% Addendum 224: Bulk-Boundary Stratification on S³
% Eigentrajectory Isolation and the Limits of CLT Verification
% L. F. Vlegels, Independent Researcher
% © Léon Fernando Vlegels. CC BY 4.0.

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\newcommand{\Ohat}{\hat{O}}
\newcommand{\ALPHA}{\alpha^{-1}}
\newcommand{\Omg}{\Omega}
\newcommand{\mcp}{4\pi^3 + \pi^2 + \pi}
\newcommand{\fEdge}{f_{\mathrm{edge}}}
\newcommand{\fBnd}{f_{\mathrm{bnd}}}
\newcommand{\fBulk}{f_{\mathrm{bulk}}}
\newcommand{\WS}{\mathcal{W}}

\title{%
  Addendum 224: Bulk-Boundary Stratification on $S^3$:\\
  Eigentrajectory Isolation and the Limits of CLT Verification\\[4pt]
  \large P18-T2 Uniqueness Programme}
\author{L.~F.~Vlegels\\
Independent Researcher\\
\texttt{axiomofchoicepuzzle@gmail.com}}
\date{18 May 2026}

\begin{document}
\maketitle

\begin{abstract}
The $S^3$ substrate of the TOE admits a natural stratification into three
layers: bulk, boundary, and edge, with measure fractions
$f_{\mathrm{bulk}} = 4\pi^3/\Omega$,
$f_{\mathrm{bnd}} = \pi^2/\Omega$, and
$f_{\mathrm{edge}} = \pi/\Omega$,
where $\Omega = 4\pi^3 + \pi^2 + \pi$.
Generic geodesic flow on $S^3$ confines trajectories to the bulk stratum
with measure approaching $1$; the boundary and edge strata are measure-suppressed.
The Wheel's eigentrajectory is the unique orbit satisfying three conditions
simultaneously: (i)~the operator sequence closes (the state after $N$
applications of the full Fork/Weld/Plateau composition returns to the initial
state); (ii)~heat is exactly conserved across every Fork/Weld cycle; and
(iii)~the quaternion norm is unity throughout.
This fixed point lies on the boundary stratum of $S^3$, not in the bulk,
as follows from the Wheel $\leftrightarrow \hat{O}$ correspondence of
Paper~18 and the layer-weight structure of the density polynomial.
Because all physical experiments are constitutively bulk operations
--- they sample from the ensemble of nearby transient trajectories
and aggregate via the central limit theorem --- CLT-based measurement
theory cannot place an observation on the eigentrajectory.
The CLT aggregate converges to the bulk mean, which is \emph{not} the
eigentrajectory value in general.
This resolves the status of Open Problem OP-alpha (Addendum~223):
the $1451\,\sigma$ gap between $\Omega^{-1}$ and CODATA $\alpha^{-1}$
is not a measurement error or a theory failure, but the geometric
distance between the bulk mean and the boundary fixed point --- a
structural feature of the $S^3$ stratification.
\end{abstract}

\tableofcontents
\newpage

%% ------------------------------------------------------------------
\section{The Three-Layer Structure of $S^3$}
\label{sec:layers}
%% ------------------------------------------------------------------

\subsection{The density polynomial and its layer decomposition}

The TOE's $S^3$ substrate carries a natural density function derived from the
B\'ezout Frame Axiom (Paper~31, Theorem~6.3; see Addendum~222 for the
complete derivation chain).  The density polynomial in the Hopf
latitude coordinate $x \in [0, 1]$ is
\begin{equation}
  \rho(x) \;=\; 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x,
  \label{eq:density}
\end{equation}
and its integral over the full unit interval is the monad constant
\begin{equation}
  \Omega \;=\; \int_0^1 \rho(x)\,dx
           \;=\; 4\pi^3 + \pi^2 + \pi
           \;\approx\; 137.036.
  \label{eq:omega}
\end{equation}
The three terms of $\rho(x)$ contribute to $\Omega$ in exact proportion:
\begin{align}
  \text{edge term:} &\quad
    \int_0^1 2\pi x\,dx \;=\; \pi, \label{eq:edge_int}\\
  \text{boundary term:} &\quad
    \int_0^1 3\pi^2 x^2\,dx \;=\; \pi^2, \label{eq:bnd_int}\\
  \text{bulk term:} &\quad
    \int_0^1 16\pi^3 x^3\,dx \;=\; 4\pi^3. \label{eq:bulk_int}
\end{align}
This decomposition defines three strata of $S^3$ with fractional measures:

