 54 & 1 & 25 & 28 & 54 & 1 & 3\\
\bottomrule
\end{tabular}
\caption{TASK~019 W2 same-configuration lollipop witnesses imported from TASK~014.}
\label{tab:w2}
\end{table}

\begin{remark}
This is the precise sense in which the lollipop $n-1$ count is
Omega-relative.  The operation is not ``add one more vertex''; it is
``coequalize $n$ free vertices to one shared reference.''  The invariant that
sees the operation is the collapse delta of Theorem~\ref{thm:collapse}.
\end{remark}

%==========================================================================
\section{Observer Rank and the Correction to TASK 018}
\label{sec:task018}

\begin{proposition}[Rank arithmetic on the reference graph]
\label{prop:rank}
For the complete observer graph $K_n$,
\[
  \binom{n}{2}-\beta_1(K_n)=n-1.
\]
\end{proposition}

\begin{proof}
Since $K_n$ is connected,
\[
  \beta_1(K_n)=|E|-|V|+1=\binom{n}{2}-n+1.
\]
Therefore
\[
  \binom{n}{2}-\beta_1(K_n)
  =
  \binom{n}{2}-\left(\binom{n}{2}-n+1\right)
  =
  n-1.
\]
\end{proof}

\begin{corollary}[Collapse-delta reading of the rank value]
\label{cor:rank}
The TASK~018 rank value agrees with the Omega collapse delta:
\[
  \binom{n}{2}-\beta_1(K_n)
  =
  \delta_\Omega(\{O_1,\ldots,O_n\},K_n)
  =
  n-1.
\]
\end{corollary}

\begin{remark}[Status of the physical interpretation]
The equality is structural.  The physical interpretation still requires a
typed bridge from P27 observer disagreement to the distinguished endpoint set
$A=\{O_1,\ldots,O_n\}$.  P27 supplies the motivation: at one observer there is
no disagreement, at two observers communication creates time, and $\Omega$ is
the consensus state.  This addendum does not by itself derive the full
P27-to-P34 Kleisli morphism identification left open in TASK~018.
\end{remark}

\begin{openproblem}[Observer-category to collapse-map functor]
\label{op:functor}
Construct a functor
\[
  F : \mathcal{O}\mathrm{bs} \to \mathcal{C}\mathrm{ollapse}
\]
from the observer category --- with observers as objects and pairwise
disagreements as morphisms --- to the category of collapse maps
$\bigl(A \to \{\Omega\}\bigr)$, such that $F$ is natural in $n$ and such that
the rank of the induced collapse delta recovers the P27 reconciliation rank.
This functor would upgrade Corollary~\ref{cor:rank} from a numerical
coincidence to a sourced P27 bridge.  Until it is derived, the two $n-1$
counts are identified only structurally, not physically.
\end{openproblem}

\begin{proposition}[The rejected relative-H1 theorem]
\label{prop:reject}
For the quotient graph $K_n^\Omega$,
\[
  \rank H_1(K_n^\Omega,\Omega;\mathbb{k})=\binom{n}{2}.
\]
In particular, this rank is not $n-1$ except accidentally at no value
$n\geq2$.
\end{proposition}

\begin{proof}
$K_n^\Omega$ has one vertex, namely $\Omega$, and $\binom{n}{2}$ loop edges.
In the relative pair $(K_n^\Omega,\{\Omega\})$,
\[
  C_1(K_n^\Omega,\Omega;\mathbb{k})\cong \mathbb{k}^{\binom{n}{2}},
  \qquad
  C_0(K_n^\Omega,\Omega;\mathbb{k})=0.
\]
The boundary map is therefore the zero map into zero, and all loop-edge
generators survive in $H_1$.
\end{proof}

\begin{remark}[Graph cone 1-skeleton]
For the graph cone 1-skeleton, the relative chain group has
$\binom{n}{2}+n$ one-cells and $n$ relative zero-cells.  The $n$ spoke
columns have full rank, so the relative $H_1$ rank is again
\[
  \binom{n}{2}+n-n=\binom{n}{2}.
\]
Thus neither Omega graph model supports the original relative-H1 theorem.
\end{remark}

\subsection{TASK 018 status}

TASK~018 should be revised to:
\[
  \textbf{Case B+: rank-level value confirmed; coefficient level open.}
\]
The numerical value $n-1$ is confirmed by a cleaner mechanism, but the
earlier Case~A$'$ synthesis remains over-strong because it did not close the
P34 coefficient-module or P27/P34 identification gaps.

