% Addendum 186: SR2 Closure --- R8 Forces Additive Composition of \hat{O} Sectors;
%   Kleisli Composition is Inadmissible.
%
%   P18 §4.4 records the assembly-uniqueness question (P18-T2) as open and
%   names three independent sub-results (SR1, SR2, SR3) that together would
%   constitute a proof. SR2 asks: is additive composition the only admissible
%   way to combine the five \hat{O} sectors, or can Kleisli (monadic) composition
%   --- as supplied by Paper 34's monad (S³, return, bind) --- also serve?
%
%   This paper closes SR2. The argument:
%
%   Proposition 4.1 (Additive satisfies R8):
%     Under additive assembly \hat{O}_+ = A + \beta M, the moment operator
%     M = \sum_n (\mu_n/\mu_0) P_n contributes an additive term
%     \beta(\mu_n/\mu_0) to E_n with a UNIFORM coefficient \beta across all
%     modes n. R8's moment hierarchy linearity is satisfied.
%
%   Theorem 4.2 (SR2 Closure --- Kleisli violates R8):
%     Kleisli sequential composition of the \hat{O} sectors violates R8 in
%     two cases that jointly exhaust all possible Kleisli assemblies:
%     (A) Commuting-sector case: the effective coupling to \mu_n in the
%         Kleisli eigenvalue E_n^K = \beta G_n (\mu_n/\mu_0) is modulated
%         by G_n = product of all other sectors' eigenvalues at mode n,
%         which varies with n. The coupling is mode-dependent, not uniform;
%         R8 moment-linearity is violated.
%     (B) Non-commuting-sector case: the sectors of \hat{O} are mutually
%         non-commuting --- [D^2_{B^4}, \gamma T_{\rm cycle}] \neq 0 and
%         [\zeta\mathcal{R}, \Delta_{S^3}] \neq 0 --- so any Kleisli
%         sequential product mixes the radial modes \{\psi_n\}. The
%         eigenstates of the Kleisli assembly are not the modes of M, and the
%         mode-moment correspondence n <-> \mu_n required by R8 is absent.
%
%   Corollary 4.3 (SR2 Closure):
%     R8's moment hierarchy linearity is structurally incompatible with
%     Kleisli composition of the \hat{O} sectors. Additive composition is
%     the only admissible composition under R8. SR2 is closed.
%
%   The open sub-results SR1 and SR3 are identified and scoped.
%
%   Numerical companion: verify/verify_P186.py (mp.dps=55, all 12 checks
%   pass). Key checks: \mu_n moments computed by closed form (16\pi^3/(n+4)
%   + 3\pi^2/(n+3) + 2\pi/(n+2)); additive E_n^+ = a_n + \beta m_n has
%   constant coupling \beta; Kleisli E_n^K = a_n \cdot \beta m_n has
%   mode-dependent coupling \beta a_n; 3x3 commutator [D^2_{\rm model},
%   T_{\rm model}] is nonzero.
%
% L.\ F.\ Vlegels, Independent Researcher
% Copyright: L\'eon Fernando Vlegels. License: CC BY 4.0. May 2026.

\documentclass[12pt,a4paper]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{mathtools}
\usepackage{geometry}
\usepackage{booktabs}
\usepackage{enumitem}
\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref}
\geometry{margin=1in}

\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{assertion}{Assertion}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{openproblem}[theorem]{Open Problem}

% --- TOE shorthand ---
\newcommand{\Ohat}{\hat{O}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\HH}{\mathbb{H}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\NN}{\mathbb{N}}
\newcommand{\calR}{\mathcal{R}}
\newcommand{\calM}{\mathcal{M}}
\newcommand{\ForkOp}{\mathrm{Fork}}
\newcommand{\WeldOp}{\mathrm{Weld}}
\newcommand{\PlateauOp}{\mathrm{Plateau}}
\newcommand{\OscOp}{\mathrm{Oscillate}}
\newcommand{\PertOp}{\mathrm{Perturb}}
\newcommand{\bind}{\mathbin{\triangleright\!\!=}}
\newcommand{\KlComp}{\mathbin{\circ_K}}
\newcommand{\ALPHAINV}{\alpha^{-1}}
\newcommand{\mun}{\mu_n}
\newcommand{\muzero}{\mu_0}
\newcommand{\munscore}{m_n}

\title{SR2 Closure: R8 Forces Additive Composition\\
of the Five $\hat{O}$ Sectors;\\
Kleisli Composition is Inadmissible\\[0.5em]
\large Addendum 186 to the TOE Corpus}

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

\date{May 2026}

\begin{document}
\maketitle
\tableofcontents
\newpage

%==========================================================================
\begin{abstract}
Paper~18's assembly-uniqueness open problem (P18-T2, \S4.4) is that the design
requirements (R1)--(R8) may admit alternative admissible assemblies of the
master operator $\Ohat$ beyond the canonical additive form.  A proof of
uniqueness requires three independent sub-results: SR1~(each $R_i$'s operator
is the unique one in its category), SR2~(additive composition is the only
admissible composition), and SR3~(the four coefficients are the only
spectrum-matching assignment).  Paper~34 identifies Kleisli composition ---
via its monad $(S^3, \mathsf{return}, \bind)$ --- as the leading alternative
to additive composition, making SR2 the tractable sub-result.

This paper closes SR2.  We show that requirement R8 (moment hierarchy:
connect spectrum to $\mu_n$) forces additive composition and rules out Kleisli
sequential composition in two cases that jointly exhaust all possibilities.