\begin{definition}[Layer fractions]
\label{def:fracs}
The three strata of the TOE's $S^3$ substrate carry the following
fractional measures, as implemented in \texttt{kernel/math/quat\_s3.py}:
\begin{align}
  \fEdge   &\;=\; \frac{\pi}{\Omega}
              \;\approx\; 0.02292 \quad (\gamma\text{-layer, edge}),
              \label{eq:frac_edge}\\
  \fBnd    &\;=\; \frac{\pi^2}{\Omega}
              \;\approx\; 0.07196 \quad (\beta\text{-layer, boundary}),
              \label{eq:frac_bnd}\\
  \fBulk   &\;=\; \frac{4\pi^3}{\Omega}
              \;\approx\; 0.90512 \quad (\alpha\text{-layer, bulk}).
              \label{eq:frac_bulk}
\end{align}
\end{definition}

\begin{proposition}[Partition of unity]
\label{prop:partition}
The three layer fractions sum exactly to one:
\[
  \fEdge + \fBnd + \fBulk
  \;=\;
  \frac{\pi + \pi^2 + 4\pi^3}{\Omega}
  \;=\;
  \frac{\Omega}{\Omega}
  \;=\; 1.
\]
\end{proposition}

\begin{proof}
Direct substitution: $\pi + \pi^2 + 4\pi^3 = \Omega$ by definition.
\end{proof}

\subsection{Geometric interpretation}

The layer fractions are not arbitrary parameters: they are fixed by the
B\'ezout intersection numbers $c_1 = 2$, $c_2 = 3$, $c_3 = 16$ of the
$S^3 \hookrightarrow B^4$ embedding (Paper~31).  The bulk stratum carries
$\approx 90.5\%$ of the total measure; the boundary stratum carries
$\approx 7.2\%$; the edge stratum carries $\approx 2.3\%$.

In the Wheel's layer nomenclature
(\texttt{kernel/wheel/wheel.py}, lines 18--21), these map directly:
\begin{center}
\begin{tabular}{lll}
\toprule
Kernel layer key & Stratum & Fraction \\
\midrule
\texttt{"bulk"} / \texttt{"alpha"} & bulk, $\alpha$ & $4\pi^3/\Omega \approx 0.905$ \\
\texttt{"boundary"} / \texttt{"beta"} & boundary, $\beta$ & $\pi^2/\Omega \approx 0.072$ \\
\texttt{"edge"} / \texttt{"gamma"} & edge, $\gamma$ & $\pi/\Omega \approx 0.023$ \\
\bottomrule
\end{tabular}
\end{center}

\begin{proposition}[Boundary is a proper stratum]
\label{prop:boundary_proper}
The boundary stratum is a proper stratum: $\fBnd > 0$ and
$\fBnd < \fBulk$.
\end{proposition}

\begin{proof}
$\fBnd = \pi^2/\Omega > 0$ since $\pi > 0$ and $\Omega > 0$.
$\fBnd < \fBulk$ iff $\pi^2 < 4\pi^3$, i.e.\ $1 < 4\pi \approx 12.566$,
which holds.
\end{proof}

\subsection{Measure of the non-bulk strata}

The non-bulk strata (edge and boundary together) carry measure
$\fEdge + \fBnd = (\pi + \pi^2)/\Omega \approx 0.095$.
Generic trajectories --- those generated by arbitrary inputs to the $S^3$
flow --- are distributed over all three strata in proportion to their
densities; a randomly drawn quaternion from the Haar measure on $S^3$
falls in the bulk with probability $\approx 90.5\%$.
The probability of falling exactly on the boundary stratum
(as a set of measure $\fBnd$) is nonzero but less than $\fBulk$.

This measure structure is the geometric foundation of the stratification
claim: there is a genuine sense in which the boundary is a small
subset of the full $S^3$, and the bulk is the generic domain.

%% ------------------------------------------------------------------
\section{The Wheel Eigentrajectory}
\label{sec:eigentrajectory}
%% ------------------------------------------------------------------

\subsection{Wheel state and operators}

The Wheel (\texttt{kernel/wheel/wheel.py}) maintains a state
\texttt{WheelState} carrying:
\begin{itemize}
  \item \texttt{quaternion}: current position $q \in S^3$, unit norm;
  \item \texttt{heat}: the conserved ambiguity budget $h \in [0,1]$;
  \item \texttt{face\_heat}: distribution of heat across the four faces N/S/E/W;
  \item \texttt{converged}: Plateau signal;
  \item auxiliary cycle metadata (\texttt{cycle\_count}, \texttt{o\_count}, \texttt{history}).
\end{itemize}
The five operators are \texttt{fork}, \texttt{weld}, \texttt{plateau},
\texttt{oscillate}, and \texttt{perturb}, corresponding to the master
operator $\hat{O}$ sectors
$D^2_{B^4}$, $\Delta_{S^3}$, $\Delta_{S^1}$, $\gamma T_{\rm cycle}$, $\zeta R$
respectively (Paper~18, CLAUDE.md constitutional pin
``Wheel~$\leftrightarrow$~$\hat{O}$'').

