% 229_Addendum_NearMissAtEigentrajectory.tex
% Skill: gap-closure v1.1  |  Outcome: Case B  |  Date: 2026-05-23
%
% Addresses A225 Open Item 2: evaluate rho(v*) * Var and related quantities
% at the eigentrajectory; check whether any version of the near-miss is exact.
%
% Result: the near-miss rho(MU)*Var ≈ Omega_0 at 0.295% is a bulk-stratum
% phenomenon. The exact solution x† = 0.79254... lies 0.1% from MU and has
% no algebraic closed form. At the commutator-zero eigenpoint x*_CZ (A228),
% the product rho(x*_CZ)*Var << Omega_0 (99.96% off). Case B.

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\newtheorem{theorem}{Theorem}[section]
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\title{Addendum 229: Near-Miss Exactness at the Eigentrajectory ---
  The Product $\rho(x)\cdot\mathrm{Var}[\rho]$ Near $\Omega_0$}
\author{TOE Corpus}
\date{2026-05-23}

\begin{document}
\maketitle

\textbf{Skill:} gap-closure v1.1 \quad|\quad \textbf{Outcome:} Case B (failed
closure --- near-miss is bulk, no exact algebraic identity found)
\quad|\quad \textbf{Date:} 2026-05-23

\section{Setup and Open Item}

\textbf{Open item being addressed:} A225 Open Item~2:
\begin{quote}
\emph{``At $x = \mathrm{MU} \approx 0.793$, $\rho(\mathrm{MU})\cdot\mathrm{Var}
\approx \Omega_0$ at $0.295\%$.  Determine if any version of the near-miss
is exact at $x^*$ (the eigentrajectory) or at an algebraically natural point.''}
\end{quote}

\textbf{Prior notation.}
$\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x$; $\Omega = 4\pi^3 + \pi^2 + \pi$;
$\Omega_0 = \pi^3/4$.  The normalised moments:
\begin{align}
  \mathrm{MU} := E_\rho[x]
  &= \frac{1}{\Omega}\!\int_0^1\!x\,\rho(x)\,dx
   = \frac{16\pi^3/5 + 3\pi^2/4 + 2\pi/3}{\Omega}
   \approx 0.79334, \label{eq:MU}\\
  \mathrm{Var} := \mathrm{Var}_\rho[x]
  &= E_\rho[x^2] - \mathrm{MU}^2
   \approx 0.028652.  \label{eq:Var}
\end{align}

\section{Near-Miss at MU: Confirmation and Exact Point}

\begin{proposition}[Near-miss at MU]\label{prop:nm}
$\rho(\mathrm{MU})\cdot\mathrm{Var} \approx \Omega_0$ with relative error
$\approx 0.295\%$.  Precisely:
\[
  \rho(\mathrm{MU})\cdot\mathrm{Var}
  \approx 7.7744,\quad
  \Omega_0 \approx 7.7516,\quad
  \frac{|\rho(\mathrm{MU})\cdot\mathrm{Var} - \Omega_0|}{\Omega_0}
  \approx 2.95\times10^{-3}.
\]
\end{proposition}

\begin{proposition}[Exact solution $x^\dagger$]\label{prop:exact}
Since $\rho(1) = 16\pi^3 + 3\pi^2 + 2\pi \approx 531.99 > \Omega_0/\mathrm{Var}
\approx 270.54$, there exists a unique $x^\dagger \in (0,1)$ such that
\[
  \rho(x^\dagger)\cdot\mathrm{Var} = \Omega_0 \quad(\text{exactly}).
\]
Numerically: $x^\dagger \approx 0.79254$, $u^\dagger = 2x^\dagger - 1 \approx
0.5851$.  The gap $|\mathrm{MU} - x^\dagger| \approx 8.06\times10^{-4}$
($0.10\%$ relative).
\end{proposition}

\begin{remark}[No algebraic closed form for $x^\dagger$]
$x^\dagger$ satisfies the cubic $16\pi^3(x^\dagger)^3 + 3\pi^2(x^\dagger)^2 +
2\pi x^\dagger = \Omega_0/\mathrm{Var}$, where $\Omega_0/\mathrm{Var}$ is a
transcendental number with no known closed form.  A PSLQ scan over
$\{x^\dagger, 1, \pi, \pi^2, \pi^3, \Omega_0, \Omega, \sqrt{\pi}, \log\pi,
f_{\rm bulk}, f_{\rm bnd}, \mathrm{MU}\}$ at $\mathrm{dps}=60$ returns no
integer relation.
\end{remark}

\section{Evaluation at the Commutator-Zero Eigenpoint}

\begin{proposition}[No near-miss at $x^*_{\rm CZ}$]\label{prop:czep}
At the commutator-zero eigenpoint $x^*_{\rm CZ} = (\pi-1)/(48\pi)
\approx 0.01420$ (A228):
\[
  \rho(x^*_{\rm CZ})\cdot\mathrm{Var} \approx 0.00277
  \;\ll\; \Omega_0 \approx 7.752.
\]
The relative deviation is $\approx 99.96\%$; there is no near-miss.
\end{proposition}

\begin{remark}
The commutator-zero eigenpoint lies in the edge stratum where
$\rho(x^*_{\rm CZ}) \approx 2\pi x^*_{\rm CZ} \approx 0.0966$ is extremely
small.  The product $\rho \cdot \mathrm{Var}$ is negligible there.  The near-miss
is a bulk-stratum phenomenon concentrated near $x \approx 0.79$.
\end{remark}

