% 236_Addendum_BiharmonicHeatKernel.tex
% Skill: gap-closure v1.1  |  Outcome: Case B  |  Date: 2026-05-23
%
% Investigates whether T = Delta_{S^3} o D^2_{B^4} has a non-trivial fixed-point
% density in L^2([0,1], w dx) (w = sqrt(x(1-x))), and what that means for the
% OP-alpha blocker (A224 Remark 3.6).
%
% Result: Case B — T has no non-trivial L^2 fixed point.  The OP-alpha blocker
% is deepened: x* cannot be defined via a T-eigenstate density.
% 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 236: Biharmonic Heat Kernel Fixed-Point Analysis\\
  and the OP-alpha Blocker}
\author{TOE Corpus}
\date{2026-05-23}

\begin{document}
\maketitle

\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case B (deepened blocker --- no $L^2$ fixed point for $T$)
\quad|\quad \textbf{Date:} 2026-05-23

\section{Overview and Main Results}

\paragraph{Problem.}
A224 Remark~3.6 flagged: \emph{``precise value of $|u^*|$ not derived in
closed form here --- flagged as pending a direct computation from the
$\hat{O}$ eigenspectrum.''} A228 established that the commutator-zero
eigenpoint $x^*_{\rm CZ} = (\pi-1)/(48\pi)$ is a natural algebraic
candidate for the true eigentrajectory, but noted that the T-fixed-point
equation requires the full $L^2$ eigenspectrum.  A231 proved that
$T = \weld\circ\fork$ is nilpotent in every finite Gegenbauer truncation.

The present addendum asks the next natural question: \emph{does $T$ have a
non-trivial fixed point in the full infinite-dimensional $L^2([0,1], w\,dx)$
space?}  This would close the OP-alpha blocker by providing a density $f^*$
from which $x^* = \langle f^*|x|f^*\rangle / \langle f^*|f^*\rangle$ could
be computed.

\paragraph{Main results.}
\begin{enumerate}
\item \textbf{Gegenbauer matrix formula (Theorem~\ref{thm:gegmat}):}
  The general element $[D^2]_{\rm geg}[n,k]$ is expressed in closed form
  via the monomial change-of-basis matrix $P$.  The superdiagonal element
  $[D^2]_{\rm geg}[n-2,n] = 256n^2(n^2-1)$ (A231) is recovered as a
  special case.

\item \textbf{Explicit T-recurrence (Section~\ref{sec:recurrence}):}
  The fixed-point equation $T f^* = f^*$ in Gegenbauer coefficients gives
  $c_0 = 0$ and an infinite downward recurrence expressing $c_n$ in terms
  of $c_k$ for $k \geq n+2$, with leading amplification factor
  $|T[n,n+2]| = n(n+2)\cdot 256(n+2)^2((n+2)^2-1) \sim 256\,n^6$.

\item \textbf{Finite-truncation nilpotency implies no truncated fixed point
  (Theorem~\ref{thm:nilp-nofp}):}
  In every $N$-mode Gegenbauer truncation, the only solution to
  $T^{(N)} c = c$ is $c = \mathbf{0}$ (consequence of A231 Theorem~3).

\item \textbf{Downward amplification theorem
  (Theorem~\ref{thm:amplification}):}
  The leading amplification factor satisfies
  $|T[n,n+2]| \geq 55296$ for all $n \geq 1$, grows monotonically, and
  reaches $\sim 256\,n^6$ asymptotically.  Starting from any compactly
  supported seed $c_N = 1$ ($c_k = 0$ for $k \neq N$), the induced
  lower-mode coefficients satisfy
  $|c_{N-2k}| \gtrsim (256)^k \prod_{j=0}^{k-1}(N-2j)^6$,
  which is non-$\ell^2$ for all $N \geq 2$.

\item \textbf{Case B theorem (Theorem~\ref{thm:caseB}):}
  $T = \weld\circ\fork$ has \emph{no non-trivial fixed point} in
  $L^2([0,1], w\,dx)$.  The fixed-point density does not exist as an $L^2$
  object.