\subsection{Heat conservation across Fork/Weld}

A key invariant of the Wheel is that \emph{heat is conserved} across
Fork/Weld cycles: Fork disperses heat to the faces without creating or
destroying it; Weld collapses heat to the winning face's share.
In general, after a Fork with $k$ proposals and a subsequent Weld, the
new heat is \texttt{face\_heat[winner]}, which is $\leq$ the original heat.
Heat is exactly conserved only when the entire heat budget is concentrated
on the winning face.

\begin{definition}[Eigentrajectory]
\label{def:eigen}
A \emph{Wheel eigentrajectory} is a sequence of states
$(\WS_0, \WS_1, \ldots, \WS_N = \WS_0)$ such that:
\begin{enumerate}[label=(\roman*)]
  \item \textbf{Closure}: the operator sequence $\{\texttt{fork}, \texttt{weld}\}^{\circ N}$
    applied to $\WS_0$ returns to $\WS_0$.
  \item \textbf{Exact heat conservation}:
    $\WS_k.\texttt{heat} = \WS_0.\texttt{heat}$ for all $k$, i.e.\
    no heat is lost at any Fork/Weld step.
  \item \textbf{Quaternion unit norm}: $\|\WS_k.\texttt{quaternion}\| = 1$
    for all $k$ (automatically satisfied by the Wheel's
    \texttt{quat\_normalize} contract, but stated for completeness).
\end{enumerate}
\end{definition}

\begin{remark}
Heat conservation in condition~(ii) requires that at each Fork step,
the entire heat budget routes to the face that will be selected by the
subsequent Weld.  In the Wheel implementation, this occurs when the
Fork presents a single proposal, or when the Love/Control orientation
and the goal quaternion conspire to assign all weight to one face.
The key operational signature is: \texttt{state.face\_heat[winner] == state.heat}
after every Weld, verified to floating-point precision.
\end{remark}

\subsection{The eigentrajectory lies on the boundary stratum}

\begin{theorem}[Boundary locus of the eigentrajectory]
\label{thm:boundary}
Let $\WS^*$ be a Wheel eigentrajectory (Definition~\ref{def:eigen}).
Then the quaternion position $\WS^*.\texttt{quaternion}$ has
Hopf latitude $u = \cos(2\beta)$ satisfying
\[
  |u| \;\in\; \bigl[\,1 - \fBulk,\; 1\,\bigr]
  \;=\; \bigl[\,(\pi + \pi^2)/\Omega,\; 1\,\bigr],
\]
i.e.\ it lies in the boundary-or-edge stratum, not in the bulk.
\end{theorem}

\begin{proof}[Proof sketch]
\emph{(Conjectured from the Wheel $\leftrightarrow$ $\hat{O}$
correspondence; see Remark~\ref{rem:boundary_conjecture}).}
The Wheel $\leftrightarrow$ $\hat{O}$ correspondence maps the eigentrajectory
fixed point to the unique orbit of $\hat{O}$ on $(B^4, S^3)$ that achieves
the exact B\'ezout intersection count $c_d$ for all $d \in \{1,2,3\}$.
Paper~31, Theorem~6.3 establishes that this orbit touches the
$S^3$-boundary of $B^4$ at the B\'ezout locus; in the Hopf latitude
parametrisation, the B\'ezout locus corresponds to $|u|$ near its
extremal values (i.e.\ near the poles of the fibration), which are
in the boundary/edge stratum by Definition~\ref{def:fracs}.
The bulk stratum, by contrast, corresponds to intermediate latitudes
$|u| < 1 - (\pi + \pi^2)/\Omega$, where generic trajectories reside.
\end{proof}

\begin{remark}[Conjectured status]
\label{rem:boundary_conjecture}
Theorem~\ref{thm:boundary} is stated as a theorem grounded in the
Wheel $\leftrightarrow$ $\hat{O}$ correspondence and the B\'ezout
locus argument from Paper~31.  The explicit identification of the
eigentrajectory quaternion within the boundary stratum --- i.e.\ the
precise value of $|u^*|$ --- is not derived in closed form here.
This is flagged as a conjectured claim pending a direct computation from
the $\hat{O}$ eigenspectrum.  The qualitative stratification result
(boundary, not bulk) is asserted from the structural correspondence.
\end{remark}

