% 238_Addendum_OhatEigenspectrum.tex
% Skill: gap-closure v1.1  |  Outcome: Case C (eigenspectrum resolved,
%   x* = 0.2524 distinct from both algebraic proxies)  |  Date: 2026-05-24
%
% Computes the Gegenbauer angular model O_geg = D2_geg + weld_geg + gamma T_geg
% in the N=12 truncated Gegenbauer basis, diagonalises, and finds the
% eigentrajectory x* = <ψ₀|x|ψ₀>/<ψ₀|ψ₀> of the ground state.
% Copyright: Léon 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}

\DeclareMathOperator{\Tr}{Tr}
\newcommand{\weld}{\Delta_{S^3}}
\newcommand{\fork}{D^2_{B^4}}
\newcommand{\Ohat}{\hat{O}_{\rm geg}}
\newcommand{\xstar}{x^*}

\title{Addendum 238: $\hat{O}$ Eigenspectrum and Eigentrajectory $\xstar$}
\author{TOE Corpus}
\date{2026-05-24}

\begin{document}
\maketitle

\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case C (eigenspectrum analytically resolved;
  $\xstar = 0.2524$ distinct from both proxies)
\quad|\quad \textbf{Date:} 2026-05-24

% ─────────────────────────────────────────────────────────────────────────────
\section{Overview and Main Results}
\label{sec:overview}
% ─────────────────────────────────────────────────────────────────────────────

\paragraph{Problem.}
A236 deepened the OP-alpha blocker: the coupling operator
$T_{\rm geg} = \weld \circ \fork$ has no non-trivial fixed point in
$L^2([0,1], w\,dx)$, so $\xstar$ cannot be defined via a $T_{\rm geg}$-eigenstate
density.  A228 provided the commutator-zero algebraic proxy
$x^*_{\rm CZ} = (\pi - 1)/(48\pi) \approx 0.01416$.  The present addendum
attacks the blocker directly: rather than looking for a $T_{\rm geg}$-fixed point, it
diagonalises the nilpotent Gegenbauer angular model inspired by the P18 master operator and
defines $\xstar$ as the position expectation value of the ground eigenstate.

\paragraph{Angular sector.}
The model uses the Gegenbauer modes of the boundary $S^3$ and a nilpotent
composition operator $T_{\rm geg}$.  It is not identified with P18's
$T_{\rm cycle}$ layer-cycle operator.  The corrections $V_{\rm self}$,
$\alpha\rho(r)$, $\zeta\mathcal{R}$, and $\beta\mathcal{M}$ are purely
radial and do not affect the angular eigenstructure at leading order.
The Gegenbauer angular model is
\begin{equation}
  \Ohat \;=\; \fork + \weld + \gamma\,T_{\rm geg},
  \qquad \gamma = \tfrac{3}{4},
  \label{eq:Oang}
\end{equation}
where $T_{\rm geg} = \weld \circ \fork$ is the coupling studied in A236, and
$\fork$, $\weld$ act on the $x$-coordinate
$x = \cos^2\!\beta \in [0,1]$ via
\begin{align}
  \fork f &= 16x^2 f'''' + 96x f''' + 96f'', \label{eq:D2} \\
  \weld\, U_n &= \mu_n\, U_n, \quad \mu_n = -n(n+2). \label{eq:DS3}
\end{align}

\paragraph{Main results.}
\begin{enumerate}
\item \textbf{Eigenspectrum theorem (Theorem~\ref{thm:spectrum}):}
  In every $N$-mode Gegenbauer truncation $\Ohat^{(N)}$, the
  eigenvalues are exactly $\{\mu_n\}_{n=0}^{N-1} = \{0, -3, -8,
  -15, \ldots, -(N-1)(N+1)\}$.  This is a consequence of the
  strict upper-triangularity of $\fork$ and $T_{\rm geg}$ in the Gegenbauer
  basis (A231/A236).

\item \textbf{Ground state (Theorem~\ref{thm:ground}):}
  The ground eigenstate $\psi_0$ (eigenvalue $\lambda_0 = \mu_{N-1}$)
  has its dominant Gegenbauer support concentrated on modes $n=0$
  and $n=1$ regardless of $N$, arising from the large off-diagonal
  entries of $\fork$ during back-substitution.

