% 231_Addendum_BiharmonicKernelDOF2.tex
% Skill: gap-closure v1.1  |  Outcome: Case A  |  Date: 2026-05-23
%
% Proves the general monomial formula for D^2_{B^4}(x^n), extends the
% Gegenbauer truncation to 8 modes, establishes structural nilpotency of
% T = Delta_{S^3} o D^2_{B^4} in every finite Gegenbauer truncation, analyses
% the L^2 kernel of T, and shows that DOF-2 Kleisli composition does not
% produce new fixed points relative to additive composition.
% Copyright: Leon Fernando Vlegels, MIT.

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

\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{conjecture}[theorem]{Conjecture}

\DeclareMathOperator{\Tr}{Tr}
\newcommand{\weld}{\Delta_{S^3}}
\newcommand{\fork}{D^2_{B^4}}

\title{Addendum 231: Biharmonic Kernel Structure and DOF-2 Kleisli Analysis}
\author{TOE Corpus}
\date{2026-05-23}

\begin{document}
\maketitle

\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case A (closed) \quad|\quad
\textbf{Date:} 2026-05-23

\section{Overview and Results}

This addendum extends the biharmonic analysis of $T = \weld \circ \fork$
in two directions: (i) it derives a clean general formula for the action of
$\fork$ on monomials and proves that the resulting operator is structurally
nilpotent in \emph{any} finite Gegenbauer truncation; (ii) it analyses
P18 DOF-2 (whether Kleisli/monadic composition is admissible as a
compositional alternative to additive $\hat{O}$), showing that Kleisli
composition does \emph{not} produce new fixed points for the TOE density
$\rho$ or any polynomial in the same family.

\paragraph{Main results.}
\begin{enumerate}
\item \textbf{Monomial formula (Theorem~\ref{thm:monomial}):}
  $\fork(x^n) = 16n^2(n^2-1)\,x^{n-2}$ for $n \geq 2$, and $0$ for $n = 0,1$.
\item \textbf{Structural Nilpotency Theorem (Theorem~\ref{thm:nilpotent}):}
  $[D^2]_{\rm geg}$ is strictly upper triangular with offset $\geq 2$ in any
  finite Gegenbauer truncation, so $T_{\rm geg}$ is nilpotent of order
  $\leq \lceil N/2 \rceil$ in the $N$-mode truncation.
\item \textbf{8-mode explicit matrix (Section~\ref{sec:8mode}):}
  All entries of the $8 \times 8$ matrix $T_{\rm geg}$ are computed exactly;
  nilpotency order is 4 (i.e.\ $T_{\rm geg}^4 = 0$, $T_{\rm geg}^3 \neq 0$).
\item \textbf{L² spectrum (Section~\ref{sec:L2}):}
  $\ker(T)$ on polynomials consists of solutions to $\fork(f) = c$
  (a biharmonic ODE); the spectrum of $T$ has no non-zero polynomial
  eigenfunctions.
\item \textbf{DOF-2 Kleisli (Theorem~\ref{thm:kleisli}):}
  $T_K(\rho) = \frac{48\pi}{1+16\pi}(1 - 2x)$ is linear in $x$; it is
  not proportional to $\rho_{\rm TOE}$ (cubic), so no Kleisli eigenvalue
  exists. For the entire polynomial family $ax^3 + bx^2 + cx$, the Kleisli
  fixed-point equation has no non-trivial solution.
\item \textbf{DOF-2 conclusion (Corollary~\ref{cor:dof2}):}
  Kleisli composition does not open new fixed-point solutions beyond those
  of additive $T$. DOF-2 does not alter the eigentrajectory conclusion
  from A228.
\end{enumerate}

\paragraph{Cross-references.}
A224 (eigentrajectory definition); A225 ($a_3 = \Omega_0$); A226 (commutator
analysis); A227 ($d=4$ equal-weight identity); A228 (4-mode truncation and
$x^*_{\rm CZ}$); A230 (commutator ratio $-48\pi/(\pi-1)$).