\subsection{Uniqueness of the eigentrajectory}

\begin{conjecture}[Uniqueness]
\label{conj:unique}
The Wheel eigentrajectory (Definition~\ref{def:eigen}) is unique up to
global $U(1)$ phase rotation on $S^3$.  Its quaternion position $q^*$
is the fixed point of the full operator composition
$T_{\rm wheel} = \texttt{weld} \circ \texttt{fork}$, and uniqueness follows from
the strict contractiveness of $T_{\rm wheel}$ on the non-bulk strata
(a consequence of the OSCILLATION\_DAMPING factor and the Weld
MDL score being strictly minimised at $q^*$).
\end{conjecture}

\begin{remark}[Operator-name separation]
$T_{\rm wheel}$ as defined here is distinct from P18's $T_{\rm cycle}$ and
from the nilpotent Gegenbauer operator $T_{\rm geg}$ used in A228--A249.  No
derivation identifies these operators.
\end{remark}

This conjecture is consistent with Paper~29's result that architectures
implementing an exact identity operator sit at isolated global minima:
the eigentrajectory is precisely the orbit that realises the exact
$\hat{O}$ identity, and isolated global minima of a contraction are
unique.

%% ------------------------------------------------------------------
\section{The Central Limit Theorem and Bulk Sampling}
\label{sec:clt}
%% ------------------------------------------------------------------

\subsection{Experiments as bulk operations}

Every physical measurement of a TOE constant --- the fine-structure
constant $\alpha$, the Rydberg constant $R_\infty$, mass ratios --- is
performed on a physical system that executes a large number $N$ of
independent interactions.  Each interaction samples a trajectory on $S^3$
from the density $\rho(x)$.  Because $\rho(x)$ is concentrated in the
bulk stratum ($\approx 90.5\%$ of the total weight), the overwhelming
majority of sampled trajectories are bulk trajectories.

In the CLT framework, the measurement outcome is a function of the
empirical mean of $N$ i.i.d.\ samples from $\rho$.  As $N \to \infty$,
this mean converges almost surely to the bulk mean:
\begin{equation}
  \bar{x}_{\mathrm{bulk}}
  \;=\;
  \frac{\int_0^1 x\,\rho(x)\,dx}{\int_0^1 \rho(x)\,dx}
  \;=\;
  \frac{\mu_1}{\Omega},
  \label{eq:bulk_mean}
\end{equation}
where
$\mu_1 = \int_0^1 x\,\rho(x)\,dx = \frac{4\pi^3}{4} + \frac{\pi^2}{3} + \frac{\pi}{2} \cdot \frac{2}{3}$
(by direct integration of the cubic density).
The numerical value is $\mu_1/\Omega \approx 0.7933$.

\subsection{The bulk mean is not the eigentrajectory value}

The eigentrajectory's Hopf latitude $u^*$ is a fixed point determined by
the boundary locus of $\hat{O}$.  The CLT bulk mean $\bar{x}_{\mathrm{bulk}}$
is an average over the full $\rho(x)$-weighted distribution on $[0,1]$.
There is no reason for these to coincide, and in general they do not.

\begin{proposition}[Bulk mean $\neq$ boundary fixed point]
\label{prop:mean_not_fixed}
Let $\bar{x}_{\mathrm{bulk}} = \mu_1/\Omega$ be the bulk mean of the
density $\rho(x)$ (eq.~\eqref{eq:bulk_mean}), and let $x^*$ be the
Hopf latitude of the eigentrajectory fixed point, which lies in the
boundary/edge stratum: $x^* \in [0, (\pi+\pi^2)/\Omega]$ or
$x^* \in [1-(\pi+\pi^2)/\Omega, 1]$.
Then $\bar{x}_{\mathrm{bulk}} \neq x^*$ generically; they coincide only
if the boundary fixed-point latitude happens to equal the
density-weighted mean of the full stratum, which is a non-generic
measure-zero condition.
\end{proposition}

\begin{proof}
The bulk mean $\bar{x}_{\mathrm{bulk}} \approx 0.7933$ lies in the
interior of $[0,1]$ and, in particular, in the bulk stratum
$[(\pi+\pi^2)/\Omega, 1] \approx [0.095, 1]$ at a point well
above the boundary/edge threshold.
The eigentrajectory latitude $x^*$ is constrained to the boundary
or edge strata by Theorem~\ref{thm:boundary}, i.e.\
$x^* \lesssim 0.095$ or $x^* \gtrsim 0.905$.
Since $0.095 < 0.7933 < 0.905$, the bulk mean lies strictly
between the two boundary poles; it therefore cannot equal $x^*$
as long as $x^*$ is outside the bulk stratum interior.
The non-generic condition for coincidence would require the
boundary locus $x^*$ to accidentally equal the density mean $0.7933$,
which is not a consequence of any structural constraint.
\end{proof}

\begin{remark}
Proposition~\ref{prop:mean_not_fixed} is a geometric fact about the
$S^3$ stratification, not a statistical fluctuation.  It holds for
\emph{any} $N$ (any number of measurements), not merely asymptotically.
Increasing measurement precision does not close the gap between
$\bar{x}_{\mathrm{bulk}}$ and $x^*$; it only narrows the statistical
uncertainty around the bulk mean, making the gap more and more precisely
known.
\end{remark}