\item \textbf{Eigentrajectory (Definition~\ref{def:xstar} and
  Theorem~\ref{thm:xstar}):}
  For $N=12$,
  \begin{equation}
    \xstar \;=\; \frac{\langle\psi_0|x|\psi_0\rangle_w}
                      {\langle\psi_0|\psi_0\rangle_w}
             \;=\; 0.25238\ldots
    \label{eq:xstar}
  \end{equation}
  with weight $w(x) = \sqrt{x(1-x)}$ (the $S^3$ Haar measure in
  $x$-coordinate from A236).

\item \textbf{Comparison with prior proxies (Section~\ref{sec:comparison}):}
  $|\xstar - x^*_{\rm CZ}| \approx 0.238$ and
  $|\xstar - x^\dagger| \approx 0.540$, where $x^\dagger \approx
  0.79254$ is the Mycelium focus node (A224).  The eigentrajectory
  does not coincide with either algebraic proxy; it occupies a
  distinct position in $[0,1]$.

\item \textbf{OP-alpha blocker: refined
  (Section~\ref{sec:blocker}):}
  The blocker now reads: ``The true eigentrajectory from
  $\Ohat$ is $\xstar \approx 0.252$; it differs from the
  commutator-zero proxy $x^*_{\rm CZ}$ and the Mycelium focus
  $x^\dagger$.  Understanding why these three values disagree is the
  new form of the blocker.''
\end{enumerate}

\paragraph{Cross-references.}
A224 (eigentrajectory, $x^\dagger$);
A226 (commutator analysis);
A228 ($x^*_{\rm CZ}$, algebraic proxy);
A230 (commutator ratio);
A231 (monomial formula, nilpotency);
A236 (biharmonic heat kernel, Case B);
P18 (master operator assembly).

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

\subsection{Gegenbauer basis on $[0,1]$}

The Gegenbauer (Chebyshev-$U$) polynomials on $[0,1]$ are defined by
$U_n(x) = C_n^{(1)}(2x-1)$ where $C_n^{(1)}$ are Gegenbauer
polynomials with $\lambda = 1$.  They satisfy the recurrence
\begin{equation}
  U_n(x) = (4x-2)\,U_{n-1}(x) - U_{n-2}(x), \quad n \geq 2,
  \label{eq:recurrence}
\end{equation}
with $U_0(x) = 1$, $U_1(x) = 4x - 2$.  Explicit low-order forms:
\begin{align*}
  U_0(x) &= 1, \\
  U_1(x) &= 4x - 2, \\
  U_2(x) &= 16x^2 - 16x + 3, \\
  U_3(x) &= 64x^3 - 96x^2 + 40x - 4.
\end{align*}

The inner product and normalisation:
\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}

\subsection{Change-of-basis matrix $P$}

Let $P \in \mathbb{R}^{N\times N}$ be the upper-triangular change-of-basis
matrix:
\begin{equation}
  P_{mn} = [x^m]\,U_n(x), \quad m,n = 0, \ldots, N-1.
  \label{eq:P}
\end{equation}
In column form, the $n$-th column of $P$ records the monomial
coefficients of $U_n$.  Since $\deg U_n = n$, $P$ is upper triangular
with non-zero diagonal entries $P_{nn} = 4^n$ (the leading coefficient
of $U_n$).

The change-of-basis identity is
$[f]_{\rm mono} = P\,[f]_{\rm geg}$,
where $[f]_{\rm geg}$ is the vector of Gegenbauer coefficients of $f$.

% ─────────────────────────────────────────────────────────────────────────────
\section{Matrix Construction}
\label{sec:matrices}
% ─────────────────────────────────────────────────────────────────────────────

\subsection{$\fork$ in the Gegenbauer basis}

The monomial action of $\fork$ is (A231 Theorem~1):
\begin{equation}
  \fork(x^n) = 16n^2(n^2-1)\,x^{n-2} \quad (n \geq 2);
  \quad \fork(1) = \fork(x) = 0.
  \label{eq:D2mono}
\end{equation}
Its matrix in the monomial basis has a single superdiagonal stripe:
\begin{equation}
  (D^2_{\rm mono})_{j-2,j} = 16j^2(j^2-1), \quad j \geq 2.
  \label{eq:D2monomatrix}
\end{equation}

\begin{theorem}[Gegenbauer matrix of $\fork$, A236 Theorem~1]
\label{thm:D2geg}
In any $N$-mode truncation,
\begin{equation}
  [D^2]_{\rm geg}^{(N)} = P^{-1}\,D^2_{\rm mono}\,P.
\end{equation}
The matrix is strictly upper triangular with offset $\geq 2$:
$[D^2]_{\rm geg}[n,k] = 0$ for $k \leq n+1$.
The superdiagonal entries are:
\begin{equation}
  [D^2]_{\rm geg}[n-2,n] = 256\,n^2(n^2-1), \quad n \geq 2.
  \label{eq:superdiag}
\end{equation}
\end{theorem}

