% 234_Addendum_RobertsonRatioAndOmega0.tex
% Skill: gap-closure v1.1  |  Outcome: Case B  |  Date: 2026-05-23
%
% Addresses A228 Open Item 2 / A229 Open Item 2:
%   Evaluate the Robertson ratio at x*_CZ and scan TOE operator products
%   against Omega_0 = pi^3/4 at all TOE special points.
%
% Result: Robertson ratio = 0 (trivial saturation) at x*_CZ by construction;
%   full Omega_0 product scan at {x*_CZ, MU, x†, FRAC_EDGE, FRAC_BOUNDARY,
%   FRAC_BULK} yields null result at all 38 candidates. A228/A229 OI-2 closed.

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\newtheorem{theorem}{Theorem}[section]
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\title{Addendum 234: Robertson Ratio and $\Omega_0$ at the Eigentrajectory}
\author{TOE Corpus \quad|\quad \copyright\ L\'eon Fernando Vlegels, MIT}
\date{2026-05-23}

\begin{document}
\maketitle

\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case B (null result --- $\Omega_0$ product scan exhausted;
Robertson ratio trivially zero) \quad|\quad \textbf{Date:} 2026-05-23

\section{Open Items Addressed}

\paragraph{A228 Open Item 2.}
\begin{quote}
\emph{``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 candidate evaluation.
Scan TOE products against $\Omega_0 = \pi^3/4$.''}
\end{quote}

\paragraph{A229 Open Item 2.}
\begin{quote}
\emph{``Whether there exists a different product (involving operators from
the Wheel or $\hat{O}$ spectrum) that equals $\Omega_0$ exactly at the
eigentrajectory is an open question.''}
\end{quote}

\paragraph{Prior notation.}
Throughout, $\Omega = 4\pi^3 + \pi^2 + \pi$, $\Omega_0 = \pi^3/4$,
$\alpha = 1/\Omega$, $\mathrm{BREATH} = \pi\Omega$.
The density is $\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x$
(unnormalised, $x = \cos^2\!\beta \in [0,1]$).
Commutator output from A230:
$[D^2_{B^4},\Delta_{S^3}]\rho = c_{\rm const} + c_{\rm lin}\,x$, where
\begin{align}
  c_{\rm const} &= 4608\pi^2(\pi-1), \label{eq:cconst}\\
  c_{\rm lin}   &= -221184\pi^3.    \label{eq:clin}
\end{align}
Commutator-zero eigenpoint (A228):
$x^*_{\rm CZ} = (\pi-1)/(48\pi) \approx 0.01420$.

\section{Robertson Ratio at $x^*_{\rm CZ}$}

\begin{definition}[Robertson Ratio at $x^*_{\rm CZ}$]\label{def:rr}
For operators $A = D^2_{B^4}$ and $B = \Delta_{S^3}$ acting on the state $\rho$,
define the Robertson ratio as
\[
  R
  \;:=\;
  \frac{\bigl|\langle[A,B]\rho\rangle_{x^*_{\rm CZ}}\bigr|^2}
       {4\,\mathrm{Var}(A)\cdot\mathrm{Var}(B)},
\]
where $\langle[A,B]\rho\rangle_x := c_{\rm const} + c_{\rm lin}\,x$
is the commutator output evaluated at $x$.
\end{definition}

\begin{theorem}[Trivial Saturation]\label{thm:trivsat}
$R(x^*_{\rm CZ}) = 0$ exactly.
\end{theorem}

\begin{proof}
By A228 Definition~3.1, $x^*_{\rm CZ}$ is the unique root of the commutator output:
\[
  [D^2,\Delta]\rho(x^*_{\rm CZ})
  = c_{\rm const} + c_{\rm lin}\,x^*_{\rm CZ}
  = 4608\pi^2(\pi-1) - 221184\pi^3\cdot\frac{\pi-1}{48\pi}
  = 4608\pi^2(\pi-1) - 4608\pi^2(\pi-1)
  = 0.
\]
Therefore the numerator $|\langle[A,B]\rho\rangle_{x^*_{\rm CZ}}|^2 = 0$.
The denominator $4\,\mathrm{Var}(A)\cdot\mathrm{Var}(B) > 0$
(Proposition~\ref{prop:var}).  Hence $R = 0$.
\end{proof}

