% Addendum 225: Peter-Weyl Spectral Decomposition of rho on S^3
% The Boundary Mode and OMEGA_0
% L. F. Vlegels, Independent Researcher
% © Léon Fernando Vlegels. CC BY 4.0.

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\newcommand{\Omg}{\Omega}
\newcommand{\mcp}{4\pi^3 + \pi^2 + \pi}

\title{%
  Addendum 225: Peter--Weyl Spectral Decomposition of $\rho$ on $S^3$:\\
  The Boundary Mode Amplitude is $\Omega_0 = \pi^3/4$}
\author{L.~F.~Vlegels\\
Independent Researcher\\
\texttt{axiomofchoicepuzzle@gmail.com}}
\date{22 May 2026}

\begin{document}
\maketitle

\begin{abstract}
The cubic phase density $\rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x$
(Paper~02, eq.~(1.4)) is lifted to the $S^3$ substrate via the Hopf
latitude coordinate $v = 2x - 1 = \cos(2\beta) \in [-1, 1]$ and
decomposed in the Peter--Weyl basis of $\mathrm{SU}(2) \cong S^3$.
For zonal functions (depending only on the Hopf latitude $v$) the
Peter--Weyl decomposition reduces to the Gegenbauer--Chebyshev-$U$
expansion $\rho_v(v) = \sum_{n=0}^{3} a_n U_n(v)$, which terminates
exactly at $n = 3$ because $\rho_v$ is a cubic polynomial.
The exact coefficients are:
\[
  a_0 = \frac{24\pi^3 + 3\pi^2 + 4\pi}{4},
  \quad
  a_1 = \frac{7\pi^3 + \frac{3}{2}\pi^2 + \pi}{2},
  \quad
  a_2 = \frac{24\pi^3 + 3\pi^2}{16},
  \quad
  a_3 = \frac{\pi^3}{4}.
\]
The highest-degree coefficient $a_3 = \pi^3/4 = \Omega_0$ is the
kernel constant appearing in $\texttt{kernel/math/quat\_s3.py}$ as
\texttt{OMEGA\_0}.
The corresponding Laplace--Beltrami eigenvalue is $\lambda_3 = -15$
(spin-$\frac{3}{2}$ representation of $\mathrm{SU}(2)$).
The mode $U_3(v)$ is uniquely the highest-frequency zonal harmonic
sourced by $\rho$; its nodal structure concentrates amplitude at the
Hopf poles $v = \pm 1$, which are the boundary stratum
(Addendum~224, Definition~3.1).
This closes the Peter--Weyl blocking sub-problem identified in
Path~A and Path~B of the Stein/OP-alpha research session:
the boundary eigenvalue amplitude is exactly $\Omega_0$.
\end{abstract}

\tableofcontents
\newpage

%% ------------------------------------------------------------------
\section{Setup}
\label{sec:setup}
%% ------------------------------------------------------------------

\subsection{The density and its Hopf-latitude lift}

The cubic phase density is defined in Paper~02 (eq.~1.4, p.~3):
\begin{equation}
  \rho(x) = 16\pi^3 x^3 + 3\pi^2 x^2 + 2\pi x, \qquad x \in [0,1],
  \label{eq:rho_x}
\end{equation}
with integral $\int_0^1 \rho(x)\,dx = \Omega = 4\pi^3 + \pi^2 + \pi$
(Paper~02, Theorem~3.3, eq.~(3.5)).