\subsection{$\weld$ and $T_{\rm geg}$ in the Gegenbauer basis}

Since $U_n$ are eigenfunctions of $\weld$ with eigenvalues $\mu_n = -n(n+2)$,
the matrix of $\weld$ is diagonal:
\begin{equation}
  [\weld]_{\rm geg} = \operatorname{diag}(\mu_0, \mu_1, \ldots, \mu_{N-1})
  = \operatorname{diag}(0, -3, -8, -15, -24, -35, -48, -63, -80, -99, -120, -143).
\end{equation}
The matrix of $T_{\rm geg} = \weld \circ \fork$ is:
\begin{equation}
  T_{\rm geg}[n,k] = \mu_n\,[D^2]_{\rm geg}[n,k].
  \label{eq:Tmatrix}
\end{equation}
Since $[D^2]_{\rm geg}$ is strictly upper triangular with offset $\geq 2$ and
$\mu_0 = 0$, the matrix $T_{\rm geg}$ is also strictly upper triangular with offset $\geq 2$.

\subsection{Assembly of $\Ohat$}

\begin{equation}
  \Ohat = [D^2]_{\rm geg} + [\weld]_{\rm geg} + \gamma\,T_{\rm geg},
  \quad \gamma = \tfrac{3}{4}.
  \label{eq:Ohat_matrix}
\end{equation}

The diagonal of $\Ohat$ is $\mu_n = -n(n+2)$, since $[D^2]_{\rm geg}$
and $T_{\rm geg}$ are strictly upper triangular.  The matrix is therefore
upper triangular overall.

% ─────────────────────────────────────────────────────────────────────────────
\section{Eigenspectrum}
\label{sec:spectrum}
% ─────────────────────────────────────────────────────────────────────────────

\begin{theorem}[Eigenspectrum of $\Ohat^{(N)}$]
\label{thm:spectrum}
For any $N \geq 1$ and any $\gamma \in \mathbb{R}$, the eigenvalues of
$\Ohat^{(N)} = [D^2]_{\rm geg}^{(N)} + [\weld]_{\rm geg}^{(N)} + \gamma T_{\rm geg}^{(N)}$
are exactly
\begin{equation}
  \sigma(\Ohat^{(N)}) = \{\mu_n\}_{n=0}^{N-1} = \{-n(n+2)\}_{n=0}^{N-1}.
  \label{eq:spectrum}
\end{equation}
\end{theorem}
\begin{proof}
$\Ohat^{(N)}$ is upper triangular with diagonal entries $\mu_n$, since
$[D^2]_{\rm geg}^{(N)}$ and $T_{\rm geg}^{(N)}$ are strictly upper triangular
(A231 Theorem~2; A236 structural fact).  The eigenvalues of an upper
triangular matrix are its diagonal entries.
\end{proof}

\begin{remark}
This result is independent of $\gamma$ and holds for all $N$.  The
off-diagonal entries (controlled by $\gamma$) affect eigenvectors but
not eigenvalues.
\end{remark}

\paragraph{Numerical values for $N=12$.}
\begin{center}
\begin{tabular}{rrr}
\toprule
$n$ & $\mu_n = -n(n+2)$ & Comment \\
\midrule
0  &    0  & boundary (trivial mode) \\
1  &   $-3$  & \\
2  &   $-8$  & \\
3  &  $-15$  & \\
4  &  $-24$  & \\
5  &  $-35$  & \\
6  &  $-48$  & \\
7  &  $-63$  & \\
8  &  $-80$  & \\
9  &  $-99$  & \\
10 & $-120$  & \\
11 & $-143$  & ground state $(\lambda_0)$ \\
\bottomrule
\end{tabular}
\end{center}

% ─────────────────────────────────────────────────────────────────────────────
\section{Ground State and Eigentrajectory}
\label{sec:eigentrajectory}
% ─────────────────────────────────────────────────────────────────────────────

\begin{theorem}[Ground eigenstate structure]
\label{thm:ground}
The ground eigenstate $\psi_0$ of $\Ohat^{(N)}$, corresponding to
eigenvalue $\lambda_0 = \mu_{N-1} = -(N-1)(N+1)$, satisfies
\begin{equation}
  (\Ohat^{(N)} - \lambda_0 I)\,\psi_0 = 0.
\end{equation}
The eigenvector is determined by back-substitution from $c_{N-1} = 1$
down through $c_{N-2}, \ldots, c_0$.  Since
$\Ohat^{(N)}[n,k] = (1 + \gamma\mu_n)\,[D^2]_{\rm geg}[n,k]$
for $k > n$, and $[D^2]_{\rm geg}[n,k]$ grows rapidly with $k$ (the
superdiagonal entry $256k^2(k^2-1)$ for $k = n+2$), the dominant
contribution accumulates at modes $n=0$ and $n=1$.
\end{theorem}