\bigskip

% ─────────────────────────────────────────────────────────────────────────────
\section{Part 1: General Monomial Formula for $\fork$}
\label{sec:monomial}
% ─────────────────────────────────────────────────────────────────────────────

In the Hopf $x$-coordinate ($x = \cos^2\!\beta$, $x \in [0,1]$), the
biharmonic operator on $B^4$ takes the form
\begin{equation}
  \fork f = 16x^2 f'''' + 96x f''' + 96 f''.
  \label{eq:D2-def}
\end{equation}

\begin{theorem}[Monomial Formula]\label{thm:monomial}
For $n \geq 2$:
\[
  \fork(x^n) = 16n^2(n^2-1)\,x^{n-2}.
\]
For $n = 0$ or $n = 1$: $\fork(x^n) = 0$.
\end{theorem}

\begin{proof}
For $n \geq 4$, the three terms contribute as follows (lower values are handled
by inspection since some derivatives vanish):
\begin{align*}
  16x^2 \cdot n(n-1)(n-2)(n-3)x^{n-4}
  &= 16n(n-1)(n-2)(n-3)\,x^{n-2}, \\
  96x \cdot n(n-1)(n-2)x^{n-3}
  &= 96n(n-1)(n-2)\,x^{n-2}, \\
  96 \cdot n(n-1)x^{n-2}
  &= 96n(n-1)\,x^{n-2}.
\end{align*}
Factor out $n(n-1)x^{n-2}$:
\[
  \fork(x^n) = n(n-1)x^{n-2}
    \bigl[16(n-2)(n-3) + 96(n-2) + 96\bigr].
\]
Expanding the bracket:
\begin{align*}
  16(n-2)(n-3) + 96(n-2) + 96
  &= 16(n^2 - 5n + 6) + 96n - 192 + 96 \\
  &= 16n^2 - 80n + 96 + 96n - 96 \\
  &= 16n^2 + 16n = 16n(n+1).
\end{align*}
Therefore:
\[
  \fork(x^n) = n(n-1) \cdot 16n(n+1) \cdot x^{n-2}
  = 16n^2(n-1)(n+1)\,x^{n-2} = 16n^2(n^2-1)\,x^{n-2}.
\]
For $n=2$: only the $f''$ term survives, giving $96 \cdot 2 \cdot 1 = 192 = 16 \cdot 4 \cdot 3$. For $n=3$: $16 \cdot 9 \cdot 8 \cdot x = 1152x$ (the $f''''$ term vanishes since $3! = 6$ requires $n\geq4$ for a non-zero 4th derivative; here only $f'''$ and $f''$ contribute, giving the same formula). The cases $n=0,1$ give zero since $f'' = 0$.
\end{proof}

\begin{corollary}[Degree Lowering]\label{cor:degree}
$\fork$ maps any polynomial of degree $n$ to a polynomial of degree $n-2$
(or to zero for $n \leq 1$). In particular, $\fork$ lowers polynomial degree
by exactly $2$.
\end{corollary}

\begin{remark}[Explicit values]
\begin{center}
\begin{tabular}{@{}ccc@{}}
\toprule
$n$ & $16n^2(n^2-1)$ & $\fork(x^n)$ \\
\midrule
0,1 & 0             & $0$ \\
2   & 192           & $192$ \\
3   & 1152          & $1152\,x$ \\
4   & 3840          & $3840\,x^2$ \\
5   & 9600          & $9600\,x^3$ \\
6   & 20160         & $20160\,x^4$ \\
7   & 37632         & $37632\,x^5$ \\
\bottomrule
\end{tabular}
\end{center}
\end{remark}

% ─────────────────────────────────────────────────────────────────────────────
\section{Part 2: Structural Nilpotency Theorem}
\label{sec:nilpotent}
% ─────────────────────────────────────────────────────────────────────────────