\textbf{Case~A} (commuting sectors, Proposition~\ref{prop:additive} and
Theorem~\ref{thm:kleisli}(A)): Under additive assembly, the moment operator
$M = \sum_n (\mu_n/\mu_0)\,P_n$ contributes an additive term
$\beta(\mu_n/\mu_0)$ to eigenvalue $E_n$ with a \emph{uniform} coefficient
$\beta$ for all~$n$.  Under Kleisli (sequential product) composition, the
effective coupling to $\mu_n$ is $\beta G_n$, where $G_n$ is the product of
all other sectors' eigenvalues at mode~$n$.  Since $G_n$ varies with~$n$, the
coupling is mode-dependent, violating R8's uniform linearity requirement.

\textbf{Case~B} (non-commuting sectors, Theorem~\ref{thm:kleisli}(B)):
The sectors of $\Ohat$ are mutually non-commuting: $[D^2_{B^4}, \gamma T_{\rm cycle}]
\neq 0$ and $[\zeta\calR, \Delta_{S^3}] \neq 0$.  Kleisli sequential
composition of non-commuting sectors produces mode-mixing, so the eigenstates
of the Kleisli assembly are not the radial modes $\{\psi_n\}$ that diagonalise
$M$.  The mode-moment correspondence $\psi_n \leftrightarrow \mu_n$ required
by R8 is destroyed.

\textbf{Corollary} (SR2 Closure, Corollary~\ref{cor:sr2}): R8's moment
hierarchy linearity is structurally incompatible with Kleisli composition of
the $\Ohat$ sectors.  Additive composition is the only admissible composition
under~R8.  SR1 and SR3 remain open.

Numerical companion: \texttt{verify/verify\_P186.py}
($\texttt{mp.dps}=55$; all~12 checks pass).
\end{abstract}

%==========================================================================
\section{Background and Position in the Corpus}
\label{sec:background}
%==========================================================================

\subsection{The P18-T2 assembly-uniqueness question}

Paper~18 \cite{P18} assembles the master operator
\begin{equation}
  \Ohat \;=\; D_{B^4}^2 + \Delta_{S^3} + \Delta_{S^1}
  + V_{\mathrm{self}}(r) + \lambda\rho(r) + \gamma T_{\rm cycle}
  + \zeta\calR + \beta\calM,
  \label{eq:ohat}
\end{equation}
where each sector is derived in a prior paper and the four coefficients
$(\alpha, \gamma, \zeta, \beta)$ are pinned by geometric consistency.
P18~\S4.4 records as an open problem~(P18-T2) whether the design requirements
(R1)--(R8) uniquely determine this assembly:

\begin{assertion}[P18 §4.4 open problem, verbatim]
\label{ass:p18open}
\emph{Whether the design requirements} (R1)--(R8) \emph{admit alternative
admissible assemblies of the operator $\hat{O}$ --- different orderings,
coefficient assignments, additional sectors, or operadic compositions in place
of additive composition --- is open.}
\end{assertion}

A proof of P18-T2 uniqueness would require three independent sub-results
\cite{P18T2reframed}:

\begin{itemize}[leftmargin=2em]
  \item[\textbf{SR1.}] Each $R_i$'s named operator is the unique operator in
    its category (bulk dynamics $\to$ $D^2_{B^4}$, boundary $\to$ $\Delta_{S^3}$,
    etc.).
  \item[\textbf{SR2.}] Additive composition is the only admissible composition
    of the sectors --- alternatives such as Kleisli composition (via
    Paper~34~\cite{P34}) or operadic composition (on $S^3$) are ruled out.
  \item[\textbf{SR3.}] The coefficients $(\alpha, \gamma, \zeta, \beta)$ are
    the only spectrum-matching assignment given the Hopf-induced eigenvalue
    structure.
\end{itemize}

Addendum~P184 \cite{P184} formally derived the Wheel~$\leftrightarrow$~$\Ohat$
dictionary for the canonical assembly, noting that P18-T2 uniqueness is
separate and not resolved.  The present paper addresses SR2 specifically.

\subsection{SR2: the composition question}
\label{subsec:sr2}

SR2 asks whether additive composition $(\Ohat = \sum_i K_i)$ is the only
admissible way to combine the sectors, or whether Kleisli composition ---
in which sectors are sequentially applied as Kleisli morphisms in a monad ---
provides an admissible alternative.  Additive composition is the canonical
choice of P18; Paper~34 supplies the Kleisli alternative.

The focus on Kleisli composition is motivated by the P18-T2 reframing memo
\cite{P18T2reframed}, which names ``composition via Kleisli arrows on a monad
over the $\hat{O}$-sector algebra, per Paper~34'' as the leading operadic
alternative, and by Paper~34 itself, which establishes the formal monad
$(S^3, \mathsf{return}, \bind)$ as a compositional framework for LumenOS
computations.

\begin{assertion}[SR2 is tractable]
\label{ass:sr2tractable}
SR2 is identified in \cite{P18T2reframed} as the tractable sub-result because:
(i)~additive vs.\ Kleisli composition is a concrete algebraic distinction
amenable to spectral analysis, and (ii)~Paper~34's Kleisli machinery is
already formalized, providing the precise alternative to test against R8.
SR1 and SR3 require additional representation-theoretic and spectral-rigidity
machinery not yet available in the corpus.
\end{assertion}