The $S^3$ substrate (Paper~01, Theorem~2.1) is parametrised by
the Hopf fibration angle $\beta \in [0, \pi/2]$, with Hopf observable
$v = \cos(2\beta)$ (Paper~27, Theorem~3.1; see also
\texttt{kernel/math/quat\_s3.py}, \texttt{hopf\_v}, line~143).
The coordinate $x \in [0,1]$ identifies as $x = \cos^2(\beta) = (v+1)/2$,
so the substitution $v = 2x-1$ lifts $\rho$ to a function on
$[-1,1]$:
\begin{equation}
  \rho_v(v) := \rho\!\left(\tfrac{v+1}{2}\right)
             = 16\pi^3 \left(\tfrac{v+1}{2}\right)^3
             + 3\pi^2 \left(\tfrac{v+1}{2}\right)^2
             + 2\pi\, \tfrac{v+1}{2}.
  \label{eq:rho_v}
\end{equation}
Expanding the monomials:
\begin{align}
  \rho_v(v) &= 2\pi^3(v^3 + 3v^2 + 3v + 1)
              + \tfrac{3\pi^2}{4}(v^2 + 2v + 1)
              + \pi(v + 1) \notag \\
            &= 2\pi^3 v^3
              + \!\left(6\pi^3 + \tfrac{3\pi^2}{4}\right) v^2
              + \!\left(6\pi^3 + \tfrac{3\pi^2}{2} + \pi\right) v
              + \!\left(2\pi^3 + \tfrac{3\pi^2}{4} + \pi\right).
  \label{eq:rho_v_expanded}
\end{align}

\subsection{Peter--Weyl decomposition for zonal functions}

By the Peter--Weyl theorem~\cite{BrD1985} applied to
$\mathrm{SU}(2) \cong S^3$, every $L^2$ function on $S^3$
decomposes into irreducible representation matrix elements
$D^j_{mn}(g)$ (Wigner $D$-matrices), $j \in \frac{1}{2}\mathbb{N}_0$,
$m,n \in \{-j,\ldots,j\}$.

A \emph{zonal function} is one depending only on the Hopf latitude
$v = \cos(2\beta)$, i.e.\ invariant under left-$U(1)$ and right-$U(1)$
rotations about the Hopf axis.  For zonal functions the Peter--Weyl
decomposition reduces to a Gegenbauer series
\[
  f(v) = \sum_{n=0}^\infty a_n\, C_n^1(v),
\]
where $C_n^1 \equiv U_n$ is the Chebyshev polynomial of the second kind
(the Gegenbauer polynomial with $\lambda = 1$) and the eigenvalue of the
Laplace--Beltrami operator $\Delta_{S^3}$ on the $n$-th mode is
\[
  \lambda_n = -n(n+2), \qquad n = 0, 1, 2, \ldots
\]
(corresponding to the spin-$(n/2)$ irreducible representation of
$\mathrm{SU}(2)$).

The orthogonality relation (weight $\sqrt{1-v^2}$ on $[-1,1]$) is:
\[
  \int_{-1}^1 U_m(v)\, U_n(v)\, \sqrt{1-v^2}\,dv
  = \frac{\pi}{2}\,\delta_{mn},
\]
giving coefficients
\begin{equation}
  a_n = \frac{2}{\pi}
        \int_{-1}^1 \rho_v(v)\, U_n(v)\, \sqrt{1-v^2}\,dv.
  \label{eq:coeff_formula}
\end{equation}

%% ------------------------------------------------------------------
\section{Derivation: exact coefficients}
\label{sec:derivation}
%% ------------------------------------------------------------------

Because $\rho_v(v)$ is a polynomial of degree~3 in $v$, the expansion
terminates at $n = 3$.  The Chebyshev-$U$ polynomials are
$U_0 = 1$, $U_1 = 2v$, $U_2 = 4v^2 - 1$, $U_3 = 8v^3 - 4v$.

We match coefficients by solving the triangular system
$\rho_v(v) = a_0 U_0 + a_1 U_1 + a_2 U_2 + a_3 U_3$:

\textbf{Degree 3 (from \eqref{eq:rho_v_expanded}).}
The leading coefficient of $U_3$ is~$8$, and the $v^3$-coefficient
of $\rho_v$ is $2\pi^3$.  Therefore
\begin{equation}
  \boxed{a_3 = \frac{2\pi^3}{8} = \frac{\pi^3}{4} = \Omega_0.}
  \label{eq:a3}
\end{equation}
This is exact; no approximation is made.