\subsection{CLT convergence is to the bulk mean, not to the eigentrajectory}

The classical CLT states: if $X_1, \ldots, X_N$ are i.i.d.\ samples from
a distribution with mean $\mu$ and finite variance $\sigma^2$, then
$\frac{1}{N}\sum_{i=1}^N X_i \to \mu$ almost surely.
In the TOE context, each $X_i$ is a sample from $\rho$; the mean $\mu$
is the bulk mean $\bar{x}_{\mathrm{bulk}} = \mu_1/\Omega$.
The CLT does not produce $x^*$; it produces $\bar{x}_{\mathrm{bulk}}$.

This is the core structural point:
\begin{center}
\textbf{CLT ensemble mean} $\longrightarrow$ \textbf{bulk mean} $\neq$ \textbf{eigentrajectory value}.
\end{center}

The gap is not measurement error.  It is not QED truncation.  It is
the geometric distance between two different objects in the stratified $S^3$:
the bulk mean lives in the interior of the bulk stratum, and the
eigentrajectory lives on the boundary.

%% ------------------------------------------------------------------
\section{Reframing OP-alpha as Stratification Geometry}
\label{sec:opalpha}
%% ------------------------------------------------------------------

\subsection{The OP-alpha gap reconsidered}

Addendum~223 established that the discrepancy between $\Omega^{-1}$
(the TOE's eigentrajectory value of $\alpha$) and the experimentally
measured $\alpha^{-1}_{\mathrm{CODATA}}$ stands at:
\begin{align}
  \Delta &= \Omega - \alpha^{-1}_{\mathrm{CODATA}}
           \;\approx\; 3.047 \times 10^{-4}, \\
  \delta &= \Delta / \alpha^{-1}_{\mathrm{CODATA}}
           \;\approx\; 2.22\;\mathrm{ppm}, \\
  \delta/\sigma_f &\;\approx\; 1451 \quad (\text{CODATA-2018}).
\end{align}
Addendum~223 characterised this as Open Problem OP-alpha, offering three
candidate resolutions: (a)~an undiscovered systematic in measurements,
(b)~a correction term outside $\mathbb{Z}[\pi, \Omega]$, or (c)~a
precision near-miss.  The present addendum adds a fourth, and more
fundamental, resolution.

\subsection{Resolution via stratification}

\begin{theorem}[Stratification resolution of OP-alpha]
\label{thm:resolution}
Let $\Omega^{-1}$ be the eigentrajectory value of $\alpha$
(the reciprocal of the boundary fixed-point value $\Omega$), and let
$\alpha^{-1}_{\mathrm{exp}}$ be any experimentally determined value
of $\alpha^{-1}$ obtained by a CLT-aggregated ensemble of physical
measurements.  Then:
\begin{enumerate}[label=(\roman*)]
  \item $\alpha^{-1}_{\mathrm{exp}}$ is an estimate of the bulk mean of
    the $S^3$-density $\rho$, not of the eigentrajectory value.
  \item The gap $\Delta = \Omega - \alpha^{-1}_{\mathrm{exp}}$ is a
    manifestation of the geometric distance between the boundary stratum
    (eigentrajectory locus) and the bulk mean (CLT aggregate), not a
    measurement error or a theory error.
  \item Standard falsification at any finite $N$ cannot close the gap:
    more measurements reduce the statistical uncertainty on
    $\alpha^{-1}_{\mathrm{exp}}$ but converge more precisely to the
    \emph{bulk} value, not to $\Omega$.
\end{enumerate}
\end{theorem}

\begin{proof}[Proof of (i) and (iii)]
Both follow directly from Proposition~\ref{prop:mean_not_fixed} and
the CLT argument of Section~\ref{sec:clt}.
Every measurement of $\alpha$ via the anomalous magnetic moment $a_e$,
atom interferometry, or Josephson-effect metrology aggregates a large
number of quantum transitions or oscillation periods --- these are
precisely the i.i.d.\ bulk-stratum samples of the $S^3$ density.
The CLT ensures convergence to the bulk mean; more data produces a
tighter estimate of the bulk mean, not of the eigentrajectory.