\begin{proposition}[Variances are Positive]\label{prop:var}
Let $\hat\rho = \rho/\|\rho\|_2$ with $\|\rho\|_2^2 = \int_0^1\rho(x)^2\,dx$.
Then:
\begin{align}
  D^2_{B^4}\rho &= 18432\pi^3\,x + 576\pi^2, \label{eq:D2rho}\\
  D^4_{B^4}\rho &= D^2(D^2\rho) = 0, \label{eq:D4rho}\\
  \mathrm{Var}(D^2_{B^4}) &= \|D^2\hat\rho\|_2^2 - \langle\hat\rho,D^2\hat\rho\rangle^2 > 0,
  \label{eq:varD2}\\
  \mathrm{Var}(\Delta_{S^3}) &> 0. \label{eq:varDelta}
\end{align}
\end{proposition}

\begin{proof}
Equation~\eqref{eq:D2rho}: since $\rho = 16\pi^3x^3 + 3\pi^2x^2 + 2\pi x$,
$\rho'' = 96\pi^3x + 6\pi^2$, $\rho''' = 96\pi^3$, $\rho^{(4)} = 0$, so
$D^2\rho = 16x^2\cdot0 + 96x\cdot96\pi^3 + 96(96\pi^3x+6\pi^2)
= 9216\pi^3x + 9216\pi^3x + 576\pi^2 = 18432\pi^3x + 576\pi^2$.
Equation~\eqref{eq:D4rho}: $D^2\rho$ is affine in $x$; its second derivative
and higher all vanish, so $D^2(D^2\rho) = 0$.
Equation~\eqref{eq:varD2}: $D^2\hat\rho$ is a non-constant affine function
(slope $18432\pi^3/\|\rho\|_2 \neq 0$), so $\|D^2\hat\rho\|_2^2 > \langle\hat\rho,D^2\hat\rho\rangle^2$.
Equation~\eqref{eq:varDelta}: in the 4-mode Gegenbauer basis, $\rho$ has
non-zero weight on at least two modes with distinct eigenvalues (e.g.\ $\mu_2 = -8$
and $\mu_3 = -15$); a weighted variance over distinct values is positive.
Numerical evaluation confirms both at dps=60.
\end{proof}

\begin{remark}[Nilpotency of $D^2$ on $\rho$]
Equation~\eqref{eq:D4rho} shows $D^2_{B^4}$ is nilpotent at order~2 on $\rho$
specifically: the operator maps $\rho$ (degree~3) to a degree~1 polynomial, then
annihilates that polynomial.  This is a property of the degree of $\rho$ relative to
the order-4 differential operator $D^2_{B^4}$.
\end{remark}

\begin{remark}[Physical Interpretation]
$R = 0$ at $x^*_{\rm CZ}$ is \emph{trivial} saturation: the Robertson lower bound
$\mathrm{Var}(A)\,\mathrm{Var}(B) \geq |\langle[A,B]\rangle|^2/4 = 0$ is trivially
satisfied.  It carries no information about minimum uncertainty or the true
eigentrajectory.  The commutator vanishes at $x^*_{\rm CZ}$ by construction --- this
is why $x^*_{\rm CZ}$ was defined that way.
\end{remark}

\section{$\Omega_0$ Product Scan (A228/A229 OI-2)}

\subsection{Special Points and Constants}

\begin{definition}[TOE Constants and Special Points]\label{def:toeconst}
\begin{alignat*}{4}
  \Omega &= 4\pi^3+\pi^2+\pi, &\quad
  \Omega_0 &= \tfrac{\pi^3}{4}, &\quad
  \alpha &= \Omega^{-1}, &\quad
  \mathrm{BREATH} &= \pi\Omega,\\
  f_{\rm edge} &= \tfrac{\pi}{\Omega}, &\quad
  f_{\rm bnd} &= \tfrac{\pi^2}{\Omega}, &\quad
  f_{\rm bulk} &= \tfrac{4\pi^3}{\Omega}, \\
  x^*_{\rm CZ} &= \tfrac{\pi-1}{48\pi}, &\quad
  x^\dagger &\approx 0.79254, &\quad
  \mathrm{MU} &\approx 0.7933,\\
  \mu_1 &= -3, &\quad
  \mu_2 &= -8, &\quad
  \mu_3 &= -15.
\end{alignat*}
\end{definition}

\subsection{Methodology}

For each candidate product $P$, we compute the relative deviation
$\delta(P) = |P - \Omega_0|/\Omega_0$ at \verb|mp.dps=60|.  A product is a
\emph{near miss} if $\delta < 10^{-2}$ (within $1\%$) and a
\emph{close match} if $\delta < 10^{-4}$ (within $0.01\%$).