\subsection{Gegenbauer Basis and 8-Mode Extension}
\label{sec:8mode}

The Gegenbauer basis on $S^3$ in the $x$-coordinate is $U_n(x) = C_n^{(1)}(2x-1)$
(Chebyshev-U polynomials of the second kind evaluated at $2x-1$), with the
recurrence $U_{n+1}(x) = (4x-2)U_n(x) - U_{n-1}(x)$.  The first eight are:
\begin{align*}
U_0 &= 1, \\
U_1 &= 4x - 2, \\
U_2 &= 16x^2 - 16x + 3, \\
U_3 &= 64x^3 - 96x^2 + 40x - 4, \\
U_4 &= 256x^4 - 512x^3 + 336x^2 - 80x + 5, \\
U_5 &= 1024x^5 - 2560x^4 + 2304x^3 - 896x^2 + 140x - 6, \\
U_6 &= 4096x^6 - 12288x^5 + 14080x^4 - 7680x^3 + 2016x^2 - 224x + 7, \\
U_7 &= 16384x^7 - 57344x^6 + 79872x^5 - 56320x^4 + 21120x^3 - 4032x^2 + 336x - 8.
\end{align*}
The leading coefficient of $U_n$ is $4^n$ and $U_n(0) = (-1)^n(n+1)$.

The $\weld$ (Laplace-Beltrami on $S^3$) eigenvalues are $\mu_n = -n(n+2)$:
\[
  \mu_0=0,\; \mu_1=-3,\; \mu_2=-8,\; \mu_3=-15,\; \mu_4=-24,\;
  \mu_5=-35,\; \mu_6=-48,\; \mu_7=-63.
\]

\subsection{Structural Zero Theorem}

\begin{theorem}[Structural Upper-Triangularity]\label{thm:struct}
In any $N$-mode Gegenbauer truncation (modes $0, 1, \ldots, N-1$):
\[
  [D^2]_{\rm geg}[i,j] = 0 \quad \text{whenever } j \leq i+1.
\]
Equivalently, $[D^2]_{\rm geg}$ is strictly upper triangular with offset
$\geq 2$: only entries with $j \geq i+2$ can be non-zero.
\end{theorem}

\begin{proof}
By Corollary~\ref{cor:degree}, $\fork(U_j)$ is a polynomial of degree $j-2$.
When expressed in the Gegenbauer basis, only modes $U_0, U_1, \ldots, U_{j-2}$
appear in $\fork(U_j)$.  Consequently, $[D^2]_{\rm geg}[i,j] = 0$ for $i > j-2$,
i.e.\ for $j < i+2$.
\end{proof}

\begin{theorem}[Superdiagonal-2 Formula]\label{thm:superdiag}
The immediate offset-2 diagonal of $[D^2]_{\rm geg}$ is given exactly by
\[
  [D^2]_{\rm geg}[n-2,\, n] = 256\,n^2(n^2-1), \quad n \geq 2.
\]
\end{theorem}

\begin{proof}
The leading monomial of $U_n$ is $4^n x^n$.  Applying $\fork$:
\[
  \fork(4^n x^n) = 4^n \cdot 16n^2(n^2-1)\,x^{n-2}.
\]
The leading monomial of $U_{n-2}$ is $4^{n-2} x^{n-2}$.  The projection
onto $U_{n-2}$ from the leading term alone is:
\[
  [D^2]_{\rm geg}[n-2,n] = \frac{4^n \cdot 16n^2(n^2-1)}{4^{n-2}}
  = 4^2 \cdot 16n^2(n^2-1) = 256\,n^2(n^2-1).
\]
\end{proof}

\begin{corollary}[Relation to Monomial Coefficient]
$[D^2]_{\rm geg}[n-2,n] = 16 \cdot \bigl(16n^2(n^2-1)\bigr) / 16
= 16 \times \text{biharm}(n)$, where $\text{biharm}(n) = 16n^2(n^2-1)$ is
the coefficient from Theorem~\ref{thm:monomial}.
\end{corollary}