\item \textbf{OP-alpha blocker: deepened, not closed
  (Section~\ref{sec:blocker}):}
  Since no $L^2$ density $f^*$ satisfying $T f^* = f^*$ exists, the
  eigentrajectory point $x^* = \langle f^*|x|f^*\rangle/\langle f^*|f^*\rangle$
  cannot be defined via $T$-eigenstates.  The commutator-zero candidate
  $x^*_{\rm CZ} = (\pi-1)/(48\pi)$ (A228) remains the best algebraic
  proxy, but its derivation now stands on commutator analysis alone, not
  on a spectral eigenstate.  The blocker is refined: the question shifts
  from ``compute $x^*$ from $T f^* = f^*$'' to ``identify the correct
  functional space for $T$ eigenstates, or abandon $T$-fixed-point
  framing entirely.''
\end{enumerate}

\paragraph{Cross-references.}
A224 (eigentrajectory, OP-alpha blocker); A225 ($a_3 = \Omega_0$);
A226 (commutator); A228 ($x^*_{\rm CZ}$, 4-mode nilpotency); A230
(commutator ratio); A231 (monomial formula, 8-mode nilpotency, DOF-2).

% ─────────────────────────────────────────────────────────────────────────────
\section{Setup and Notation}
\label{sec:setup}
% ─────────────────────────────────────────────────────────────────────────────

The Hopf $x$-coordinate is $x = \cos^2\!\beta \in [0,1]$.  The biharmonic
operator on $B^4$ in $x$ is
\begin{equation}
  \fork f = 16x^2 f'''' + 96x f''' + 96f'',
  \quad \fork(x^n) = 16n^2(n^2-1)\,x^{n-2} \quad (n \geq 2),
  \label{eq:D2}
\end{equation}
with $\fork(1) = \fork(x) = 0$ (A231 Theorem~1).

The Gegenbauer basis is $\{U_n(x)\}_{n \geq 0}$ where $U_n(x) =
C_n^{(1)}(2x-1)$ (Chebyshev-U at $2x-1$).  Inner product:
\begin{equation}
  \langle f, g\rangle_w = \int_0^1 f(x)\,g(x)\,w(x)\,dx,
  \quad w(x) = \sqrt{x(1-x)},
  \quad \langle U_n, U_m\rangle_w = \tfrac{\pi}{4}\,\delta_{nm}.
  \label{eq:inner}
\end{equation}
Eigenvalues of $\weld$ (Laplace-Beltrami on $S^3$) in this basis:
$\mu_n = -n(n+2)$.

The Gegenbauer matrix of $\fork$ is the infinite array
\begin{equation}
  [D^2]_{\rm geg}[n,k] = \frac{\langle U_n,\,\fork(U_k)\rangle_w}
                               {\langle U_n, U_n\rangle_w}
  = \frac{4}{\pi}\int_0^1 U_n(x)\,\fork(U_k)(x)\,w(x)\,dx.
  \label{eq:D2geg-def}
\end{equation}
The operator $T = \weld\circ\fork$ has Gegenbauer matrix
$T[n,k] = \mu_n\,[D^2]_{\rm geg}[n,k]$.

\textbf{Structural fact (A231 Theorem~2):}
$[D^2]_{\rm geg}[n,k] = 0$ whenever $k \leq n+1$; the matrix is strictly
upper triangular with offset $\geq 2$.

% ─────────────────────────────────────────────────────────────────────────────
\section{Gegenbauer Matrix Elements}
\label{sec:gegmat}
% ─────────────────────────────────────────────────────────────────────────────

\begin{theorem}[Gegenbauer matrix via change of basis]\label{thm:gegmat}
Let $P$ be the upper triangular change-of-basis matrix with $P_{mj}$
= coefficient of $x^m$ in $U_j(x)$, and let $D^2_{\rm mono}$ be the
matrix of $\fork$ in the monomial basis: $(D^2_{\rm mono})_{j-2,j} =
16j^2(j^2-1)$ for $j \geq 2$, all other entries zero.  Then in any
$N$-mode truncation:
\[
  [D^2]_{\rm geg}^{(N)} = P^{-1}\,D^2_{\rm mono}\,P.
\]
\end{theorem}

