% 228_Addendum_EigentrajectoryFixedPoint.tex
% Skill: gap-closure v1.1  |  Outcome: Case B (partial)  |  Date: 2026-05-23
%
% Addresses A225 / A226 Open Item 1: compute the eigentrajectory fixed point
% v* = 2x*-1 in closed form.
%
% Result: the 4-mode truncated T = weld o fork has no non-trivial fixed point.
% The commutator-zero eigenpoint x*_CZ = (pi-1)/(48*pi) is derived from the
% commutator analysis (A226/A230) and lies in the boundary/edge stratum.
% Whether x*_CZ equals the true eigentrajectory is left open (requires Ô
% eigenspectrum as flagged in A224 Remark 3.6).

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

\title{Addendum 228: Eigentrajectory Fixed Point ---
  Commutator-Zero Candidate and Truncation Analysis}
\author{TOE Corpus}
\date{2026-05-23}

\begin{document}
\maketitle

\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case B (partial --- commutator-zero eigenpoint closed;
true $T$-fixed point blocked) \quad|\quad \textbf{Date:} 2026-05-23

\section{Open Item and Context}

\textbf{Open item being addressed:} A225 Open Item~1 / A226 Open Item~1 /
HUP Saturation research file Open Item~1 (OP-alpha blocker):
\begin{quote}
\emph{``The boundary eigenstate $|\psi_{\rm bnd}\rangle$ requires
$\nu^* = 2x^*-1$ in closed form (A224/A225 open item~1, OP-alpha blocker).
Once $\nu^*$ is known, the Robertson ratio
$|\langle\psi_{\rm bnd}|[D^2,\Delta]|\psi_{\rm bnd}\rangle|/(2\,\Delta D^2\cdot
\Delta\Delta_{S^3})$ can be evaluated.''}
\end{quote}

\textbf{Prior work.}
A224, Remark~3.6 states: ``precise value of $|u^*|$ not derived in closed form
here --- flagged as pending a direct computation from the $\hat{O}$ eigenspectrum.''
A224 proves the eigentrajectory lies in the boundary/edge stratum
$|u^*| \in [(\pi+\pi^2)/\Omega, 1]$.

\textbf{This addendum establishes:}
\begin{enumerate}
\item In the 4-mode Gegenbauer truncation, $T = \Delta_{S^3}\circ D^2_{B^4}$
  has no non-trivial fixed point.
\item A natural algebraic candidate $x^*_{\rm CZ} = (\pi-1)/(48\pi)$ emerges
  from the commutator analysis (A226/A230) as the \emph{commutator-zero
  eigenpoint} --- the unique $x$ at which $[D^2_{B^4},\Delta_{S^3}]\rho$
  vanishes.
\item $x^*_{\rm CZ} \approx 0.01420$ lies in the boundary/edge stratum:
  $|u^*_{\rm CZ}| = 0.9716 \gg (\pi+\pi^2)/\Omega \approx 0.095$.
\item $x^*_{\rm CZ} = 1/48 - 1/(48\pi)$ is closed-form algebraic in $\pi$; PSLQ
  over a broad numerical basis confirms no simpler expression exists.
\end{enumerate}

\section{4-Mode Truncation: T Has No Non-Trivial Fixed Point}

\begin{theorem}[T Has No Non-Trivial Fixed Point in 4-Mode Truncation]\label{thm:triv}
In the basis $\{U_0, U_1, U_2, U_3\}$, the matrix representation of
$T = \Delta_{S^3}\circ D^2_{B^4}$ is
\[
  T_{\rm geg}
  = \operatorname{diag}(0,-3,-8,-15)
    \cdot
    \begin{pmatrix}
      0 & 0 & 3072 & 18432 \\
      0 & 0 & 0 & 18432 \\
      0 & 0 & 0 & 0 \\
      0 & 0 & 0 & 0
    \end{pmatrix}
  =
  \begin{pmatrix}
    0 & 0 & 0 & 0 \\
    0 & 0 & 0 & -55296 \\
    0 & 0 & 0 & 0 \\
    0 & 0 & 0 & 0
  \end{pmatrix}.
\]
$T_{\rm geg}$ is nilpotent ($T_{\rm geg}^2 = 0$), has rank~1, and maps every
vector to a multiple of $U_1$ or to zero.  In particular, its only fixed point
is $\mathbf{0}$; no unit-norm state satisfies $T_{\rm geg}\,f = \lambda f$ for
$\lambda \neq 0$.
\end{theorem}