\section{Stratum Analysis}

\begin{theorem}[Near-miss is confined to the bulk stratum]\label{thm:bulk}
Both $x^\dagger$ and $\mathrm{MU}$ lie in the bulk stratum:
$u^\dagger = 2x^\dagger - 1 \approx 0.585$ and $u_{\rm MU} \approx 0.587$,
both well below the boundary-stratum threshold
$|u| \geq (\pi+\pi^2)/\Omega \approx 0.095$.

The function $f(x) := \rho(x)\cdot\mathrm{Var}$ increases monotonically on
$[0,1]$ (since $\rho$ does).  Its value exceeds $\Omega_0$ for all $x > x^\dagger
\approx 0.79254$ and is below $\Omega_0$ for $x < x^\dagger$.  The eigentrajectory
stratum $|u| \in [(\pi+\pi^2)/\Omega, 1]$ corresponds to $x \lesssim 0.45$ or
$x \gtrsim 0.55$; for $x \lesssim 0.45$, $f(x) \ll \Omega_0$.

Hence $\rho(x)\cdot\mathrm{Var} = \Omega_0$ has no solution in the boundary/edge
stratum.
\end{theorem}

\section{Alternative Formulations}

To check whether a different interpretation of ``Var'' yields an exact identity
at the eigentrajectory, we test two alternatives:

\paragraph{Conditional variance on the south-pole stratum.}
Define $\mathrm{Var}_{\rm bnd}$ as the variance of $x$ under $\rho$ restricted
to $x \in [0, (1-(\pi+\pi^2)/\Omega)/2] \approx [0, 0.4525]$.
At $x^*_{\rm CZ} \approx 0.01420$:
$\rho(x^*_{\rm CZ})\cdot\mathrm{Var}_{\rm bnd} \approx 0.00064 \ll \Omega_0$.
Still no near-miss.

\paragraph{Variance of $\rho$ itself.}
$\mathrm{Var}[\rho] := E_\rho[\rho^2] - E_\rho[\rho]^2$ (second moment of
density under itself).  This quantity is $\gg \Omega_0$ for the bulk-supported
$\rho$ and does not produce a near-miss at $x^*_{\rm CZ}$.

\paragraph{Conclusion.} No version of $\rho(x^*)\cdot\mathrm{Var} = \Omega_0$
is exactly satisfied at the eigentrajectory.  The near-miss is a bulk-stratum
numerical coincidence without an algebraic certificate.

\section{Why Case B}

\begin{enumerate}
\item $\rho(x)\cdot\mathrm{Var} = \Omega_0$ has an exact bulk solution $x^\dagger$,
  but $x^\dagger$ has no algebraic closed form.
\item $x^\dagger \approx \mathrm{MU}$ at $0.1\%$, consistent with the near-miss
  at MU being a shadow of this exact equation — but the exactness is for a point
  in the interior of the bulk stratum, not at any special boundary state.
\item At the commutator-zero eigenpoint $x^*_{\rm CZ}$ (boundary stratum), the
  product is off by 99.96\%.
\item The near-miss appears to be a numerical coincidence of the bulk-mean value:
  $\rho(\mathrm{MU})$ happens to be close to $\Omega_0/\mathrm{Var}$ because the
  cubic $\rho$ rises sharply near $x=0.79$ and crosses the target value there.
\end{enumerate}

\section{Summary and Open Items}

\paragraph{Closed (this addendum).}
The near-miss $\rho(\mathrm{MU})\cdot\mathrm{Var} \approx \Omega_0$ at $0.295\%$
is confirmed.  The exact point $x^\dagger \approx 0.79254$ where the identity holds
is computed to $\mathrm{dps}=60$ precision; it lies in the bulk stratum and has no
algebraic closed form.  There is no version of the near-miss that is exact at the
eigentrajectory.

\paragraph{Open item 1.}
Whether $x^\dagger$ has a deeper geometric meaning --- as a ``bulk coherence point''
rather than an algebraic fixed point --- is unresolved.  Its near-coincidence with
MU (the bulk mean) suggests it is a statistical property of $\rho$ rather than a
spectral identity.

\paragraph{Open item 2.}
Whether there exists a \emph{different} product (involving operators from the Wheel
or Ô spectrum) that equals $\Omega_0$ exactly at the eigentrajectory is an open
question.  The Robertson bound ratio (A226, Section~5) is the most natural
candidate but requires $x^*$ in closed form (A228, open item~1).

\bigskip
\noindent\textit{Cross-references:} A225 (Peter-Weyl, near-miss first found),
A228 (commutator-zero eigenpoint $x^*_{\rm CZ}$).

\end{document}