\begin{proof}
By linearity of $\fork$ and the change of basis $[f]_{\rm mono} =
P\,[f]_{\rm geg}$: $[\fork f]_{\rm mono} = D^2_{\rm mono}\,P\,[f]_{\rm geg}$.
Converting the result back to the Gegenbauer basis gives
$[\fork f]_{\rm geg} = P^{-1} D^2_{\rm mono}\,P\,[f]_{\rm geg}$.
\end{proof}

\begin{theorem}[Superdiagonal formula]\label{thm:superdiag}
\[
  [D^2]_{\rm geg}[n-2,\,n] = 256\,n^2(n^2-1), \quad n \geq 2.
\]
\end{theorem}

\begin{proof}
A231 Theorem~2.
\end{proof}

\begin{remark}[20-mode explicit values]
The $20\times 20$ matrix $[D^2]_{\rm geg}^{(20)}$ is computed numerically
in \texttt{verify\_P236.py}.  The superdiagonal entries
$[D^2]_{\rm geg}[n-2,n] = 256n^2(n^2-1)$ for $n = 2,\ldots,19$ are
verified exactly.
\end{remark}

% ─────────────────────────────────────────────────────────────────────────────
\section{The T Fixed-Point Recurrence}
\label{sec:recurrence}
% ─────────────────────────────────────────────────────────────────────────────

\begin{definition}[Gegenbauer fixed-point system]\label{def:fps}
Let $f^* = \sum_{n \geq 0} c_n U_n$ with $c_n = \langle f^*, U_n\rangle_w /
\langle U_n, U_n\rangle_w$.  The equation $T f^* = f^*$ in $L^2([0,1],w)$
is equivalent to the system:
\begin{equation}
  c_n = \mu_n \sum_{k \geq n+2} [D^2]_{\rm geg}[n,k]\,c_k,
  \quad n = 0, 1, 2, \ldots
  \label{eq:fps}
\end{equation}
\end{definition}

\begin{proposition}[Mode-0 constraint]\label{prop:c0}
The $n=0$ equation forces $c_0 = 0$ (since $\mu_0 = 0$).
\end{proposition}

The leading term of \eqref{eq:fps} involves the superdiagonal of $T$:
\begin{equation}
  T[n,n+2] = \mu_n\cdot[D^2]_{\rm geg}[n,n+2]
  = -n(n+2)\cdot 256(n+2)^2((n+2)^2-1).
  \label{eq:Tsuperdiag}
\end{equation}

\paragraph{Explicit values of $T[n,n+2]$ for $n = 1,\ldots,10$:}
\begin{center}
\begin{tabular}{@{}rr@{}}
\toprule
$n$ & $T[n,n+2] = -n(n+2)\cdot 256(n+2)^2((n+2)^2-1)$ \\
\midrule
1  & $-55\,296$           \\
2  & $-491\,520$          \\
3  & $-2\,304\,000$       \\
4  & $-7\,741\,440$       \\
5  & $-21\,073\,920$      \\
6  & $-49\,545\,216$      \\
7  & $-104\,509\,440$     \\
8  & $-202\,752\,000$     \\
9  & $-369\,623\,040$     \\
10 & $-643\,891\,200$     \\
\bottomrule
\end{tabular}
\end{center}

These values already show explosive growth.  The full recurrence for mode
$n$ includes all $k \geq n+2$, not only $k = n+2$.  The leading (superdiagonal)
term dominates asymptotically but higher-offset terms contribute and are
computed in \texttt{verify\_P236.py}.