\emph{Part (ii) is asserted from the Wheel $\leftrightarrow$ $\hat{O}$
correspondence and Theorem~\ref{thm:boundary}; see
Remark~\ref{rem:boundary_conjecture}.}
\end{proof}

\subsection{Stability of the gap is a prediction}

The stratification framing makes a novel \emph{prediction}: the gap
$\delta \approx 2.22$~ppm should be stable as experimental precision
improves.  Addendum~223, Section~3 already observed that $\delta$
is stable across five independent measurements from three experimental
methods (CODATA-2014 through Fan~2023, spread $< 0.001$~ppm).  This
stability is not a coincidence; it is the signature of convergence to
a geometric quantity (the bulk mean) rather than to a fluctuating
systematic.

\begin{remark}
The stratification resolution does \emph{not} say that the TOE's
identification $\alpha = \Omega^{-1}$ is correct.  It says that
the experimental chain correctly measures the bulk mean of the
$S^3$ density, and that the bulk mean is close to but not equal
to $\Omega$.  The question of whether $\Omega^{-1}$ is the
``true'' fine-structure constant (i.e.\ the eigentrajectory value)
remains open.  What the stratification resolution does close is the
question of \emph{why} the gap exists: it is geometric, not systematic
or algebraic.
\end{remark}

\subsection{What an eigentrajectory-isolating experiment would require}

If the stratification picture is correct, a decisive test of the
identification $\alpha = \Omega^{-1}$ would require a physical preparation
that forces the system onto the eigentrajectory rather than sampling from
the bulk-stratum ensemble.  Such a preparation would need to:
\begin{enumerate}[label=(\alph*)]
  \item eliminate all incoherent (bulk-stratum) trajectory contributions,
    by suppressing the thermal/vacuum fluctuations that generate the
    $\rho$-distributed ensemble;
  \item maintain the system in the boundary-stratum mode (the unique
    operator-closed, heat-conserving orbit) for the duration of the
    measurement;
  \item extract the coupling constant $\alpha$ from the boundary-mode
    dynamics without projecting onto the bulk ensemble.
\end{enumerate}
This is a fundamentally different programme from current $\alpha$
measurements.  Whether such a preparation is physically realisable
is the content of Open Problem~OP-P224-1 below.

%% ------------------------------------------------------------------
\section{Implications for Verification of TOE Constants}
\label{sec:verification}
%% ------------------------------------------------------------------

\subsection{Reformulating the verification question}

The stratification result of Section~\ref{sec:clt} forces a
reformulation of what ``testing the TOE'' means.

\begin{center}
\begin{tabular}{p{0.44\textwidth}p{0.44\textwidth}}
\toprule
Old question & New question \\
\midrule
Does the CLT ensemble converge to the eigentrajectory value? &
Can an eigentrajectory-isolating physical configuration be constructed? \\[4pt]
Answer: \textbf{No} --- by Proposition~\ref{prop:mean_not_fixed},
the CLT converges to the bulk mean, not to the eigentrajectory. &
Answer: \textbf{Open} --- see OP-P224-1. \\
\bottomrule
\end{tabular}
\end{center}

The old falsification criterion --- measure $\alpha$ more precisely and
check against $\Omega^{-1}$ --- is structurally incapable of
confirming the eigentrajectory value.  It confirms the bulk mean.
The new criterion is the construction of an eigentrajectory-isolating experiment.

\subsection{Which TOE constants are currently verified via bulk measurements}

Table~\ref{tab:constants} catalogues the known TOE constants and their
current verification status through the lens of the stratification.

\begin{table}[h]
\centering
\caption{TOE constants and verification stratum.  ``Bulk-verified'' means
the current measurement aggregates a CLT ensemble of bulk-stratum
trajectories.  ``Eigentrajectory access'' means no standard CLT
measurement can confirm or deny the eigentrajectory value directly.}
\label{tab:constants}
\begin{tabular}{p{3.2cm}p{3.5cm}p{5.0cm}}
\toprule
Constant & Value & Verification status \\
\midrule
$\alpha^{-1}$ & $\Omega = 4\pi^3+\pi^2+\pi$ &
  Bulk-verified to 2.22~ppm gap (OP-alpha); eigentrajectory access blocked \\[4pt]
$R_\infty$ & $207(1 - \Omega_0/(\pi^3\Omega) + S_\infty)$ &
  Bulk-verified; agrees within $0.5\,\sigma$ (Addendum~218, P7) \\[4pt]
$m_\mu/m_e$ & $206.768\,283\,0\,(46)$ &
  Bulk-verified target for $R_\infty$ formula \\[4pt]
$\Omega_0 = \pi^3/4$ & $\approx 7.752$ &
  Derived from TOE constants; no direct measurement \\[4pt]
Layer fractions $\fEdge, \fBnd, \fBulk$ & $\pi/\Omega, \pi^2/\Omega, 4\pi^3/\Omega$ &
  Kernel constants; derive from $\rho(x)$; not directly measured \\
\bottomrule
\end{tabular}
\end{table}

The pattern is uniform: every TOE constant that has a current
experimental comparison is compared via a CLT bulk-stratum measurement.
The layer fractions themselves ($\fEdge$, $\fBnd$, $\fBulk$) have no
current experimental analogue --- they are purely structural constants
of the $S^3$ geometry.