\subsection{Paper 34's Kleisli machinery}
\label{subsec:p34recap}

We recall the essential constructions from Paper~34 \cite{P34}.

\begin{assertion}[P34 monad (recap)]
\label{ass:p34monad}
Paper~34 defines the monad $(T, \eta, \mu)$ over $S^3$:
\begin{itemize}
  \item Carrier: $T(S^3) = \{(q_i, w_i)_i \mid q_i \in S^3,\; w_i \geq 0,\;
    \sum_i w_i \leq 1\}$ (scored distributions over $S^3$).
  \item Unit: $\eta(q) = \delta_q$ (point distribution at $q$).
  \item Bind: $m \bind f = \eta\!\bigl(\bigcup_i w_i \cdot f(q_i)\bigr)$
    (weighted-merge and geodesic-centroid collapse).
\end{itemize}
The monad laws (left identity, right identity, associativity) are verified in
Paper~34 Theorems 3.1--3.3.
\end{assertion}

\begin{assertion}[P34 Kleisli category (recap)]
\label{ass:p34kleisli}
The Kleisli category $\mathbf{Kl}(T)$ has:
\begin{itemize}
  \item Objects: $q \in S^3$.
  \item Morphisms $q_A \to q_B$: attention steps $f : S^3 \to T(S^3)$.
  \item Kleisli composition:
    $(g \KlComp f)(q) = f(q) \bind g$.
\end{itemize}
In code, Kleisli composition is the \texttt{depends\_on} sequencing of plan
steps: if step $B$ depends on step $A$, execution is $f_{A}(q) \bind f_B$
(Paper~34 Proposition 4.1).
\end{assertion}

The extension of Paper~34's Kleisli framework to operator sectors of $\Ohat$
is the candidate alternative specified in Assertion~\ref{ass:sr2tractable}: each
sector $K_i$ is treated as a Kleisli morphism (a map from a system state to
a scored distribution over states), and sectors are composed via
$\KlComp$ rather than summed.  We formalise this in
Definition~\ref{def:kleisli-assembly} below.

%==========================================================================
\section{R8 Precisely: Moment Hierarchy Linearity}
\label{sec:r8}
%==========================================================================

\subsection{The geometric moments \texorpdfstring{$\mu_n$}{mu\_n}}
\label{subsec:moments}

The geometric density is $\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x$
(Paper~03 \cite{P03}).  The $n$-th geometric moment is
\begin{equation}
  \mu_n \;=\; \int_0^1 x^n\,\rho(x)\,dx
  \;=\; \frac{16\pi^3}{n+4} + \frac{3\pi^2}{n+3} + \frac{2\pi}{n+2},
  \qquad n \in \NN_0.
  \label{eq:mun}
\end{equation}

\begin{assertion}[Ground moment = $\alpha^{-1}$]
\label{ass:mu0}
At $n=0$: $\mu_0 = 4\pi^3 + \pi^2 + \pi = \ALPHAINV$ (Paper~02--03 \cite{P03}).
Numerically $\mu_0 \approx 137.036$.
\end{assertion}

\begin{assertion}[Moment sequence is decreasing]
\label{ass:mudecreasing}
Since $\rho(x) \geq 0$ on $[0,1]$ and $x \in [0,1]$, the factor $x^n$ is
pointwise non-increasing in $n$.  Hence $\mu_0 > \mu_1 > \mu_2 > \cdots > 0$.
Numerically: $\mu_0 \approx 137.04$, $\mu_1 \approx 108.7$, $\mu_2 \approx 90.2$,
$\mu_3 \approx 76.6$, $\mu_4 \approx 66.3$ (verified in
\texttt{verify\_P186.py}, Checks~1--2).
\end{assertion}

\subsection{The moment operator \texorpdfstring{$\mathcal{M}$}{M}}
\label{subsec:mop}

Let $\mathcal{H}$ be the Hilbert space of $L^2$-sections of the spinor bundle
over $B^4$ (P18 \S2.3).  The radial decomposition $\psi(r,\theta) = \sum_{n,l,m}
R_{nl}(r)\,Y_{lm}(\theta)\,\chi_s$ gives a complete orthonormal system
$\{\psi_{n,l,m}\}$.  We write $\{\psi_n\}$ for the $l=0$, $m=0$ sector
(s-wave modes), which diagonalise the radial part of $D^2_{B^4}$.

\begin{assertion}[Moment operator from P18]
\label{ass:mop}
From P18 \S3.8: the moment operator is
\begin{equation}
  \calM \;=\; \sum_{n=0}^{\infty} \frac{\mu_n}{\mu_0}\,P_n,
  \label{eq:mop}
\end{equation}
where $P_n = |\psi_n\rangle\langle\psi_n|$ is the orthogonal projection onto
the $n$-th radial mode.  Eigenvalue of $\calM$ on $\psi_n$: $\mu_n/\mu_0 = m_n$
(the normalised $n$-th moment).  The operator $\calM$ is diagonal in the
mode basis, bounded, and positive.
\end{assertion}

\begin{remark}[Scaled notation]
\label{rem:scaledM}
We write $m_n = \mu_n/\mu_0$ for the normalised moments, so $m_0 = 1$,
$m_1 \approx 0.793$, $m_2 \approx 0.658$, etc.  The contribution of $\beta\calM$
to the eigenvalue at mode $n$ is $\beta m_n$.
\end{remark}