\subsection{The 8-Mode $[D^2]_{\rm geg}$ Matrix}

Applying $\fork$ to each $U_j$ using Theorem~\ref{thm:monomial} and the
change-of-basis matrix $P$ (columns = $U_n$ in monomial form), the full
$8 \times 8$ matrix $[D^2]_{\rm geg} = P^{-1} D^2_{\rm mono} P$ is:

\[
[D^2]_{\rm geg} = \begin{pmatrix}
0 & 0 & 3072 & 18432 & 76800 & 233472 & 602112 & 1351680 \\
0 & 0 & 0 & 18432 & 98304 & 356352 & 983040 & 2334720 \\
0 & 0 & 0 & 0 & 61440 & 307200 & 1029120 & 2672640 \\
0 & 0 & 0 & 0 & 0 & 153600 & 737280 & 2347008 \\
0 & 0 & 0 & 0 & 0 & 0 & 322560 & 1505280 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 602112 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix}
\]
(rows/columns indexed $0, \ldots, 7$; all lower-left entries are zero by
Theorem~\ref{thm:struct}).

The explicit values of $\fork(U_j)$ in the Gegenbauer basis for $j = 4, 5, 6, 7$:
\begin{align*}
\fork(U_4) &= 76800\,U_0 + 98304\,U_1 + 61440\,U_2, \\
\fork(U_5) &= 233472\,U_0 + 356352\,U_1 + 307200\,U_2 + 153600\,U_3, \\
\fork(U_6) &= 602112\,U_0 + 983040\,U_1 + 1029120\,U_2 + 737280\,U_3 + 322560\,U_4, \\
\fork(U_7) &= 1351680\,U_0 + 2334720\,U_1 + 2672640\,U_2 + 2347008\,U_3
             + 1505280\,U_4 + 602112\,U_5.
\end{align*}

\subsection{Nilpotency of $T_{\rm geg}$}

\begin{theorem}[Structural Nilpotency]\label{thm:nilpotent}
Let $T_{\rm geg}^{(N)} = \operatorname{diag}(\mu_0, \ldots, \mu_{N-1}) \cdot [D^2]_{\rm geg}^{(N)}$
be the $N$-mode truncation of $T = \weld \circ \fork$.  Then:
\begin{enumerate}
\item $T_{\rm geg}^{(N)}$ is strictly upper triangular with offset $\geq 2$.
\item All eigenvalues of $T_{\rm geg}^{(N)}$ are zero (nilpotent).
\item The nilpotency order satisfies $(T_{\rm geg}^{(N)})^k = 0$ for
  $k \geq \lceil N/2 \rceil$.
\item This holds in \emph{every} finite Gegenbauer truncation; it is not an
  artifact of the 4-mode case (A228).
\end{enumerate}
\end{theorem}

\begin{proof}
Statement (1): By Theorem~\ref{thm:struct}, $[D^2]_{\rm geg}^{(N)}$ has
offset $\geq 2$.  Multiplying by $\operatorname{diag}(\mu)$ from the left
preserves this structure ($\mu_0 = 0$ kills row~0 entirely, adding a further
zero; remaining rows inherit the offset from $[D^2]_{\rm geg}$).

Statement (2): Strictly upper triangular matrices (of any offset) have all
eigenvalues zero.

Statement (3): A matrix with offset $\geq r$ satisfies $M^k = 0$ for
$k \geq \lceil N/r \rceil$; here $r = 2$.

Statement (4): The proof uses only the degree-lowering property of $\fork$
(Corollary~\ref{cor:degree}), which is an algebraic identity, not a truncation artifact.
\end{proof}

\begin{remark}[8-Mode Nilpotency Order]
For the $8$-mode case, the longest chain of non-zero products in $T_{\rm geg}$
runs along modes $1 \to 3 \to 5 \to 7$ (each step jumps by at least 2):
\[
  T[1,3]\cdot T[3,5]\cdot T[5,7]
  = (-55296)(-2304000)(-21073920) \neq 0.
\]
Hence $T_{\rm geg}^3 \neq 0$.  No 4-step chain is possible (it would require
reaching mode $\geq 9$, outside the truncation), so $T_{\rm geg}^4 = 0$.
The nilpotency order is exactly $4$ in the 8-mode truncation.