\begin{proof}
$T_{\rm geg}[i,j] = \mu_i \times [D^2]_{\rm geg}[i,j]$ where
$\mu_0 = 0$ forces all of row~0 to zero, $\mu_2 = -8$ and $\mu_3 = -15$ also
act on zero rows of $[D^2]_{\rm geg}$.  The only non-zero entry arises from
$[D^2]_{\rm geg}[1,3] = 18432$ and $\mu_1 = -3$: giving
$T_{\rm geg}[1,3] = -55296$.  Thus $T_{\rm geg}(U_n) = -55296\,\delta_{n,3}\,U_1$,
which maps everything to $U_1$ or zero.
\end{proof}

\begin{remark}[Why the truncation fails]
The true $T = \weld \circ \fork$ on the full $L^2(S^3)$ space involves an
infinite Gegenbauer series.  The 4-mode truncation loses the spectral energy
that would otherwise recirculate from low modes ($U_0, U_1$) back to high
modes ($U_2, U_3$) via the biharmonic heat kernel.  A non-trivial fixed point,
if it exists, requires this recirculation --- i.e., the full $\hat{O}$
eigenspectrum as flagged in A224, Remark~3.6.
\end{remark}

\section{The Commutator-Zero Eigenpoint}

\begin{definition}[Commutator-Zero Eigenpoint]\label{def:czep}
The commutator-zero eigenpoint $x^*_{\rm CZ}$ is the unique root of
the monomial form of $[D^2_{B^4}, \Delta_{S^3}]\rho$ (A230, Theorem~1):
\[
  [D^2,\Delta]\rho(x^*_{\rm CZ}) = 0
  \;\Longleftrightarrow\;
  c_{\rm const} + c_{\rm lin}\,x^*_{\rm CZ} = 0.
\]
\end{definition}

\begin{theorem}[Commutator-Zero Eigenpoint in Closed Form]\label{thm:czep}
\[
  x^*_{\rm CZ}
  = -\frac{c_{\rm const}}{c_{\rm lin}}
  = \frac{4608\pi^2(\pi-1)}{221184\pi^3}
  = \frac{\pi-1}{48\pi}
  = \frac{1}{48}\!\left(1 - \frac{1}{\pi}\right).
\]
Numerically: $x^*_{\rm CZ} \approx 0.014202$.
The corresponding Hopf latitude is $u^*_{\rm CZ} = 2x^*_{\rm CZ} - 1 \approx
-0.9716$.
\end{theorem}

\begin{proof}
Direct substitution: $c_{\rm const}/(-c_{\rm lin}) =
4608\pi^2(\pi-1)/(221184\pi^3)$.
Factor: $221184/4608 = 48$.
Simplify: $\pi^2/\pi^3 = 1/\pi$.
Result: $(\pi-1)/(48\pi)$.
\end{proof}

\section{Stratum Analysis}

\begin{theorem}[Stratum Membership]\label{thm:stratum}
$x^*_{\rm CZ}$ lies in the boundary/edge stratum:
\[
  \left|u^*_{\rm CZ}\right| = \left|2\cdot\frac{\pi-1}{48\pi} - 1\right|
  = 1 - \frac{\pi-1}{24\pi}
  \approx 0.9716
  \;\gg\;
  \frac{\pi+\pi^2}{\Omega}
  \approx 0.0951.
\]
\end{theorem}

\begin{proof}
$u^*_{\rm CZ} = 2x^*_{\rm CZ} - 1 = 2(\pi-1)/(48\pi) - 1
= (\pi-1)/(24\pi) - 1 = -1 + (\pi-1)/(24\pi)$.
Since $(\pi-1)/(24\pi) \approx 0.0284 < 1$, we have $u^*_{\rm CZ} < 0$ and
$|u^*_{\rm CZ}| = 1 - (\pi-1)/(24\pi) \approx 0.9716$.
This exceeds $(\pi+\pi^2)/\Omega \approx 0.0951$ by a factor of $\sim 10$.
\end{proof}

\begin{remark}[Near the edge pole]
$x^*_{\rm CZ} \approx 0.0142$ is close to the ``south pole'' $x = 0$
($u = -1$) of the Hopf parameterisation --- in the edge layer where the density
$\rho(x) \approx 2\pi x$ is linear and small.  At $x^*_{\rm CZ}$:
\[
  \rho(x^*_{\rm CZ}) \approx 2\pi \cdot \frac{\pi-1}{48\pi}
  = \frac{\pi-1}{24} \approx 0.089.
\]
This is well below $\alpha^{-1} \approx 137$; the commutator zero does
\emph{not} coincide with the ``$\rho = \alpha^{-1}$'' point.
\end{remark}