\subsection{Moment-linearity: the spectral requirement of R8}
\label{subsec:r8def}

\begin{assertion}[R8 from P18 §1.2]
\label{ass:r8}
From P18 \S1.2: requirement R8 is \emph{``Moment hierarchy: Connect spectrum
to $\mu_n$.''} The named operator implementing R8 is $\beta\calM$ with
$\beta = 3\pi/20$ (P18 Proposition~4.2 \cite{P18}).
\end{assertion}

The word ``connect spectrum to $\mu_n$'' admits several interpretations.
We formalise the one that follows from the P18 construction:

\begin{definition}[R8 moment-linearity]
\label{def:r8ml}
An assembled operator $\Ohat_{\mathrm{asm}}$ satisfies \emph{R8
moment-linearity} if there exists a universal constant $\beta \neq 0$ such
that, for every radial mode $n \geq 0$,
\begin{equation}
  \langle\psi_n \mid \Ohat_{\mathrm{asm}} \mid \psi_n\rangle
  \;=\; a_n + \beta\,m_n,
  \label{eq:r8ml}
\end{equation}
where $a_n$ is the contribution from all sectors other than $\beta\calM$, and
the coupling coefficient $\beta$ is the \emph{same} for all $n$.
\end{definition}

\begin{remark}[Why uniformity of $\beta$ is the key]
\label{rem:uniformity}
Definition~\ref{def:r8ml} captures the P18 construction precisely: the moment
operator $\calM = \sum_n m_n P_n$ contributes an additive correction
$\beta m_n$ to $E_n$ with the same proportionality constant $\beta$ for every
mode.  This is the \emph{linear} coupling that P18 names ``moment hierarchy'':
the spectral ladder $\{E_n\}$ reflects the moment ladder $\{\mu_n\}$ through a
fixed linear coefficient.  An assembly that modulates $\beta$ by a
mode-dependent factor $G_n$ satisfies a different (nonlinear) coupling.
\end{remark}

%==========================================================================
\section{Kleisli Composition of Sectors: Definition and Cases}
\label{sec:kleisli-def}
%==========================================================================

\subsection{Kleisli assembly of operator sectors}
\label{subsec:kleisli-assembly-def}

The five Wheel-mapped sectors of $\Ohat$ are (CLAUDE.md Pin~5 \cite{CLAUDE}):
\begin{equation}
  \ForkOp \leftrightarrow D^2_{B^4}, \quad
  \WeldOp \leftrightarrow \Delta_{S^3}, \quad
  \PlateauOp \leftrightarrow \Delta_{S^1}, \quad
  \OscOp \leftrightarrow \gamma T_{\rm cycle}, \quad
  \PertOp \leftrightarrow \zeta\calR.
  \label{eq:wheel-dict}
\end{equation}
Together with $\beta\calM$ (R8), $V_{\mathrm{self}}$, and $\lambda\rho$, these
eight objects are the sectors of $\Ohat$.  In the canonical additive assembly
they are summed.  A Kleisli assembly sequences them as morphisms.

\begin{definition}[Kleisli assembly of $\Ohat$ sectors]
\label{def:kleisli-assembly}
A \emph{Kleisli assembly} of the $\Ohat$ sectors is an operator of the form
\begin{equation}
  \Ohat_K \;=\; K_{i_N} \circ K_{i_{N-1}} \circ \cdots \circ K_{i_1},
  \label{eq:kleisli-assembly}
\end{equation}
where $\{K_{i_j}\}$ is an ordered selection of sectors (with scalar multiples
as in the canonical assembly), and $\circ$ denotes sequential composition of
linear operators on $\mathcal{H}$ --- the operator-algebra realisation of
Kleisli composition from Paper~34 Definition~4.1.  In this realisation,
$K_{j+1} \circ K_j$ is the operator $f_{j+1} \circ f_j$ in the Kleisli
category $\mathbf{Kl}(T)$, with each $K_i$ acting as a Kleisli morphism that
maps a system state to a distribution over subsequent states.
\end{definition}

\begin{remark}[The operator-algebra realisation]
\label{rem:realization}
Paper~34 \S4 defines Kleisli composition as $(g \KlComp f)(q) = f(q) \bind g$.
In the operator-algebra context, where $f$ and $g$ are linear maps on
$\mathcal{H}$, this reduces to sequential application: the output of $f$
becomes the input of $g$, i.e., $g \circ f$ in the usual sense of operator
composition.  Definition~\ref{def:kleisli-assembly} adopts this realisation,
which is the natural extension of Paper~34's computational Kleisli composition
to the operator-algebra setting referenced in Assertion~\ref{ass:sr2tractable}.
\end{remark}

\subsection{Two cases}
\label{subsec:cases}

We distinguish two cases based on whether the sectors in the Kleisli chain
commute:

\begin{definition}[Case A and Case B]
\label{def:cases}
\begin{itemize}[leftmargin=2em]
  \item \textbf{Case~A} (commuting): all sectors in the Kleisli chain commute
    with $\calM$ and with each other.  In this case the radial modes $\{\psi_n\}$
    remain eigenstates of $\Ohat_K$.
  \item \textbf{Case~B} (non-commuting): at least one pair of adjacent sectors
    in the Kleisli chain does not commute.  In this case mode-mixing occurs.
\end{itemize}
These two cases are exhaustive: every Kleisli assembly falls into Case~A or
Case~B.
\end{definition}

The rest of this paper shows that both cases violate R8 moment-linearity.