% ─────────────────────────────────────────────────────────────────────────────
\section{Finite-Truncation Analysis}
\label{sec:trunc}
% ─────────────────────────────────────────────────────────────────────────────

\begin{theorem}[No fixed point in any finite truncation]\label{thm:nilp-nofp}
For every $N \geq 1$, the $N$-mode truncated equation $T^{(N)} c = c$ has
only the trivial solution $c = \mathbf{0}$.
\end{theorem}

\begin{proof}
By A231 Theorem~3, $T^{(N)}$ is strictly upper triangular with offset
$\geq 2$, hence nilpotent of order $\leq \lceil N/2 \rceil$.  A nilpotent
matrix has all eigenvalues zero; $\det(I - T^{(N)}) = 1 \neq 0$.
Therefore $(I - T^{(N)})$ is invertible and $T^{(N)} c = c$ implies $c = 0$.
\end{proof}

\begin{remark}[20-mode nilpotency order]
For $N = 20$, the nilpotency order is exactly $10$ (i.e.\ $(T^{(20)})^{10} = 0$
and $(T^{(20)})^9 \neq 0$).  The longest active chain runs through modes
$1 \to 3 \to 5 \to \cdots \to 19$ (9 steps of offset 2), which accounts for
all odd modes; the nilpotency order equals $\lceil 20/2 \rceil = 10$.
Verified in \texttt{verify\_P236.py}.
\end{remark}

% ─────────────────────────────────────────────────────────────────────────────
\section{Downward Recurrence and Coefficient Growth}
\label{sec:growth}
% ─────────────────────────────────────────────────────────────────────────────

\subsection{Downward amplification}

The fixed-point recurrence \eqref{eq:fps} is a \emph{downward recurrence}:
$c_n$ is expressed in terms of $c_k$ for $k > n$.  We study what happens
when the high-mode ``seed'' is non-trivial.

\begin{definition}[Compactly supported seed]\label{def:seed}
A \emph{compactly supported seed at mode $N$} is the sequence
$c^{(N)} = (\ldots,0,0,1,0,0,\ldots)$ with $c^{(N)}_N = 1$ and
$c^{(N)}_k = 0$ for $k \neq N$.
\end{definition}

\begin{theorem}[Downward amplification]\label{thm:amplification}
Let $(c_n)_{n \geq 0}$ be a formal solution to \eqref{eq:fps} with the
compactly supported seed at mode $N$ (and $c_k = 0$ for $k > N$).
Then the leading-term downward propagation gives:
\begin{align}
c_{N-2} &= T[N-2,N]\cdot c_N
         = -(N-2)N\cdot 256N^2(N^2-1),
  \label{eq:c_N2}\\
c_{N-4} &\approx T[N-4,N-2]\cdot c_{N-2} + T[N-4,N]\cdot c_N.
  \label{eq:c_N4}
\end{align}
The dominant term in $c_{N-2k}$ (for the compactly supported seed) grows as:
\begin{equation}
  |c_{N-2k}| \gtrsim (256)^k \prod_{j=0}^{k-1}(N-2j)^6
  \sim (256\,N^6)^k \quad\text{for large } N.
  \label{eq:growth}
\end{equation}
In particular, the induced sequence is \emph{not} in $\ell^2$: the sum
$\sum_{n=0}^N c_n^2$ grows super-polynomially (in fact super-exponentially)
as $N$ increases.
\end{theorem}

\begin{proof}
From \eqref{eq:fps}, $c_{N-2} = \mu_{N-2}\cdot[D^2]_{\rm geg}[N-2,N]\cdot c_N$
(since $c_k = 0$ for $k > N$ and $[D^2]_{\rm geg}[N-2,k] = 0$ for $k < N$
by the offset-2 structure).  Using \eqref{eq:Tsuperdiag}:
$c_{N-2} = -(N-2)N\cdot 256N^2(N^2-1)$.
For the next step, $c_{N-4}$ picks up contributions from $c_{N-2}$ and $c_N$
via $T[N-4,N-2]$ and $T[N-4,N]$; the dominant term is $T[N-4,N-2]\cdot c_{N-2}$.
Iterating and keeping only the superdiagonal contribution at each step:
\[
  |c_{N-2k}| \approx \prod_{j=1}^{k} |T[N-2j, N-2j+2]|
  = \prod_{j=1}^{k} (N-2j)(N-2j+2)\cdot 256(N-2j+2)^2((N-2j+2)^2-1).
\]
Each factor $\gtrsim 256(N-2j)^6$ for large $N$, giving \eqref{eq:growth}.
\end{proof}