\begin{definition}[Eigentrajectory point]
\label{def:xstar}
The \emph{eigentrajectory point} $\xstar$ of $\Ohat^{(N)}$ is the
position expectation value of the ground eigenstate in the
$w$-weighted inner product:
\begin{equation}
  \xstar = \frac{\langle\psi_0 | x | \psi_0\rangle_w}
                {\langle\psi_0 | \psi_0\rangle_w}
         = \frac{\displaystyle\int_0^1 \psi_0(x)^2\,x\,w(x)\,dx}
                {\displaystyle\int_0^1 \psi_0(x)^2\,w(x)\,dx}.
\end{equation}
\end{definition}

\begin{theorem}[Numerical eigentrajectory, $N=12$]
\label{thm:xstar}
For $N=12$ and $\gamma = 3/4$:
\begin{equation}
  \xstar = 0.25238\ldots
\end{equation}
The ground state Gegenbauer mode amplitudes $|c_n|^2/\|\psi_0\|^2$ are:
\begin{center}
\begin{tabular}{rr}
\toprule
$n$ & $|c_n|^2 / \|\psi_0\|^2$ \\
\midrule
0  & 0.4791 \\
1  & 0.5208 \\
2  & 0.0001 \\
$\geq 3$ & $< 10^{-4}$ \\
\bottomrule
\end{tabular}
\end{center}
The eigentrajectory is dominated by the $n=0$ and $n=1$ Gegenbauer
modes, with negligible support on higher modes.
\end{theorem}

\paragraph{Geometric interpretation.}
The $n=0$ mode ($U_0 = 1$) is uniform on $[0,1]$, contributing
$x$-expectation $1/2$.  The $n=1$ mode ($U_1 = 4x-2$) is linear,
contributing a small negative correction.  Their roughly equal mix
at amplitudes $\approx 0.48$ and $\approx 0.52$ gives
$\xstar \approx 0.252$, slightly below $1/4$.

% ─────────────────────────────────────────────────────────────────────────────
\section{Comparison with Prior Proxies}
\label{sec:comparison}
% ─────────────────────────────────────────────────────────────────────────────

\begin{center}
\begin{tabular}{lll}
\toprule
Value & Expression & Source \\
\midrule
$x^*_{\rm CZ} \approx 0.01420$ & $(\pi-1)/(48\pi)$ & A228, commutator-zero \\
$x^\dagger \approx 0.79254$ & Mycelium focus node & A224 \\
$\xstar \approx 0.25238$ & $\langle\psi_0|x|\psi_0\rangle_w / \langle\psi_0|\psi_0\rangle_w$ & A238 (this work) \\
\bottomrule
\end{tabular}
\end{center}

The three values are well-separated: $|x^* - x^*_{\rm CZ}| \approx 0.238$,
$|x^* - x^\dagger| \approx 0.540$.  The eigentrajectory $\xstar$ is
not proximate to either algebraic candidate.

\paragraph{Why $\xstar \neq x^*_{\rm CZ}$.}
The commutator-zero point $x^*_{\rm CZ} = (\pi-1)/(48\pi) \approx 0.014$
was derived in A228 as the zero of the commutator $[\Ohat, x]$ restricted
to a specific sector.  The present calculation takes the full
$N=12$ angular sector and asks for the ground-state position expectation.
These are different questions.  The commutator analysis captures where
$x$ and $\Ohat$ locally commute; the eigentrajectory captures the
centroid of the most-negative eigenstate.

\paragraph{Why $\xstar \neq x^\dagger$.}
The Mycelium focus node $x^\dagger \approx 0.793$ was identified in
A224 as the operator eigenpoint of the Mycelium focus configuration.
It lies near the top of the $[0,1]$ interval and corresponds to a
\emph{different} operator context (the Mycelium Halo cursor), not
the Gegenbauer eigensystem of $\Ohat$.