%==========================================================================
\section{The SR2 Theorem}
\label{sec:sr2theorem}
%==========================================================================

\subsection{Additive assembly satisfies R8}
\label{subsec:additive-ok}

\begin{proposition}[Additive assembly satisfies R8]
\label{prop:additive}
Let $A = D^2_{B^4} + \Delta_{S^3} + \Delta_{S^1} + V_{\mathrm{self}} +
\lambda\rho + \gamma T_{\rm cycle} + \zeta\calR$ (all sectors of $\Ohat$ except $\beta\calM$).
The additive assembly $\Ohat_+ = A + \beta\calM$ satisfies R8
moment-linearity (Definition~\ref{def:r8ml}) with coupling coefficient $\beta$.
\end{proposition}

\begin{proof}
For any mode $\psi_n$ in the radial basis:
\begin{equation}
  \langle\psi_n \mid \Ohat_+ \mid \psi_n\rangle
  \;=\; \langle\psi_n \mid A \mid \psi_n\rangle
  + \beta\,\langle\psi_n \mid \calM \mid \psi_n\rangle
  \;=\; a_n + \beta\,m_n,
  \label{eq:additive-eigenvalue}
\end{equation}
where $a_n = \langle\psi_n \mid A \mid \psi_n\rangle$ and $\calM\psi_n =
m_n\psi_n$ (since $\calM = \sum_k m_k P_k$ is diagonal in the mode basis).
The moment coupling coefficient $\beta = 3\pi/20$ appears with the same value
for every $n$. \qquad $\checkmark$
\end{proof}

\begin{remark}[Physical grounding of R8 in additive assembly]
\label{rem:physical}
In P18's additive assembly, the moment coupling creates a spectral ladder:
\begin{equation}
  E_n^+ - E_{n-1}^+ \;=\; (a_n - a_{n-1}) + \beta\,(m_n - m_{n-1}).
  \label{eq:spectral-ladder}
\end{equation}
The increment $\beta(m_n - m_{n-1})$ is a universal offset (same proportionality
$\beta$ at every step), so the moment hierarchy appears as an \emph{additive
modulation} of the spectral ladder with fixed gain $\beta$.  This is precisely
what P18 uses to identify the gravitational coupling: $\log(M_\mathrm{Pl}/m_e)
= \beta\,\mu_1/\mu_0 / (1 - \alpha^2 \mu_1/\mu_0)$ (P18 Theorem~5.3
\cite{P18}).
\end{remark}

\subsection{Kleisli assembly violates R8}
\label{subsec:kleisli-bad}

\begin{theorem}[Kleisli composition violates R8 moment-linearity]
\label{thm:kleisli}
Let $\Ohat_K$ be any Kleisli assembly of the $\Ohat$ sectors in the sense of
Definition~\ref{def:kleisli-assembly}, with $\beta\calM$ appearing somewhere
in the chain.  Then $\Ohat_K$ violates R8 moment-linearity.
\end{theorem}

\begin{proof}
We treat Cases~A and~B separately.

\medskip
\noindent\textbf{Case~A: Commuting sectors.}

Assume all sectors in the Kleisli chain commute with each other and with $\calM$.
Write the chain as
\[
  \Ohat_K \;=\; B_{\mathrm{after}} \circ (\beta\calM) \circ B_{\mathrm{before}},
\]
where $B_{\mathrm{before}} = K_{j-1} \circ \cdots \circ K_1$ and
$B_{\mathrm{after}} = K_N \circ \cdots \circ K_{j+1}$ are products of sectors
appearing before and after $\beta\calM$ in the chain.  Since all sectors commute
and share the mode basis $\{\psi_n\}$, write
\[
  B_{\mathrm{before}}\psi_n = F_n^{<}\,\psi_n, \qquad
  B_{\mathrm{after}}\psi_n = F_n^{>}\,\psi_n,
\]
where $F_n^{<}$ and $F_n^{>}$ are the eigenvalues of $B_{\mathrm{before}}$ and
$B_{\mathrm{after}}$ at mode $n$.  Applying $\Ohat_K$ to $\psi_n$:
\begin{equation}
  \Ohat_K\psi_n
  \;=\; B_{\mathrm{after}}\bigl(\beta\calM\bigl(F_n^{<}\psi_n\bigr)\bigr)
  \;=\; B_{\mathrm{after}}\bigl(\beta m_n F_n^{<}\psi_n\bigr)
  \;=\; \beta m_n F_n^{<} F_n^{>}\,\psi_n.
  \label{eq:kleisli-eigenvalue}
\end{equation}
So the eigenvalue of $\Ohat_K$ at mode $n$ is
\begin{equation}
  E_n^K \;=\; \beta\,m_n\,G_n, \qquad G_n \;:=\; F_n^{<}\,F_n^{>}.
  \label{eq:kleisli-en}
\end{equation}
The effective coupling of $E_n^K$ to the moment $m_n$ is the coefficient
$\beta G_n$, which varies with $n$ as $G_n$ varies.  Specifically:
\begin{itemize}
  \item For the canonical D$^2_{B^4}$ contribution: $F_n^{>} = (2n+2)^2 = 4(n+1)^2$,
    which grows as $n^2$.  Hence $G_n \sim 4(n+1)^2 \cdot (\text{other factors})$,
    an $n$-dependent quantity.
  \item Since $G_n \neq G_m$ for $n \neq m$ (the spectral ladder of $D^2_{B^4}$
    alone gives distinct values), the coupling $\beta G_n$ is not a universal
    constant.
\end{itemize}
Therefore the condition~\eqref{eq:r8ml} fails: there is no constant $\bar\beta$
independent of $n$ such that $E_n^K = a_n' + \bar\beta m_n$ for all $n$,
because the ``$a_n'$'' term and the ``$\bar\beta m_n$'' term are not
additively separated --- they appear as a product $G_n m_n$.  Case~A violates
R8 moment-linearity.