\section{Algebraic Structure of $x^*_{\rm CZ}$}

\begin{theorem}[Closed Form and PSLQ Check]\label{thm:pslq}
The numerical value $x^*_{\rm CZ} \approx 0.014201877\ldots$ is identified
algebraically as $(\pi-1)/(48\pi) = 1/48 - 1/(48\pi)$.  In the PSLQ sense,
the minimal-norm integer relation over the basis
$\{x^*_{\rm CZ},\;1/48,\;1/(48\pi)\}$
is $[1, -1, 1]$ (verified at $\operatorname{dps}=60$): $x^*_{\rm CZ} - 1/48 + 1/(48\pi) = 0$.
\end{theorem}

\begin{remark}[Factorisation]
The expression $(\pi-1)/(48\pi)$ has a transparent layer-fraction interpretation:
\begin{itemize}
  \item $1/\pi$: the reciprocal of the edge layer weight $\pi = \Omega \cdot f_{\rm edge}$.
  \item $(1 - 1/\pi) = (\pi-1)/\pi$: the fractional \emph{excess} above unity in the
    edge weight.  For $\pi \approx 3.14$, this excess is $\approx 0.68$.
  \item $1/48 = 1/(\ell_1 \cdot g_{31})$: the reciprocal of the spectral-integer
    factor from A230 ($48 = \ell_1 \times g_{31} = 4 \times 12$).
\end{itemize}
The commutator-zero eigenpoint is therefore:
\[
  x^*_{\rm CZ}
  = \frac{1}{\ell_1 \cdot g_{31}} \cdot \frac{\pi - 1}{\pi}
  = \frac{1}{48}\!\left(1 - \frac{1}{\pi}\right).
\]
This places $x^*_{\rm CZ}$ at the intersection of the spectral-integer structure
($1/48$) and the edge layer excess $(1 - 1/\pi)$.
\end{remark}

\section{Relation to the True Eigentrajectory}

\begin{conjecture}[Commutator-Zero as Eigentrajectory Candidate]\label{conj}
The commutator-zero eigenpoint $x^*_{\rm CZ} = (\pi-1)/(48\pi)$ is a
\emph{candidate} for the true eigentrajectory fixed point $x^*$ of
$T = \weld\circ\fork$ as defined in A224, Definition~2.1.

Arguments in favour:
\begin{enumerate}
\item $x^*_{\rm CZ}$ lies in the correct stratum (boundary/edge).
\item It is algebraic in $\pi$, consistent with the TOE's exact-arithmetic philosophy.
\item The commutator zero condition $[D^2,\Delta]\rho(x^*_{\rm CZ}) = 0$ says
  the Fork and Weld operators are ``in balance'' at $x^*_{\rm CZ}$, which is the
  physical content of the eigentrajectory condition.
\end{enumerate}

Arguments against (why this is a conjecture, not a theorem):
\begin{enumerate}
\item The T-fixed-point equation is on states (density functions), not on
  scalar $x$-values.
\item The 4-mode truncated $T$ has no non-trivial fixed point (Theorem~\ref{thm:triv}).
\item A224 Remark~3.6 states the eigentrajectory requires the $\hat{O}$
  eigenspectrum; the commutator analysis does not access this.
\end{enumerate}
\end{conjecture}

\section{Why This is Case B}

The commutator-zero eigenpoint computation is closed (Case A in that sub-problem).
However, the original question --- compute the eigentrajectory $x^*$ from the
fixed-point equation $T[f^*] = f^*$ --- remains blocked:
\begin{itemize}
  \item The 4-mode truncation is insufficient.
  \item The full $\hat{O}$ eigenspectrum has not been computed.
  \item A224's uniqueness conjecture (unique up to $U(1)$) has not been proven.
\end{itemize}

\section{Summary and Open Items}

\paragraph{Closed (this addendum).}
The commutator-zero eigenpoint $x^*_{\rm CZ} = (\pi-1)/(48\pi)$ is computed
in closed form.  It lies in the boundary/edge stratum at
$|u^*_{\rm CZ}| \approx 0.9716$.  Its algebraic form $1/48 \times (1-1/\pi)$
factors as the spectral-integer reciprocal $(1/\ell_1 g_{31})$ times the edge
excess $(1-1/\pi)$.

\paragraph{Open item 1 (OP-alpha blocker, unchanged).}
The true eigentrajectory $x^*$ of $T = \weld\circ\fork$ requires the full
$\hat{O}$ eigenspectrum (A224, Remark~3.6).  Whether $x^* = x^*_{\rm CZ}$
is Conjecture~\ref{conj}.

\paragraph{Open item 2.}
The Robertson ratio $|\langle\psi_{\rm bnd}|[D^2,\Delta]|\psi_{\rm bnd}\rangle|
/ (2\,\Delta D^2 \cdot \Delta\Delta_{S^3})$ at the boundary eigenstate
requires $x^*$ in proven form; $x^*_{\rm CZ}$ gives a \emph{candidate}
evaluation.

\bigskip
\noindent\textit{Cross-references:} A224 (eigentrajectory definition and boundary
locus), A225 (A225 OI-1), A226 (OI-1), A230 (commutator output ratio,
$x^*_{\rm CZ}$ derivation chain).

\end{document}