The full $8 \times 8$ $T_{\rm geg}$ matrix is:
\[
T_{\rm geg} = \begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & -55296 & -294912 & -1069056 & -2949120 & -7004160 \\
0 & 0 & 0 & 0 & -491520 & -2457600 & -8232960 & -21381120 \\
0 & 0 & 0 & 0 & 0 & -2304000 & -11059200 & -35205120 \\
0 & 0 & 0 & 0 & 0 & 0 & -7741440 & -36126720 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & -21073920 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{pmatrix}
\]
\end{remark}

% ─────────────────────────────────────────────────────────────────────────────
\section{Part 3: L² Spectrum of $T$}
\label{sec:L2}
% ─────────────────────────────────────────────────────────────────────────────

\begin{proposition}[No Polynomial Eigenfunctions of $T$]\label{prop:L2spec}
$T = \weld \circ \fork$ has no non-zero polynomial eigenfunction on $[0,1]$.
\end{proposition}

\begin{proof}
Suppose $T(p) = \lambda p$ for a polynomial $p$ of degree $n$.  By
Corollary~\ref{cor:degree}, $\fork(p)$ has degree $n-2$, and $\weld(\cdot)$
preserves degree (Laplace-Beltrami maps degree-$k$ polynomials to degree-$k$
polynomials in the $x$-coordinate).  Hence $T(p)$ has degree $n-2 < n$.
For $T(p) = \lambda p$ we need $n - 2 = n$ (degree matching), which is
impossible.  Alternatively, $\lambda = 0$ and $T(p)$ is identically zero,
but then $p$ is in $\ker(T)$, not an eigenfunction for $\lambda \neq 0$.
\end{proof}

\begin{remark}[Structural vs.\ Spectral nilpotency]
The finite-truncation nilpotency (Theorem~\ref{thm:nilpotent}) combined with
Proposition~\ref{prop:L2spec} shows that $T$ has no non-zero spectrum on
polynomial functions.  The situation on the full $L^2(S^3)$ (beyond
polynomials) requires separate treatment; as an unbounded operator involving
6th-order derivatives, $T$ may have non-polynomial spectrum.
\end{remark}

\begin{proposition}[Kernel of $T$ on Polynomials]
A polynomial $f$ lies in $\ker(T)$ if and only if $\fork(f) \in \ker(\weld)$.
Since $\ker(\weld) = \{\text{constants}\}$ (the $n=0$ Gegenbauer mode with
$\mu_0 = 0$), this requires $\fork(f) = c$ for some constant $c$, i.e.:
\[
  16x^2 f'''' + 96x f''' + 96 f'' = c.
\]
The homogeneous equation ($c=0$) has the Euler-type solutions $1, x, x^{-1},
\ln(x)$ (where the latter two are excluded on $[0,1]$ for polynomial or
smooth solutions).  For $c \neq 0$, a particular solution is $f_p = cx^2/192$.
\end{proposition}

% ─────────────────────────────────────────────────────────────────────────────
\section{Part 4: DOF-2 --- Kleisli vs.\ Additive Composition}
\label{sec:dof2}
% ─────────────────────────────────────────────────────────────────────────────

\subsection{Setup}

P18, \S4.4 identifies DOF-2 as the open question of whether additive composition
is the only admissible composition for $\hat{O}$, or whether Kleisli/monadic
composition (via the distribution monad) is also admissible.

\paragraph{Additive $T$ (current):}
$T_{\rm add} = \weld \circ \fork$ is linear.  From A228, $T_{\rm add}$ is
nilpotent in every finite Gegenbauer truncation and has no non-zero polynomial
eigenfunction.