\medskip
\noindent\textbf{Case~B: Non-commuting sectors.}

We show that at least one pair of adjacent $\Ohat$ sectors does not commute.
This is sufficient because mode-mixing from any non-commuting pair destroys the
common mode basis, making Definition~\ref{def:r8ml} inapplicable.

\medskip
\noindent\emph{Pair 1: $[D^2_{B^4},\, \gamma T_{\rm cycle}] \neq 0$.}

$D^2_{B^4}$ is the squared Dirac operator with radial eigenvalues $E_{n,l} =
(2n+l+2)^2$ at angular momentum $l$ (P18 Proposition~3.1).  The operator $T$
is the layer-cycle: $T^3 = I$, cycling bulk $\to$ boundary $\to$ fibre $\to$
bulk (P18 \S3.6).  The three layers have different functions of $l$ (bulk is
$l=0$ s-waves; boundary encodes $l \geq 1$ modes; fibre encodes winding modes).
$T$ maps $\psi_{n,l}$ to states in a different layer, which generally have a
different eigenvalue of $D^2_{B^4}$.  Therefore:
\begin{equation}
  D^2_{B^4}(T\psi_{n,l}) \;\neq\; T(D^2_{B^4}\psi_{n,l})
  \quad\Longrightarrow\quad
  [D^2_{B^4},\, \gamma T_{\rm cycle}] \;\neq\; 0.
  \label{eq:comm1}
\end{equation}
Numerically: modelling $D^2$ as $\mathrm{diag}(E_0, E_1, E_2) = \mathrm{diag}(4, 9, 16)$
(l=0 s-wave, $n=0,1,2$) and $T$ as the $3\times3$ cyclic permutation, the
commutator $[D^2, T]$ has entry $(0,1)$ equal to $(E_0 - E_1) = -5 \neq 0$
(verified in \texttt{verify\_P186.py}, Check~9).

\medskip
\noindent\emph{Pair 2: $[\zeta\calR,\, \Delta_{S^3}] \neq 0$.}

The renormalization operator $\calR = r\partial/\partial r + \Delta_\psi$
(P18 \S3.7).  The boundary Laplacian in the full radial decomposition contains
a term $(1/r^2)\Delta_{S^3}$.  Computing the commutator with $r\partial/\partial r$:
\begin{equation}
  \bigl[r\tfrac{\partial}{\partial r},\; (1/r^2)\Delta_{S^3}\bigr]
  \;=\; -\frac{2}{r^2}\,\Delta_{S^3},
  \label{eq:comm2}
\end{equation}
which is nonzero wherever $\Delta_{S^3}$ acts nontrivially (i.e., on all modes
with $l \geq 1$).

Therefore, any Kleisli chain that includes both $(D^2_{B^4}, \gamma T_{\rm cycle})$ or
$(\zeta\calR, \Delta_{S^3})$ as adjacent factors will produce mode-mixing.
Since $\Ohat$ requires all eight sectors and a Kleisli chain must include each
sector at most once, every Kleisli assembly of $\Ohat$ falls into Case~B and
contains at least one non-commuting adjacent pair.

\medskip
\noindent\emph{Consequence of mode-mixing.}

When $[K_{j+1}, K_j] \neq 0$, the eigenstates $\{\phi_n\}$ of the product
$K_{j+1} \circ K_j$ are generally not the modes $\{\psi_n\}$ of $\calM$.
Write $\phi_k = \sum_n c_{kn}\psi_n$ (mode expansion in the $\calM$-basis).
Applying $\calM$ to $\phi_k$:
\begin{equation}
  \calM\,\phi_k \;=\; \sum_n c_{kn}\,m_n\,\psi_n,
\end{equation}
which is a superposition of different modes with different moment weights $m_n$.
The eigenstate $\phi_k$ of the Kleisli assembly does not have a definite
moment-mode index $n$; it mixes modes that have different $\mu_n$ weights.
There is therefore no well-defined coupling of $\phi_k$ to any single $\mu_n$,
and R8's mode-moment correspondence $\psi_n \leftrightarrow \mu_n$ is absent.
Case~B violates R8 moment-linearity.

\medskip
Combining Cases~A and~B: every Kleisli assembly of $\Ohat$ sectors violates
R8 moment-linearity.
\end{proof}

\begin{remark}[Why Case B subsumes all real Kleisli assemblies]
\label{rem:caseb-dominant}
Case~A (commuting sectors) is a degenerate scenario included for completeness.
In practice, the sectors of $\Ohat$ are mutually non-commuting: $D^2_{B^4}$ and
$\gamma T_{\rm cycle}$ alone suffice to put every Kleisli assembly of the full eight sectors
into Case~B.  The commuting case arises only if one considers a truncated chain
(e.g., $\beta\calM$ and $V_{\mathrm{self}}$ only), but such a chain does not
constitute an assembly of all sectors required by~(R1)--(R8).
\end{remark}

\begin{remark}[The multiplicative vs.\ additive obstruction]
\label{rem:obstuction}
The core distinction between additive and Kleisli composition is algebraic:
\begin{align}
  \text{Additive:} \quad & E_n^+ \;=\; a_n + \beta m_n
    \quad\text{(sum; } \beta \text{ is a universal additive offset),}
    \label{eq:additive-struct}\\
  \text{Kleisli (Case~A):} \quad & E_n^K \;=\; G_n \cdot \beta m_n
    \quad\text{(product; } \beta G_n \text{ is a mode-dependent coefficient).}
    \label{eq:kleisli-struct}
\end{align}
R8 requires the additive structure~\eqref{eq:additive-struct}; the Kleisli
structure~\eqref{eq:kleisli-struct} fails this requirement because $G_n$ is
not constant.  The obstruction is structural: it follows from the definition of
sequential operator composition and is independent of the specific values of
the TOE constants.
\end{remark}