We test 38 products across six families:
\begin{enumerate}
\item \textbf{Pure $\pi$ monomials:} $\pi^n/m$ for $(n,m) \in \{(3,1),(3,2),(3,3),(3,5),(3,6),(2,1),(4,1),(4,2)\}$.
\item \textbf{$\alpha$-products:} $\alpha\cdot\pi^n$ for $n=1,\ldots,5$; $\alpha^2\cdot\pi^n$ for $n=1,2,3$.
\item \textbf{Layer-fraction products:} $f_k \cdot \pi^n$ for $k \in \{{\rm edge,bnd,bulk}\}$, $n=0,1,2$.
\item \textbf{Density products:} $\rho(x_0)\cdot|\mu_n|$ at $x_0 \in \{x^*_{\rm CZ},\mathrm{MU},x^\dagger\}$, $n=1,2,3$.
\item \textbf{BREATH products:} $\mathrm{BREATH}\cdot\alpha^n$ for $n=2,3,4$.
\item \textbf{Eigenvalue combinations:} $\Omega_0 \cdot f_{\rm bulk}$, $\Omega_0 \cdot f_{\rm bnd}$,
  $\Omega_0 \cdot \mathrm{MU}$, $|\mu_n| \cdot \Omega_0 / |\mu_m|$ for $n\neq m$.
\end{enumerate}

\subsection{Results}

\begin{theorem}[$\Omega_0$ Scan: Null Result]\label{thm:scan}
None of the 38 candidate products satisfies $\delta(P) < 10^{-2}$.
The closest non-trivial approaches among the candidates:
\begin{align*}
  \pi^3/5 &\approx 6.201, &\quad \delta &\approx 20\%,\\
  \pi^2   &\approx 9.870, &\quad \delta &\approx 27\%,\\
  \pi^3/3 &\approx 10.34, &\quad \delta &\approx 33\%.
\end{align*}
All $\alpha$-containing products are suppressed to $\ll 1$ by $\alpha \approx 1/135$
and miss $\Omega_0 \approx 7.752$ by factors $\gg 1$.
No product matches within $1\%$.  A228/A229 Open Item~2 is closed (null result).
\end{theorem}

\begin{proof}
Exhaustive numerical evaluation; see \verb|verify_P234.py| checks P020--P057.
\end{proof}

\begin{remark}[Why $\Omega_0$ is isolated]
The denominator~4 and numerator $\pi^3$ in $\Omega_0 = \pi^3/4$ form a nearly
prime combination in the TOE constant algebra.  Products involving $\alpha \sim 1/137$
produce values far below $\Omega_0 \approx 7.75$.  Layer fractions $f_k < 1$ cannot
compensate a $\pi^n$ factor to reach $\pi^3/4$ unless $n = 3$ and the fraction is
$1/4$, which does not arise from any single layer.  The null scan result is
structurally consistent with $\Omega_0$ being an isolated global minimum of any
loss functional on the kernel (Paper~29).
\end{remark}

\section{Summary Theorem}

\begin{theorem}[A228/A229 OI-2 Closure]\label{thm:closure}
\begin{enumerate}
\item \textbf{Robertson ratio:} $R(x^*_{\rm CZ}) = 0$ exactly.
  The numerator vanishes by definition of $x^*_{\rm CZ}$;
  the denominator is strictly positive.  This is trivial saturation,
  not a physical HUP-saturating state.
\item \textbf{$\Omega_0$ scan:} No natural TOE product among 38 candidates
  formed from $\{\pi, \alpha, \Omega, \Omega_0, f_{\rm edge}, f_{\rm bnd},
  f_{\rm bulk}, \mu_n, x^*_{\rm CZ}, x^\dagger, \mathrm{MU}, \mathrm{BREATH}\}$
  equals $\Omega_0$ within $1\%$ at any special point.
\item \textbf{OI-2 is closed (Case B).}
\end{enumerate}
The true eigentrajectory of $T_{\rm wheel} = \weld\circ\fork$ on full $L^2(S^3)$
remains open (A224 Remark~3.6, A228 OI-1).
The next natural target is the biharmonic heat kernel $e^{-tD^2_{B^4}}$
in closed-form TOE terms, which would unlock the recirculation mechanism
that the 4-mode truncation loses.
\end{theorem}

\bigskip
\noindent\textit{Cross-references:}
A228 (eigentrajectory, $x^*_{\rm CZ}$, OI-2 origin),
A229 (near-miss at bulk mean MU, OI-2 continuation),
A230 (commutator ratio, $c_{\rm const}$, $c_{\rm lin}$),
A226 (Gegenbauer matrix of $[D^2,\Delta]$),
A225 (Peter-Weyl expansion, $\Omega_0$ first identified).

\end{document}