\paragraph{Kleisli $T_K$:}
Under Kleisli (probability/distribution monad), Fork normalises its output
before passing to Weld:
\[
  T_K(\rho)(x) = \weld\!\left(\frac{\fork(\rho)}{Z[\rho]}\right)(x),
  \quad Z[\rho] = \int_0^1 \fork(\rho)(x)\,dx.
\]
$T_K$ is \emph{nonlinear} in $\rho$ (the denominator $Z[\rho]$ depends on
$\rho$).  A Kleisli fixed point $T_K(\rho^*) = \rho^*$ would satisfy
$T_{\rm add}(\rho^*) = Z[\rho^*]\,\rho^*$, a nonlinear eigenvalue equation.

\subsection{$\fork(\rho_{\rm TOE})$ in Closed Form}

The TOE density is $\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x$ (P02, eq.\ 1.4).
Applying Theorem~\ref{thm:monomial} term-by-term:
\begin{align}
  \fork(16\pi^3 x^3) &= 16\pi^3 \cdot 1152\,x = 18432\pi^3 x, \notag\\
  \fork(3\pi^2 x^2)  &= 3\pi^2 \cdot 192 = 576\pi^2, \notag\\
  \fork(2\pi x)      &= 0 \quad (\text{linear term killed}). \notag
\end{align}
Therefore:
\begin{equation}
  \fork(\rho) = 576\pi^2 + 18432\pi^3\,x.
  \label{eq:D2rho}
\end{equation}
This is a \emph{linear polynomial} in $x$.

\subsection{Normalization $Z$ and Kleisli Output}

\begin{equation}
  Z = \int_0^1 \fork(\rho)\,dx = 576\pi^2 + 9216\pi^3 = 576\pi^2(1 + 16\pi).
  \label{eq:Z}
\end{equation}

Expanding \eqref{eq:D2rho} in the Gegenbauer basis with $U_1 = 4x-2$:
\[
  18432\pi^3 x = 4608\pi^3\,U_1 + 9216\pi^3\,U_0.
\]
So $\fork(\rho) = (576\pi^2 + 9216\pi^3)\,U_0 + 4608\pi^3\,U_1 = Z\,U_0 + 4608\pi^3\,U_1$.

\subsection{$T(\rho)$ and Kleisli Output}

Applying $\weld$ with eigenvalues $\mu_0 = 0$, $\mu_1 = -3$:
\[
  T(\rho) = \weld(\fork(\rho)) = 0 \cdot Z\,U_0 + (-3) \cdot 4608\pi^3\,U_1
  = -13824\pi^3\,U_1.
\]
Converting to monomials via $U_1 = 4x - 2$:
\begin{equation}
  T(\rho) = 27648\pi^3 - 55296\pi^3\,x.
  \label{eq:Trho}
\end{equation}

\begin{theorem}[Kleisli Output for $\rho_{\rm TOE}$]\label{thm:TK-rho}
\[
  T_K(\rho) = \frac{T(\rho)}{Z}
  = \frac{27648\pi^3 - 55296\pi^3 x}{576\pi^2(1+16\pi)}
  = \frac{48\pi}{1+16\pi}\,(1 - 2x).
\]
\end{theorem}

\begin{proof}
Direct division of \eqref{eq:Trho} by \eqref{eq:Z}:
$27648\pi^3 / (576\pi^2) = 48\pi$ and
$55296\pi^3 / (576\pi^2) = 96\pi$, both divided by $(1+16\pi)$.
\end{proof}

\begin{remark}[Degree mismatch]
$T_K(\rho)$ is \emph{linear} in $x$; $\rho$ is \emph{cubic} in $x$.
No scalar $\lambda$ satisfies $T_K(\rho)(x) = \lambda\rho(x)$ for all $x$.
The zero of $T_K(\rho)$ is at $x = 1/2$, not at $x^*_{\rm CZ} \approx 0.0142$.
\end{remark}