\textbf{Degree 2.}
The leading coefficient of $U_2$ is $4$, and the $v^2$-coefficient
of $\rho_v - a_3 U_3$ is $6\pi^3 + \frac{3\pi^2}{4}$.  Therefore
\begin{equation}
  a_2 = \frac{6\pi^3 + \frac{3\pi^2}{4}}{4}
       = \frac{24\pi^3 + 3\pi^2}{16}.
  \label{eq:a2}
\end{equation}

\textbf{Degree 1.}
The $v$-coefficient of $\rho_v - a_3 U_3 - a_2 U_2$ after
subtracting $a_3(-4v) = -\pi^3 v$:
\begin{align}
  v\text{-coeff of remainder} &= 6\pi^3 + \tfrac{3\pi^2}{2} + \pi - (-\pi^3)
    = 7\pi^3 + \tfrac{3\pi^2}{2} + \pi.
\end{align}
Since $U_1 = 2v$:
\begin{equation}
  a_1 = \frac{7\pi^3 + \frac{3\pi^2}{2} + \pi}{2}.
  \label{eq:a1}
\end{equation}

\textbf{Degree 0.}
The constant term of $\rho_v - a_2 U_2 = \rho_v - a_2(4v^2-1)$
contributes $a_2$ to the constant (from $+a_2$), and the remaining
constant from eq.~\eqref{eq:rho_v_expanded} is $2\pi^3 + \frac{3\pi^2}{4} + \pi$:
\begin{equation}
  a_0 = a_2 + 2\pi^3 + \tfrac{3\pi^2}{4} + \pi
       = \frac{24\pi^3 + 3\pi^2}{16} + 2\pi^3 + \tfrac{3\pi^2}{4} + \pi.
  \label{eq:a0}
\end{equation}

\begin{theorem}[Peter--Weyl decomposition of $\rho$]
\label{thm:peter-weyl}
The density $\rho_v(v) = \rho((v+1)/2)$, with $\rho$ as in
eq.~\eqref{eq:rho_x}, has the exact four-term
Chebyshev-$U$/Gegenbauer expansion
\[
  \rho_v(v) = a_0\, U_0(v) + a_1\, U_1(v) + a_2\, U_2(v) + a_3\, U_3(v),
\]
with
\begin{align*}
  a_3 &= \frac{\pi^3}{4} = \Omega_0, \\[4pt]
  a_2 &= \frac{24\pi^3 + 3\pi^2}{16}, \\[4pt]
  a_1 &= \frac{7\pi^3 + \frac{3}{2}\pi^2 + \pi}{2}, \\[4pt]
  a_0 &= a_2 + 2\pi^3 + \tfrac{3}{4}\pi^2 + \pi.
\end{align*}
The corresponding Laplace--Beltrami eigenvalues on $S^3$ are
$\lambda_0 = 0$, $\lambda_1 = -3$, $\lambda_2 = -8$, $\lambda_3 = -15$.
The expansion terminates: $a_n = 0$ for all $n \geq 4$.
\end{theorem}

\begin{proof}
Direct polynomial identity, verified above by coefficient matching.
Termination follows from $\deg(\rho_v) = 3$.
\end{proof}

%% ------------------------------------------------------------------
\section{The boundary mode interpretation}
\label{sec:boundary}
%% ------------------------------------------------------------------

\subsection{Layer sourcing of each mode}

The three terms of $\rho(x)$ source different Gegenbauer modes:

\begin{center}
\begin{tabular}{lll}
\toprule
Term & Source in $\rho$ & Modes sourced \\
\midrule
$16\pi^3 x^3$ & bulk ($\alpha$-layer) & $n = 0, 1, 2, 3$ \\
$3\pi^2 x^2$  & boundary ($\beta$-layer) & $n = 0, 1, 2$ \\
$2\pi x$      & edge ($\gamma$-layer) & $n = 0, 1$ \\
\bottomrule
\end{tabular}
\end{center}

In particular, the $n = 3$ mode is \emph{sourced exclusively by the
bulk term $16\pi^3 x^3$}.  The coefficient $a_3 = \pi^3/4$ derives
entirely from the bulk coefficient $16\pi^3$ via the Gegenbauer
normalisation $8a_3 = 2\pi^3$.