\subsection{SR2 closure corollary}
\label{subsec:sr2-closure}

\begin{corollary}[SR2 Closure]
\label{cor:sr2}
Under the design requirement R8 (moment hierarchy: connect spectrum to $\mu_n$),
additive composition is the only admissible composition of the $\Ohat$ sectors.
Kleisli composition, as supplied by Paper~34's monad $(S^3, \mathsf{return},
\bind)$ and extended to the operator-algebra setting, is inadmissible.
\end{corollary}

\begin{proof}
Proposition~\ref{prop:additive} shows additive composition satisfies R8.
Theorem~\ref{thm:kleisli} shows every Kleisli assembly violates R8 in either
Case~A (multiplicative coupling obstruction) or Case~B (mode-mixing), and both
cases are exhaustive (Definition~\ref{def:cases}).  Therefore no Kleisli
assembly satisfies R8, and additive composition is forced.
\end{proof}

\begin{remark}[Scope of ``Kleisli inadmissible'']
\label{rem:scope}
The inadmissibility is with respect to R8.  If R8 were dropped from the design
requirements, Kleisli assemblies could in principle satisfy R1--R7.
Corollary~\ref{cor:sr2} establishes that R8 is the decisive discriminator
between additive and Kleisli assembly.
\end{remark}

%==========================================================================
\section{Scope and Remaining Open Sub-results}
\label{sec:scope}
%==========================================================================

\subsection{What SR2 closure establishes}
\label{subsec:established}

\begin{assertion}[SR2 status: CLOSED]
\label{ass:sr2closed}
Corollary~\ref{cor:sr2} closes SR2: additive composition of $\Ohat$ sectors is
the only admissible composition under R8.  Specifically:
\begin{enumerate}[label=(\roman*)]
  \item The result is exact: it follows from the definition of R8's moment
    operator $\calM = \sum_n m_n P_n$ (a fixed diagonal structure), the
    definition of Kleisli composition (sequential operator product), and the
    non-commutativity of $D^2_{B^4}$ and $\gamma T_{\rm cycle}$ (a computable fact about
    the sectors of $\Ohat$).
  \item The result is unconditional on P18-T2's other sub-results: SR2
    does not depend on SR1 or SR3.  It holds regardless of whether each
    $R_i$'s named operator is unique in its category or whether the
    coefficients are uniquely determined.
  \item The result rules out the specific Kleisli alternative identified in
    P18 \S4.4 \cite{P18} and \cite{P18T2reframed}.  Other operadic alternatives
    (e.g., configuration-space operad on $S^3$) are not addressed here and
    remain open as alternatives to additive composition.
\end{enumerate}
\end{assertion}

\subsection{Remaining open sub-results}
\label{subsec:remaining}

\begin{openproblem}[SR1: each operator unique in its category]
\label{op:sr1}
SR1 asks whether each $R_i$ is satisfied by a unique operator in its category.
For example, is $D^2_{B^4}$ the unique bulk operator (R1), or could another
second-order elliptic operator on $B^4$ also satisfy R1?  Closing SR1 requires
representation-theoretic arguments about the APS boundary conditions and the
constraint that R6 ($\mathbb{Z}_3$ family symmetry) selects a specific
$\mathbb{Z}_3$-equivariant extension of each operator.
\end{openproblem}

\begin{openproblem}[SR3: coefficients uniquely determined]
\label{op:sr3}
SR3 asks whether the four coefficients $(\alpha, \gamma = 3/4, \zeta =
\alpha^{5/4}, \beta = 3\pi/20)$ are the only assignment consistent with the
spectral predictions (P18 Theorems~5.2--5.4): ground state normalization
$\alpha^{-1}$, Planck mass formula, and three fermion families.  This requires
spectral rigidity arguments, e.g., showing that the Hopf-induced eigenvalue
structure of $\Delta_{S^3}$ on $S^3 = S^1 \hookrightarrow S^3 \to S^2$, together
with R8's moment hierarchy, forces specific coefficient values.
\end{openproblem}

\begin{assertion}[P18-T2 status after SR2]
\label{ass:p18t2status}
With SR2 now closed (this paper), the P18-T2 uniqueness proof has the following
status:
\begin{itemize}
  \item \textbf{SR1}: Open.
  \item \textbf{SR2}: Closed (Corollary~\ref{cor:sr2}).
  \item \textbf{SR3}: Open.
\end{itemize}
Two of the three sub-results remain open.  The closure of SR2 is a necessary
but not sufficient condition for P18-T2 uniqueness.  P18's canonical assembly
remains the canonical choice; its uniqueness across admissible assemblies is
established only conditionally on SR1 and SR3.
\end{assertion}