\subsection{Kleisli Fixed-Point Analysis for the Polynomial Family}

\begin{theorem}[No Kleisli Fixed Point in Polynomial Family]\label{thm:kleisli}
The Kleisli fixed-point equation
\[
  T_{\rm add}(\rho_c) = Z[\rho_c]\,\rho_c
\]
has no non-trivial solution in the family $\rho_c(x) = ax^3 + bx^2 + cx$
(3 free parameters, $a \neq 0$).
\end{theorem}

\begin{proof}
For $\rho_c = ax^3 + bx^2 + cx$:
\[
  \fork(\rho_c) = 1152a\,x + 192b
  \quad\Rightarrow\quad
  Z[\rho_c] = \int_0^1 \fork(\rho_c)\,dx = 576a + 192b.
\]
In Gegenbauer coordinates ($1152ax = 288a\,U_1 + 576a\,U_0$):
\[
  T(\rho_c) = \weld\bigl((576a + 192b)\,U_0 + 288a\,U_1\bigr)
  = 0 + (-3) \cdot 288a\,U_1 = -864a\,(4x-2) = 1728a - 3456a\,x.
\]
The Kleisli fixed-point equation $T(\rho_c) = Z[\rho_c]\,\rho_c$ in monomials:
\[
  1728a - 3456a\,x = (576a + 192b)\,(ax^3 + bx^2 + cx).
\]
The constant term on the left is $1728a$; the constant term on the right is
$(576a+192b)\cdot 0 = 0$ (since $\rho_c$ has no constant term).  Therefore:
\[
  1728a = 0 \implies a = 0.
\]
With $a = 0$: $T(\rho_c) = 0$ and $Z[\rho_c]\,\rho_c = 192b\,(bx^2+cx)$.
For $T = Z\rho_c$ we need $192b\,(bx^2+cx) = 0$, so either $b = 0$ (giving
$\rho_c = cx$ with $Z = 0$, trivial) or $\rho_c = 0$.
\end{proof}

\begin{corollary}[DOF-2 Conclusion]\label{cor:dof2}
Kleisli composition (DOF-2) does not produce any new non-trivial fixed point
for the operator $T = \weld \circ \fork$ within the polynomial family
$\rho_c = ax^3 + bx^2 + cx$ that includes the TOE density $\rho_{\rm TOE}$.
In particular:
\begin{enumerate}
\item The TOE density $\rho_{\rm TOE}$ is \emph{not} a Kleisli fixed point.
\item The Kleisli output $T_K(\rho_{\rm TOE}) = \frac{48\pi}{1+16\pi}(1-2x)$
  is linear and non-proportional to $\rho_{\rm TOE}$.
\item DOF-2 does not alter the eigentrajectory conclusion from A228: the only
  $T$-fixed point (additive or Kleisli) in the polynomial family is trivial.
\end{enumerate}
\end{corollary}

% ─────────────────────────────────────────────────────────────────────────────
\section{Part 5: Eigentrajectory Implications}
\label{sec:eigentrajectory}
% ─────────────────────────────────────────────────────────────────────────────

\begin{proposition}[$x^*_{\rm CZ}$ and Kleisli]\label{prop:xstar-kleisli}
The commutator-zero eigenpoint $x^*_{\rm CZ} = (\pi-1)/(48\pi)$ (A228,
A230) does \emph{not} appear naturally in the Kleisli fixed-point analysis.
\begin{enumerate}
\item The zero of $T_K(\rho_{\rm TOE})$ is at $x = 1/2$, not $x^*_{\rm CZ}$.
\item The Kleisli fixed-point equation for polynomial $\rho_c$ imposes
  $a = 0$ as the first constraint, eliminating $x^*_{\rm CZ}$ (which was
  derived from the cubic term $a_3 = \Omega_0$).
\item $x^*_{\rm CZ}$ remains relevant to the commutator-zero condition
  $[D^2_{B^4}, \weld]\rho(x^*_{\rm CZ}) = 0$ (A226, A228), but this is
  a condition on the additive commutator, not the Kleisli normalisation.
\end{enumerate}
\end{proposition}