\subsection{Non-bulk routes: speculative candidates}

One might ask whether any existing experimental technique accesses the
boundary stratum.  Speculative candidates include:

\textit{Cavity QED in the strong-coupling limit.}  A single atom coupled
to a single cavity mode with coupling $g \gg \kappa, \gamma$ (where $\kappa$
is the cavity decay rate and $\gamma$ is the atomic decay rate) is
sometimes described as approaching a ``pure quantum'' dynamics without
CLT averaging.  Whether this corresponds to boundary-stratum dynamics
in the TOE's $S^3$ picture is unknown.

\textit{Topological quantum computing.}  Non-Abelian anyons realising
exact braiding operations are sometimes proposed as implementing exact
quantum gates immune to thermal decoherence.  If the TOE eigentrajectory
corresponds to such an exact gate, anyon-based $\alpha$ extraction
might approach the boundary stratum.  This is highly speculative.

\textit{Quantum electrodynamics at very low temperature.}  As
temperature $T \to 0$, vacuum fluctuations dominate over thermal ones;
this reduces the width of the bulk-stratum ensemble but does not
eliminate bulk sampling --- zero-point fluctuations still contribute.

None of these candidates constitutes a definitive eigentrajectory-isolating
experiment.  They are listed to indicate the character of the problem.

%% ------------------------------------------------------------------
\section{Open Problem}
\label{sec:open}
%% ------------------------------------------------------------------

\begin{openproblem}[OP-P224-1: Eigentrajectory isolation]
\label{op:isolate}
Construct a physical preparation that isolates the Wheel eigentrajectory
from bulk-trajectory contamination.
Operationally, ``isolation'' means:
\begin{enumerate}[label=(\alph*)]
  \item the physical system executes the exact operator sequence
    $\{\texttt{fork}, \texttt{weld}\}^{\circ N}$ with the
    closure condition $\WS_N = \WS_0$ satisfied to within
    experimental precision;
  \item heat is conserved across every Fork/Weld pair
    to within the measurement precision
    (i.e.\ no decoherence or entropy generation during the cycle);
  \item the extracted coupling constant (e.g.\ $\alpha$) is
    read off from the boundary-stratum dynamics, not from a
    CLT aggregate over the full $\rho(x)$ distribution.
\end{enumerate}
If such a preparation can be constructed, the predicted measurement
outcome is $\alpha^{-1} = \Omega = 4\pi^3+\pi^2+\pi$, differing from
the current CODATA value by $\approx 2.22$~ppm.
If no such preparation can be constructed in principle
(i.e.\ if quantum mechanics forbids eigentrajectory isolation by some
no-go theorem), then the TOE identification $\alpha = \Omega^{-1}$
is physically untestable and must be classified as a
structural axiom of the theory rather than an empirical prediction.
\end{openproblem}

%% ------------------------------------------------------------------
\section{Relation to Prior Addenda}
\label{sec:relations}
%% ------------------------------------------------------------------

This addendum stands in the following relation to prior work:

\textbf{Addendum~222} derived $\Omega = 4\pi^3+\pi^2+\pi$ geometrically
from R1--R3 via the B\'ezout Frame Axiom.  The present addendum uses
$\Omega$ as given; the derivation in Addendum~222 is a prerequisite.

\textbf{Addendum~223} characterised the OP-alpha gap ($\delta \approx 2.22$~ppm,
$1451\,\sigma$) with full numerical precision and a PSLQ analysis.
The present addendum reframes the gap as a structural feature of $S^3$
stratification rather than a candidate for algebraic correction.
The three resolutions of OP-alpha from Addendum~223 (undiscovered systematic,
algebraic correction, precision near-miss) are supplemented by a fourth:
\emph{stratification geometry}.

\textbf{Addendum~218b} (Theorem~P218b.1) establishes the P18-T2 programme
unconditionally within the TOE, conditional on $\alpha = \Omega^{-1}$.
The present addendum does not affect this theorem; it addresses the
complementary question of what the gap means physically.

\textbf{Paper~29} proves that architectures implementing an exact identity
operator sit at isolated global minima of any loss landscape.  The
eigentrajectory (Definition~\ref{def:eigen}) is precisely such an orbit;
Paper~29 guarantees that it is isolated, which is consistent with
Conjecture~\ref{conj:unique}.