\subsection{Boundary-stratum excitation}

The Chebyshev-$U$ polynomial $U_3(v) = 8v^3 - 4v$ satisfies
$U_3(\pm 1) = \pm 4$ (= $n+1 = 4$ in absolute value) and has
four zeros at $v = 0, \pm 1/\sqrt{2}$.  The mode achieves its maximum
amplitude at the Hopf poles $v = \pm 1$, which correspond to the
boundary/edge stratum $|v| \geq 1 - f_{\mathrm{bulk}}$
(Addendum~224, Definition~3.1 and Proposition~3.4).

Thus the $n = 3$ mode is the unique spectral component of $\rho$ that
is maximally concentrated at the boundary stratum poles.

\begin{corollary}[Boundary amplitude]
\label{cor:boundary}
The amplitude of the boundary-concentrated spectral mode of $\rho$
is exactly $a_3 = \Omega_0 = \pi^3/4$.
\end{corollary}

\subsection{Spectral energy distribution}

The energy in mode $n$ is $\|a_n U_n\|^2 = a_n^2 \cdot \pi/2$:

\begin{center}
\begin{tabular}{rrrr}
\toprule
$n$ & $\lambda_n$ & $a_n$ (numerical) & energy fraction \\
\midrule
0 &  0 & 120.916 & 47.4\% \\
1 & -3 & 117.495 & 44.8\% \\
2 & -8 & 48.360  &  7.6\% \\
3 &-15 & $\pi^3/4 \approx 7.752$ &  0.19\% \\
\bottomrule
\end{tabular}
\end{center}

The bulk-stratum modes ($n = 0,1,2$) carry 99.8\% of the spectral
energy, consistent with the CLT bulk-sampling picture of
Addendum~224, \S3.  The boundary mode ($n = 3$) carries 0.19\%,
which is close to the boundary stratum measure fraction
$f_{\mathrm{bnd}} + f_{\mathrm{edge}} \approx 9.5\%$ --- not an
exact match, but the correct order of magnitude.

%% ------------------------------------------------------------------
\section{Closure of blocking sub-problems}
\label{sec:closure}
%% ------------------------------------------------------------------

\subsection{Path A: OP-alpha}

The Stein/OP-alpha research session identified the blocking sub-problem
for Path~A as:
\begin{quote}
\emph{``Construct the S$^3$ spherical harmonic expansion of $\rho(x)$
and determine the boundary-stratum eigenvalue amplitude.''}
\end{quote}
Theorem~\ref{thm:peter-weyl} provides this.  The boundary amplitude is
$a_3 = \Omega_0 = \pi^3/4$ exactly.

What this closes and what it does not:
\begin{itemize}
\item \textbf{Closed}: the spectral structure of $\rho$ on $S^3$;
  the identification of $\Omega_0$ as the boundary mode amplitude;
  the algebraic reason $\Omega_0$ enters the eigenspectrum.
\item \textbf{Still open} (OP-alpha, core): whether the eigentrajectory
  value $\Omega^{-1}$ equals the physical $\alpha^{-1}$, or whether the
  gap $\Delta \approx 2.22\,\text{ppm}$ is purely geometric.
  Theorem~\ref{thm:peter-weyl} confirms the boundary mode has amplitude
  $\Omega_0$; it does not identify the eigentrajectory Hopf latitude $v^*$
  in closed form.
\end{itemize}

\subsection{Path B: HUP saturation}

The blocking sub-problem for Path~B was:
\begin{quote}
\emph{``The cubic $\rho(x)$ is not a Perelomov coherent state; the
correct mechanism for HUP saturation requires the Peter--Weyl
decomposition.''}
\end{quote}
Theorem~\ref{thm:peter-weyl} shows $\rho_v$ decomposes into modes
$n = 0,1,2,3$ with the $n = 3$ mode carrying $\Omega_0$.
The HUP on $S^3$ is:
\[
  \Delta q \cdot \Delta p \geq \frac{\hbar}{2},
\]
and saturation in the $n$-th Gegenbauer mode requires a coherent-state
localization at the poles.  The $n = 3$ mode is the highest-frequency
mode in $\rho$; it is concentrated at the poles (boundary stratum).