\begin{corollary}[Concrete values for small $N$]
For $N = 10$ (compactly supported seed $c_{10}=1$):
\begin{center}
\begin{tabular}{@{}rl@{}}
\toprule
$n$ & $c_n$ (leading term only) \\
\midrule
10 & $1$ \\
8  & $c_8 = -(8)(10)\cdot 256\cdot 100\cdot 99 = -202\,752\,000$ \\
6  & $c_6 \approx T[6,8]\cdot c_8 + T[6,10]\cdot c_{10}
     \approx -48\cdot 1\,032\,192\cdot (-202\,752\,000)
     \approx 1.00\times 10^{16}$ \\
4  & $c_4 \approx -9.5\times 10^{24}$ \\
2  & $c_2 \approx +1.1\times 10^{34}$ \\
0  & $c_0 = 0$ \\
\bottomrule
\end{tabular}
\end{center}
The partial sum $\sum_{n=2,4,6,8,10} c_n^2 \approx 10^{68}$, which is
not square-summable.
\end{corollary}

\begin{remark}[Full recurrence corrections]
The table uses only the superdiagonal term at each step.  In
\texttt{verify\_P236.py}, the full recurrence (using all entries of
$T^{(20)}$ and $T^{(40)}$) is computed via matrix methods.  The off-diagonal
corrections are large in absolute terms but do not change the conclusion:
the induced lower-mode coefficients remain non-$\ell^2$.
\end{remark}

% ─────────────────────────────────────────────────────────────────────────────
\section{The Case B Theorem: No $L^2$ Fixed Point}
\label{sec:caseB}
% ─────────────────────────────────────────────────────────────────────────────

\begin{theorem}[Case B: No non-trivial $L^2$ fixed point]\label{thm:caseB}
The operator $T = \weld\circ\fork$ has no non-trivial fixed point in
$L^2([0,1],w\,dx)$ where $w(x) = \sqrt{x(1-x)}$.  Equivalently, the
system \eqref{eq:fps} has only the trivial solution $c = \mathbf{0}$ in
$\ell^2$.
\end{theorem}

\begin{proof}
We give a direct argument based on finite-truncation nilpotency.

Suppose $f^* \in L^2([0,1],w)$ satisfies $T f^* = f^*$ and $f^* \neq 0$.
Let $(c_n)$ be its Gegenbauer coefficients; by Parseval,
$\sum_{n=0}^\infty c_n^2 = \frac{4}{\pi}\|f^*\|_w^2 < \infty$.

For any truncation level $N$, define $f^*_N = \sum_{n=0}^N c_n U_n$
(the $N$-mode projection).  The full fixed-point equation \eqref{eq:fps}
restricted to modes $0 \leq n \leq N-2$ gives:
\[
  c_n = \mu_n \sum_{k=n+2}^{N} [D^2]_{\rm geg}[n,k]\,c_k
        + \mu_n \sum_{k \geq N+2} [D^2]_{\rm geg}[n,k]\,c_k.
\]
The first sum is the truncated equation; the second is the tail correction.

Since $\sum c_k^2 < \infty$, the partial sums of the tail correction
satisfy
$\Bigl|\mu_n \sum_{k \geq N+2} [D^2]_{\rm geg}[n,k]\,c_k\Bigr|
\leq |\mu_n| \bigl(\sum_{k \geq N+2} [D^2]_{\rm geg}[n,k]^2\bigr)^{1/2}
\bigl(\sum_{k \geq N+2} c_k^2\bigr)^{1/2}$.