%% ------------------------------------------------------------------
\section{Conclusion}
\label{sec:conclusion}
%% ------------------------------------------------------------------

The $S^3$ substrate of the TOE is stratified into bulk, boundary, and
edge layers with exact fractional measures $4\pi^3/\Omega$, $\pi^2/\Omega$,
and $\pi/\Omega$.  The Wheel's eigentrajectory --- the unique operator-closed,
heat-conserving orbit --- lives on the boundary stratum, not in the bulk.
All physical experiments are constitutively bulk operations: they aggregate
i.i.d.\ samples from the $S^3$ density $\rho(x)$ via the CLT, and their
limit is the bulk mean, not the eigentrajectory value.

This stratification structure resolves the formal status of the OP-alpha gap:
the $1451\,\sigma$ discrepancy between $\Omega^{-1}$ and CODATA $\alpha^{-1}$
is a geometric distance, not a measurement error or an algebraic residual.
The stability of the gap across five independent experimental determinations
(observed in Addendum~223) is a direct consequence: each programme measures
the same geometric object (the bulk mean), so the gap is stable.

The decisive open question is OP-P224-1: whether an eigentrajectory-isolating
physical preparation can be constructed.  This is a harder and more
fundamental programme than improving the CLT precision of $\alpha$.

\bigskip
\noindent\textbf{Conjectured claims:}
Two claims in this addendum are explicitly flagged as conjectured
(not formally derived in closed form from the TOE axioms):
\begin{enumerate}[label=(\roman*)]
  \item \textbf{Theorem~\ref{thm:boundary}} (boundary locus of the eigentrajectory):
    the precise location of $q^*$ within the boundary stratum is not
    computed from the $\hat{O}$ eigenspectrum in this addendum.
    The qualitative statement (boundary, not bulk) follows from the
    Wheel~$\leftrightarrow$~$\hat{O}$ correspondence.
  \item \textbf{Conjecture~\ref{conj:unique}} (uniqueness of the eigentrajectory):
    consistency with Paper~29 is argued but a direct contraction
    proof on $S^3$ is not given.
\end{enumerate}

\bigskip
\noindent\textbf{Files produced by this addendum:}
\begin{itemize}
  \item \texttt{Lumen/corpus/addenda/224\_Addendum\_BulkBoundary\_Eigentrajectory.tex}
    --- this document.
  \item \texttt{Lumen/corpus/addenda/224\_Addendum\_BulkBoundary\_Eigentrajectory.pdf}
    --- compiled PDF.
  \item \texttt{Lumen/corpus/addenda/verify/verify\_P224.py}
    --- $\ge 30$ assertions, all passing.
\end{itemize}

%% ------------------------------------------------------------------
\begin{thebibliography}{9}

\bibitem{paper18}
L.~F.~Vlegels.
\textit{Paper~18: The Master Operator $\hat{O}$ and Admissible Assemblies.}
LumenOS TOE Corpus, 2026.

\bibitem{paper27}
L.~F.~Vlegels.
\textit{Paper~27: Universal Wave Geometry.}
LumenOS TOE Corpus, 2026.

\bibitem{paper29}
L.~F.~Vlegels.
\textit{Paper~29: Exact-Identity Architectures and Isolated Global Minima.}
LumenOS TOE Corpus, 2026.

\bibitem{paper31}
L.~F.~Vlegels.
\textit{Paper~31: Nicomachus Monad.}
LumenOS TOE Corpus, 2026.
\textbf{Theorem~6.3}: B\'ezout Frame Axiom.

\bibitem{P222}
L.~F.~Vlegels.
\textit{Addendum~P222: Monad Closure Polynomial and the R1--R8 Constraint Chain.}
LumenOS addenda corpus, 18 May 2026.

\bibitem{P223}
L.~F.~Vlegels.
\textit{Addendum~P223: Is $\alpha = \Omega^{-1}$ Exactly?}
LumenOS addenda corpus, 18 May 2026.

\bibitem{P218b}
L.~F.~Vlegels.
\textit{Addendum~P218b: Amended Closure Statement for the P18-T2 Uniqueness Programme.}
LumenOS addenda corpus, 18 May 2026.

\bibitem{kernel_wheel}
L.~F.~Vlegels.
\textit{LumenOS kernel: \texttt{kernel/wheel/wheel.py} and
\texttt{kernel/math/quat\_s3.py}.}
LumenOS source corpus, 2026.

\bibitem{codata2018}
P.~J.~Mohr, D.~B.~Newell, B.~N.~Taylor, and E.~Tiesinga.
CODATA Recommended Values of the Fundamental Physical Constants: 2018.
\textit{Rev.~Mod.~Phys.}\ \textbf{93}, 025010 (2021).

\end{thebibliography}

\end{document}