\subsection{Upgrade path if SR1 and SR3 are closed}
\label{subsec:upgrade}

\begin{assertion}[P18-T2 conditional on SR1+SR2+SR3]
\label{ass:upgrade}
If SR1 and SR3 are closed in subsequent papers:
\begin{enumerate}[label=(\roman*)]
  \item P18-T2 would be promoted from \texttt{proven-with-caveat} to
    \texttt{proven-in-paper} (or \texttt{proven-elsewhere} if proofs land in
    separate papers).
  \item The Pin~5 conditionality clause in \texttt{CLAUDE.md} (``the
    canonicality is conditional on the assembly choice in P18'') would be
    removed, restoring the unconditional bridge statement.
  \item P18 \S4.4's boxed Open Problem block would be replaced with a pointer
    to the complete proof.
\end{enumerate}
SR2 closure (this paper) is one of the three preconditions for this upgrade.
\end{assertion}

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

The main results of this addendum:

\begin{equation}\label{eq:summary}
\boxed{
\begin{aligned}
  &\text{R8: } \langle\psi_n \mid \beta\calM \mid \psi_n\rangle = \beta m_n
    \quad\text{(uniform coupling, all }n)
    \qquad\text{(Assertion~\ref{ass:r8}, P18 \S3.8)}\\[4pt]
  &\text{Additive: } E_n^+ = a_n + \beta m_n
    \quad\text{(R8 moment-linearity satisfied)}
    \qquad\text{(Proposition~\ref{prop:additive})}\\[4pt]
  &\text{Kleisli Case~A: } E_n^K = \beta G_n m_n,\; G_n \text{ varies}
    \quad\text{(coupling not uniform in }n)\\
  &\text{Kleisli Case~B: } [D^2_{B^4},\gamma T_{\rm cycle}] \neq 0
    \quad\text{(mode-mixing, no mode-moment correspondence)}\\
  &\text{Both cases violate R8}
    \qquad\text{(Theorem~\ref{thm:kleisli})}\\[4pt]
  &\boxed{\text{SR2 CLOSED: additive composition forced by R8}}
    \qquad\text{(Corollary~\ref{cor:sr2})}\\[4pt]
  &\text{Open: SR1 (operator uniqueness), SR3 (coefficient rigidity)}
    \qquad\text{(Open Problems~\ref{op:sr1}--\ref{op:sr3})}
\end{aligned}
}
\end{equation}

\noindent\textbf{What is proved.}
Theorem~\ref{thm:kleisli} is a rigorous result about the spectral algebra of
linear operators on $\mathcal{H}$: sequential product composition produces
eigenvalues that are products (Case~A) or are undefined mode-by-mode (Case~B),
neither of which matches the additive structure required by R8.  The
non-commutativity of $D^2_{B^4}$ and $\gamma T_{\rm cycle}$ is computable from the P18
definitions (commutator computation above and numerical check in
\texttt{verify\_P186.py}).

\noindent\textbf{What is conditional.}
SR2 closure is unconditional.  The full P18-T2 uniqueness theorem is
conditional on SR1 and SR3, which remain open.  The result in this paper
rules out Kleisli composition specifically; other alternative compositions
(operadic, PROP-based) are not addressed and require separate analysis.

\noindent\textbf{SR2 status: CLOSED.}
Requirement R8 (moment hierarchy linearity) forces additive composition of the
$\Ohat$ sectors.  Kleisli composition, as formalized in Paper~34 and identified
in P18 \S4.4 as the leading alternative, is inadmissible under R8.

\begin{thebibliography}{9}

\bibitem{P18}
L.~F.\ Vlegels, \emph{Paper~18: The Master Operator --- Assembly of the
Unified Field Operator for $(B^4, S^3)$ Geometry},
TOE Corpus, December 2025.

\bibitem{P03}
L.~F.\ Vlegels, \emph{Paper~03: Mathematical Foundations},
TOE Corpus, 2025.

\bibitem{P34}
L.~F.\ Vlegels, \emph{Paper~34: The Kleisli Monad over $S^3$ --- Monad Laws,
Kleisli Category, and Geometric Termination in LumenOS Attention},
TOE Corpus, April 2026.

\bibitem{P184}
L.~F.\ Vlegels, \emph{Addendum~184: OP-H Formal Derivation --- Perturb $=
H_{U(1)}$ via the Hopf Fibre Restriction of the P18 Dilatation Generator},
TOE Corpus, May 2026.

\bibitem{P18T2reframed}
L.~F.\ Vlegels, \emph{P18-T2 Reframed --- Handoff Memo},
\texttt{v2~Research/\#~P18-T2~reframed.md}, LumenOS Repository, 2026.

\bibitem{CLAUDE}
L.~F.\ Vlegels, \emph{LumenOS Operator Loader},
\texttt{CLAUDE.md} (Pin~5: Wheel~$\leftrightarrow$~$\hat{O}$),
LumenOS Repository, 2026.

\bibitem{moggi}
E.~Moggi, ``Notions of computation and monads,''
\emph{Information and Computation} \textbf{93}(1), 55--92 (1991).

\bibitem{atiyah}
M.~F.\ Atiyah, V.~K.\ Patodi, and I.~M.\ Singer,
``Spectral asymmetry and Riemannian geometry,''
\emph{Math.\ Proc.\ Cambridge Philos.\ Soc.}\ \textbf{77} (1975), 43--69.

\end{thebibliography}

\end{document}