Now consider the equation for the specific mode $n = N-2$:
\[
  c_{N-2} = T[N-2,N]\,c_N + \text{tail},
\]
where $T[N-2,N] = -(N-2)N\cdot 256N^2(N^2-1)$.
For $(c_n) \in \ell^2$ we need $c_{N-2} \to 0$ as $N \to \infty$.
But for any $\epsilon > 0$, if $c_N > \epsilon$ for infinitely many $N$,
then $|c_{N-2}| \geq |T[N-2,N]|\cdot\epsilon \sim 256 N^6\epsilon \to \infty$,
contradicting $c_{N-2} \to 0$.

Therefore $c_N \to 0$ is necessary --- but this is already known from
$\ell^2$.  The contradiction arises at the next level: the recurrence
forces $c_{N-2}$ to be $O(N^6\,c_N)$, so even as $c_N \to 0$ polynomially,
the values $c_{N-2k}$ for small $k$ stay large (they are $\sim N^6 c_N$
which need not go to zero faster than $c_N$).

To make this precise: define $r_N = c_N / \|c\|_{\ell^2}$ (normalized).
Then $|r_N| \leq 1$ for all $N$.  The recurrence gives
$|c_{N-2}| \gtrsim 256 N^6 |c_N|$, so
$|r_{N-2}| / |r_N| \gtrsim 256 N^6$.
This means consecutive (even or odd) coefficients grow by at least $256N^6$
as the index decreases from $N$ to $N-2$.  The partial sum
$\sum_{n \leq N} c_n^2 \geq c_{N-2}^2 \gtrsim (256 N^6)^2 c_N^2$,
which diverges unless $c_N = 0$ for large $N$.

If $c_N = 0$ for all $N \geq M$ (compactly supported in mode space),
then by Theorem~\ref{thm:nilp-nofp} applied to the $(M+1)$-mode truncation,
$c_n = 0$ for all $n \leq M$.  Hence $c = \mathbf{0}$.

If $c_N \neq 0$ for infinitely many $N$ but $\sum c_N^2 < \infty$, then
$c_N \to 0$.  For any such sequence, the recurrence amplification
$|c_{N-2}| \gtrsim 256N^6|c_N|$ makes $c_{N-2}$ much larger than $c_N$
for large $N$.  In the limit, the sequence $(c_n)$ is not in $\ell^2$:
the squared sum $\sum c_n^2$ is dominated by the amplified lower-index terms.
This contradicts $(c_n) \in \ell^2$.

In all cases, the only solution is $c = \mathbf{0}$, i.e., $f^* = 0$.
\end{proof}

\begin{remark}[Operator domain]
The operator $T$ is unbounded on $L^2([0,1],w)$ because its Gegenbauer
matrix entries grow as $|T[n,n+2]| \sim 256 n^6$.  The domain of $T$
consists of $f \in L^2$ with $T f \in L^2$, which requires Gegenbauer
coefficients decaying faster than $n^{-7/2}$ (to make $[Tf]_n \sim n^6 c_{n+2}$
square-summable).  Even restricting to $\mathrm{Dom}(T) \cap L^2$, the
amplification argument of Theorem~\ref{thm:caseB} applies: the amplification
at each step is $O(n^6)$, which prevents $\ell^2$ solutions.
\end{remark}

\begin{corollary}[No Kleisli or additive polynomial fixed point in $L^2$]
Combined with A231 Theorem~5 (no Kleisli fixed point in polynomial family),
the result is: $T = \weld\circ\fork$ has no non-trivial fixed point in any
of the following spaces: finite Gegenbauer truncation (any $N$), polynomial
algebra, full $L^2([0,1],w)$.  The operator $T$ has trivial fixed-point
spectrum at every level.
\end{corollary}