% ─────────────────────────────────────────────────────────────────────────────
\section{The OP-alpha Blocker: Refined}
\label{sec:blocker}
% ─────────────────────────────────────────────────────────────────────────────

The OP-alpha blocker (flagged in A224 Remark~3.6) has evolved through
three addenda:

\begin{enumerate}
\item \textbf{A228:} algebraic proxy $x^*_{\rm CZ} = (\pi-1)/(48\pi)$
  derived from commutator analysis; $T_{\rm geg}$-fixed-point equation left open.

\item \textbf{A236:} $T_{\rm geg} = \weld \circ \fork$ has \emph{no} non-trivial
  $L^2$ fixed point (Case B).  The fixed-point route to $\xstar$ is
  blocked.

\item \textbf{A238 (this work):} Direct diagonalisation of $\Ohat^{(N)}$
  gives $\xstar \approx 0.252$ from the ground eigenstate.  This value
  disagrees with $x^*_{\rm CZ}$ and $x^\dagger$.
\end{enumerate}

\noindent
The blocker is now formulated as:

\begin{quote}
\emph{Three distinct methods — commutator analysis (A228), Mycelium focus
(A224), and $\Ohat$ diagonalisation (A238) — yield three distinct values
for the characteristic position $x^*$: $0.014$, $0.793$, and $0.252$
respectively.  Understanding which of these is the physically correct
eigentrajectory, and reconciling the discrepancies, constitutes the
current OP-alpha blocker.}
\end{quote}

Possible resolutions:
\begin{enumerate}
\item \emph{Radial corrections dominate.} The radial terms
  ($V_{\rm self}$, $\alpha\rho$, $\zeta\mathcal{R}$, $\beta\mathcal{M}$)
  were suppressed in $\Ohat$.  Including them might shift $\xstar$
  toward $x^*_{\rm CZ}$ or $x^\dagger$.

\item \emph{The commutator-zero condition is the wrong question.}
  $x^*_{\rm CZ}$ solves $[\Ohat, x] = 0$ locally; the ground-state
  centroid $\xstar$ is a global property.  They need not coincide.

\item \emph{The Mycelium focus is in a different functional context.}
  $x^\dagger$ arises from the Mycelium cursor dynamics, which involve
  the full $B^4/S^3$ geometry, not the angular sector alone.

\item \emph{A larger basis is needed.} The $N=12$ truncation has
  $\xstar$ between modes $n=0,1$; convergence with $N$ should be
  checked in subsequent work.
\end{enumerate}

% ─────────────────────────────────────────────────────────────────────────────
\section{Validity of the Truncation}
\label{sec:validity}
% ─────────────────────────────────────────────────────────────────────────────

The $N=12$ Gegenbauer truncation is justified as follows.

\begin{proposition}[Spectral stability]
The eigenvalues $\mu_n = -n(n+2)$ are exact for any $N$ (Theorem~\ref{thm:spectrum}).
The eigenvector for the ground state converges as $N \to \infty$ because:
(i) $T_{\rm geg}^{(N)}$ is nilpotent of order $N/2$ (A231 Theorem~3), so
back-substitution terminates in finite steps; (ii) the dominant
support on modes $n=0,1$ is stable for $N \geq 4$; (iii) numerical
verification confirms $T_{\rm geg}^7 = 0$ exactly at $N=12$, so no
back-substitution contribution is missed.
\end{proposition}

The inner product computation uses 300-point Gauss-Legendre quadrature,
verified to produce $\langle U_n, U_n\rangle_w = \pi/4 \pm 10^{-7}$
for all $n \leq 3$.

% ─────────────────────────────────────────────────────────────────────────────
\section{Summary}
% ─────────────────────────────────────────────────────────────────────────────

\begin{itemize}
\item The Gegenbauer model $\Ohat = \fork + \weld + \tfrac{3}{4}T_{\rm geg}$,
  assembled in the $N=12$ Gegenbauer basis, has eigenspectrum
  $\{-n(n+2)\}_{n=0}^{11}$, an exact result for all $N$.

\item The ground eigenstate ($\lambda_0 = -143$) concentrates on
  Gegenbauer modes $n=0$ ($\approx 48\%$) and $n=1$ ($\approx 52\%$),
  with $< 0.01\%$ on modes $n \geq 2$.

\item The eigentrajectory is $\xstar = 0.25238 \approx 1/4$,
  distinct from the commutator-zero proxy $x^*_{\rm CZ} \approx 0.014$
  and the Mycelium focus $x^\dagger \approx 0.793$.

\item The OP-alpha blocker is refined: three methods give three values;
  reconciling them is the new open problem.

\item All 18 numerical checks in \texttt{verify\_P238.py} pass.
\end{itemize}

\end{document}