%==========================================================================
\section{The D(n) Gap}
\label{sec:dn}

\begin{proposition}[No ordinary graph torsion]
\label{prop:notorsion}
For every finite graph $G$, $H_1(G;\ZZ)$ is a free abelian group.  Therefore
ordinary graph homology cannot produce a $\ZZ$-versus-$\RR$ torsion gap.
\end{proposition}

\begin{proof}
A graph is a one-dimensional CW-complex.  Its cellular chain complex is
\[
  0\to C_1(G;\ZZ)\xrightarrow{\partial} C_0(G;\ZZ)\to 0,
\]
where both chain groups are free abelian.  Since there is no $C_2$, one has
$H_1(G;\ZZ)=\Ker(\partial)$, a subgroup of a free abelian group.  Subgroups of
free abelian groups are free abelian.  Hence no torsion subgroup appears.
\end{proof}

\begin{remark}[Status of $D(n)=\beta_1(K_n)-1$]
The first-pass verdict assigns $D(n)$ to a planar outer-face/cycle obstruction
or to a future non-trivial coefficient-system calculation.  It is not
explained by the Omega collapse delta, by the beta1 difference between cone
and quotient, or by ordinary $\ZZ$ versus $\RR$ graph homology.  A TOE-native
homological explanation would have to specify the P34 associativity-defect
module and the action of $\pi_1(K_n)$ on that module.
\end{remark}

%==========================================================================
\section{Open Problems}
\label{sec:open}

\begin{openproblem}[P34 coefficient module for $D(n)$]
Define the coefficient module $A$ for the P34 associativity defect, including
the target group, the cocycle condition, and the action of $\pi_1(K_n)$.
Only then can one test whether $D(n)=\beta_1(K_n)-1$ is a TOE-native
coefficient-module phenomenon rather than a generic planar outer-face count.
\end{openproblem}

\begin{openproblem}[Functorial observer bridge]
Construct a functor from the observer category, with observers as objects and
disagreements/reconciliations as morphisms, to the category of collapse maps
\[
  A\to \{\Omega\}.
\]
This would upgrade the collapse-delta interpretation from structural analogy
to a sourced P27/P34 bridge.
\end{openproblem}

\begin{openproblem}[P27 textual revision]
Determine whether P27's Time-from-Disagreement theorem should include a
remark distinguishing pairwise communication from the Omega collapse delta.
The theorem statement need not change, but the $n>2$ extension should not be
cited through TASK~018's quarantined Case~A$'$ language.
\end{openproblem}

%==========================================================================
\section{Conclusion}
\label{sec:conclusion}

The first-pass computation supports a partial unification.  The lollipop
compactification jump and the observer-rank value share a precise invariant:
\[
  \rank\Ker\bigl(H_0(\{O_1,\ldots,O_n\})\to H_0(X)\bigr)=n-1.
\]
This is the Omega collapse delta.  It is a coequalizer-style operation on
components, not an additional peer observer and not a relative $H_1$ theorem.

The $D(n)=\beta_1(K_n)-1$ phenomenon remains separate.  It may yet connect to
the TOE through a P34 coefficient module, but ordinary graph homology does not
provide that bridge.  The correct status is therefore Case~B+: the rank-level
value is confirmed, while the torsion/coefficient layer remains open.

\begin{thebibliography}{99}

\bibitem{P27}
L.\ F.\ Vlegels,
\textit{The Second Observer: Why Existence Requires Witness, Why Witness Requires Two,
and How Intersubjectivity Generates Geometry},
TOE Corpus, Paper~27 (\texttt{27\_Paper\_SecondObserver.tex}),
lines~362 and~386 cited for the two $\Omega$ passages.

\bibitem{P34}
L.\ F.\ Vlegels,
\textit{The Kleisli Monad over $S^3$: Categorical Composition of Observer States},
TOE Corpus, Paper~34.

\bibitem{TASK014}
Codex,
\textit{TASK\_014 lollipop-bounded-arrangement report and data},
\texttt{Codex/Read/TASK\_014\_lollipop-bounded-arrangement\_report.md},
2026.

\bibitem{TASK018}
Codex,
\textit{TASK\_018 P27 multi-observer reconciliation report and staged draft},
\texttt{Codex/Read/TASK\_018\_p27-multi-observer-reconciliation\_report.md},
2026.

\bibitem{TASK019}
Codex,
\textit{TASK\_019 Omega-relative observer graph W1--W4 artifacts},
\texttt{Codex/Read/TASK\_019\_W1\_data.json},
\texttt{TASK\_019\_W2\_data.json},
\texttt{TASK\_019\_W3\_collapse\_delta.md},
\texttt{TASK\_019\_W4\_verdict.md},
2026.

\end{thebibliography}

\end{document}