\begin{remark}[General structural observation]
The Kleisli normalisation $Z[\rho] = \int_0^1 \fork(\rho)\,dx$ acts as a
global scale factor derived from the \emph{bulk} (higher-order terms in
$\rho$).  For $\rho_{\rm TOE}$, $Z = 576\pi^2(1+16\pi)$ is dominated by
$16\pi \approx 50$, so $Z \approx 576\pi^2 \cdot 50 \approx 284{,}000$.
The Kleisli equation $T(\rho) = Z\rho$ requires $T(\rho)$ to be much larger
than $T(\rho) = 27648\pi^3 - 55296\pi^3 x$ (which is $O(\pi^3) \approx 31$),
creating a parametric mismatch of order $16\pi$ that prevents a fixed point.
\end{remark}

\begin{remark}[Relationship to A228 Conjecture~5.1]
A228 Conjecture~5.1 identified $x^*_{\rm CZ}$ as the candidate eigentrajectory
for the \emph{additive} $T$.  The present result shows that switching to
Kleisli composition does not rescue this conjecture: the Kleisli equation
has no solution in the polynomial family at all, not just no solution at
$x^*_{\rm CZ}$.  The eigentrajectory fixed-point problem remains open for
the full $L^2$ operator.
\end{remark}

% ─────────────────────────────────────────────────────────────────────────────
\section{Summary and Open Items}
\label{sec:summary}
% ─────────────────────────────────────────────────────────────────────────────

\paragraph{Closed (this addendum).}
\begin{enumerate}
\item The formula $\fork(x^n) = 16n^2(n^2-1)x^{n-2}$ is proven in full
  generality.
\item Structural nilpotency of $T_{\rm geg}^{(N)}$ in any finite-mode
  truncation is established algebraically.
\item The 8-mode $T_{\rm geg}$ matrix is computed explicitly;
  its nilpotency order is 4.
\item The superdiagonal formula $[D^2]_{\rm geg}[n-2,n] = 256n^2(n^2-1)$
  provides a closed-form expression for the first non-zero diagonal.
\item DOF-2: Kleisli composition does not produce new polynomial fixed points.
  $T_K(\rho_{\rm TOE}) = \frac{48\pi}{1+16\pi}(1-2x)$ is definitively
  non-proportional to $\rho_{\rm TOE}$.
\end{enumerate}

\paragraph{Open item 1 (OP-alpha blocker, from A228, unchanged).}
The true eigentrajectory $x^*$ of $T = \weld\circ\fork$ on the full $L^2(S^3)$
requires the complete $\hat{O}$ eigenspectrum (A224 Remark 3.6).  Whether
$x^* = x^*_{\rm CZ} = (\pi-1)/(48\pi)$ remains Conjecture~5.1 of A228.

\paragraph{Open item 2.}
The L² spectrum of the unbounded operator $T = \weld\circ\fork$ beyond the
polynomial subspace is not addressed here.  Specifically, whether there exist
smooth (non-polynomial) functions $f \in L^2([0,1], w\,dx)$ satisfying
$T(f) = \lambda f$ for $\lambda \neq 0$ is open.

\paragraph{Open item 3.}
The DOF-2 analysis here covers only the polynomial family $ax^3+bx^2+cx$.
An exhaustive analysis of Kleisli fixed points in $L^2(S^3)$ (beyond
polynomials) is a separate research item.

\bigskip
\noindent\textit{Cross-references:} A224 (eigentrajectory, boundary locus),
A225 ($a_3 = \Omega_0$, Gegenbauer coefficients), A226 (commutator analysis),
A227 ($d=4$ equal-weight identity, $[D^2]_{\rm geg}[0,3]=[D^2]_{\rm geg}[1,3]$),
A228 (4-mode truncation, $x^*_{\rm CZ}$),
A230 (commutator ratio $-48\pi/(\pi-1)$).

\end{document}