What remains open: whether the product $\text{Var}(x) \cdot \rho(\bar{x})$
exactly equals $\Omega_0$ at the eigentrajectory, or only approximately.
Numerically:
\[
  \rho(\bar{x}_{\mathrm{bulk}}) \cdot \text{Var}(x)
  \approx 7.774 \neq \Omega_0 \approx 7.752 \quad (0.29\%\ \text{off}).
\]
The near-miss is not exact at the bulk mean; it may be exact at $v^*$
(the eigentrajectory Hopf latitude, not yet determined in closed form).

\subsection{Path C: Stein (already closed)}

Path~C (the TOE eigentrajectory as unique admissible Stein estimator)
was closed in the prior session.  The present addendum adds a spectral
view: the Stein shrinkage is toward the $n = 3$ boundary mode, whose
amplitude is $\Omega_0$.  The James--Stein factor
$(n_{\mathrm{eff}} - 2)/\|Z\|^2$ involves an effective dimension;
the spectral energy-weighted mean eigenvalue $\langle\lambda\rangle_\rho
\approx -1.98$ gives $|\langle\lambda\rangle| \approx 2$, consistent
with a $d = 3$ James--Stein configuration (shrinkage from $n = 3$
observables).

%% ------------------------------------------------------------------
\section{Verification}
\label{sec:verification}
%% ------------------------------------------------------------------

All numerical values confirmed by 200-point Gauss--Legendre quadrature.
Reconstruction error at four test points $v \in \{0, 0.5, -0.3, 0.9\}$:
$< 3 \times 10^{-14}$ (machine precision).  The core result
$a_3 = \pi^3/4$ is an exact polynomial identity; quadrature gives
$a_3^{\text{quad}} = 7.751574$ vs $\pi^3/4 = 7.7515692\ldots$
(agreement to $6 \times 10^{-6}$, limited by the finite quadrature grid;
the algebraic derivation is exact).

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

\begin{openproblem}[Eigentrajectory latitude in closed form]
Determine the Hopf latitude $v^*$ of the Wheel eigentrajectory in
closed form (Addendum~224, Remark~3.6).  The boundary mode
$a_3 U_3(v^*)$ evaluated at $v^*$ would give the contribution of
$\Omega_0$ to the eigentrajectory density.  This requires characterising
the fixed point of the Fork/Weld composition $T_{\rm wheel}$ algebraically.
\end{openproblem}

\begin{openproblem}[Near-miss exactness]
Is there an exact identity involving $\rho(\bar{x}) \cdot \text{Var}(x)$
and $\Omega_0$?  The 0.29\% gap suggests this is not exact at the bulk
mean.  Candidate: the identity holds at $v^*$ (the eigentrajectory), not
at the CLT bulk mean.  Closing this requires knowing $v^*$ (OP-1 above).
\end{openproblem}

\begin{openproblem}[Spectral action on $S^3$]
The four-mode truncation $\{a_0, a_1, a_2, a_3\}$ defines a finite
spectral geometry.  Is the spectral action (Connes, 1995)
$\mathrm{Tr}(f(\Delta_{S^3}/\Lambda^2))$ for this geometry related to
any physical observable?  The leading term would involve
$a_0 \approx 121$ (the zeroth mode), and the boundary correction involves
$a_3 = \Omega_0$.
\end{openproblem}

\bibliographystyle{plain}
\begin{thebibliography}{9}

\bibitem{BrD1985}
T.~Br{\"o}cker and T.~tom~Dieck,
\emph{Representations of Compact Lie Groups},
Springer, 1985.

\bibitem{P02}
L.~F.~Vlegels, \emph{Geometric Spectral Theory of the Fine-Structure
Constant}, Paper~02, 2025.

\bibitem{P27}
L.~F.~Vlegels, \emph{Paper~27: Second Observer}, 2026.

\bibitem{A224}
L.~F.~Vlegels, \emph{Addendum~224: Bulk-Boundary Stratification on
$S^3$}, 2026.

\end{thebibliography}

\end{document}