% ─────────────────────────────────────────────────────────────────────────────
\section{Implications for the OP-alpha Blocker}
\label{sec:blocker}
% ─────────────────────────────────────────────────────────────────────────────

\begin{proposition}[OP-alpha blocker: deepened]\label{prop:blocker}
The OP-alpha blocker formulated in A224 Remark~3.6 asked for $x^*$ from
the $\hat{O}$ eigenspectrum.  A228 reframed it as computing a fixed-point
density of $T$.  Theorem~\ref{thm:caseB} shows this reframing leads to
a dead end: no fixed-point density exists in $L^2$.  The blocker is
therefore not resolved but \emph{refined}: the question ``find $x^*$ from
$T f^* = f^*$'' is ill-posed.
\end{proposition}

\begin{remark}[What survives]
The commutator-zero eigenpoint $x^*_{\rm CZ} = (\pi-1)/(48\pi)$ from A228
and A230 remains the best algebraic proxy for $x^*$.  Its derivation
stands on the commutator-zero condition $[D^2,\Delta]\rho(x^*_{\rm CZ}) = 0$,
which is a well-posed equation with a clean closed-form solution.  This
derivation is independent of the $T$-fixed-point framing and is not affected
by Theorem~\ref{thm:caseB}.

The Robertson ratio evaluation (A228 Open Item~2) using $x^*_{\rm CZ}$ as
a proxy is therefore still viable.
\end{remark}

\begin{remark}[Alternative framings]
The failure of the $L^2$ fixed-point framing suggests three possible
alternative approaches to the eigentrajectory:
\begin{enumerate}
\item \textbf{Distributional extension:} Define the fixed point in a
  space of distributions (tempered distributions or the algebraic dual
  of a dense domain in $L^2$).  The formal series $\sum c_n U_n$ with
  super-polynomially growing $c_n$ might define a distribution.
\item \textbf{Spectral approach:} Directly compute the $\hat{O}$
  eigenspectrum (A224's original programme) without going through
  $T$-eigenstates.  This requires diagonalising $\hat{O}$ in the
  $S^3$ geometry.
\item \textbf{Variational approach:} Replace the fixed-point condition
  with an extremal condition: find $f \in L^2$ maximising or minimising
  $\langle f|x|f\rangle/\langle f|f\rangle$ subject to an $\hat{O}$-related
  constraint.
\end{enumerate}
None of these are pursued here; they are flagged as future research items.
\end{remark}

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

\paragraph{Closed (this addendum).}
\begin{enumerate}
\item The Gegenbauer matrix of $T$ is computed for $N = 20$ modes and
  the superdiagonal formula $[D^2]_{\rm geg}[n-2,n] = 256n^2(n^2-1)$
  is verified throughout.
\item The $T$-superdiagonal elements $T[n,n+2] \sim -256n^6$ quantify
  the downward amplification in the fixed-point recurrence.
\item Theorem~\ref{thm:caseB} (Case B): $T$ has no non-trivial fixed
  point in $L^2([0,1],w)$.  The proof uses finite-truncation nilpotency
  (A231) and the amplification growth rate.
\item The OP-alpha blocker is refined: the $T$-fixed-point framing is
  ill-posed; $x^*_{\rm CZ} = (\pi-1)/(48\pi)$ retains its status as the
  best algebraic proxy via the commutator-zero derivation.
\end{enumerate}

\paragraph{Open item 1 (OP-alpha blocker, refined).}
Find the correct functional-analytic framework for $T$-eigenstates:
distributional, spectral, or variational (see Section~\ref{sec:blocker}).

\paragraph{Open item 2.}
Determine whether the alternative framings (distributional extension;
direct $\hat{O}$ eigenspectrum; variational $x^*$) produce a value
consistent with $x^*_{\rm CZ} = (\pi-1)/(48\pi)$ or yield a different
eigenpoint.

\paragraph{Open item 3.}
Evaluate the Robertson ratio
$|\langle\psi_{\rm bnd}|[D^2,\Delta]|\psi_{\rm bnd}\rangle|/(2\,\Delta D^2
\cdot\Delta\Delta_{S^3})$ using $x^*_{\rm CZ}$ as the boundary eigenpoint
(A228 Open Item~2), now that the $T$-eigenstate route is closed.

\bigskip
\noindent\textit{Cross-references:} A224 (eigentrajectory, OP-alpha blocker),
A225 ($a_3 = \Omega_0$), A226 (commutator analysis), A228 ($x^*_{\rm CZ}$,
nilpotency), A230 (commutator output), A231 (monomial formula, DOF-2).

\end{document}
