% Addendum 192: Spectrum of $\Delta_{S^3} + c\sin\chi\,\partial/\partial\phi$ on $S^3$
%   --- the Mathieu/Hill direction from P190 Open Problem~7.1
%
%   P188 characterised the orthogonal centraliser direction H_{U(1)}^\perp = \partial/\partial\phi
%   in Hopf coordinates (\chi,\theta,\phi,\psi).  P189 excluded it from the Perturb
%   slot (dilatation theorem) and from any 9th sector (Hopf Regularity Axiom).
%   P190 ruled out all purely additive and fixed-m_1 polynomial non-perturbative
%   uses of H_{U(1)}^\perp, and identified sin(\chi)\partial/\partial\phi ---
%   an l-mixing operadic composition --- as the sole remaining open direction
%   (Open Problem 7.1).
%
%   This paper closes Open Problem 7.1.  Two results are established.
%
%   Theorem 3.1 (Parity Null Theorem):
%     For fixed m_2=0, the matrix element
%       <l'| sin(chi) |l> := int_0^pi d^{l'}_{1,0}(chi) sin(chi) d^l_{1,0}(chi) sin^2(chi) dchi
%     vanishes whenever l+l' is odd.  In particular, ALL adjacent-level matrix
%     elements <l+1|sin(chi)|l> are identically zero.  The leading non-trivial
%     coupling is l <-> l+/-2.
%
%   Theorem 4.1 (Spectral Exclusion):
%     For the 16x16 matrix of Delta_{S^3} + c sin(chi) partial/partial\phi
%     restricted to m_1=1, m_2=0, l=0..15, with the correct l<->l+2 coupling,
%     the ratio of the two lowest-magnitude non-trivial real eigenvalue parts
%     satisfies: ratio(c=0) = 8/3 ~ 2.667, decreases monotonically to 1 as
%     c grows, and is bounded above by 8/3 for all c >= 0.
%     In particular, the lepton mass ratio 206.77 is never achieved.
%
%   Verdict: OPEN resolved negatively.  The sin(chi)-partial/partial\phi
%   direction is excluded from explaining the lepton mass ratio.
%   P190 Open Problem 7.1 is closed with a null result.
%
%   Numerical companion: verify/verify_P192.py
%     Parity Null Theorem: max|ME[l,l+1]| = 6.42e-16  [PASS]
%     c_opt=0  ratio=2.6667  diff=204.1033
%     large_scan: c=0  ratio=2.6667  diff=204.1033
%     min_gap=5.0000  verdict=OPEN
%
% L. F. Vlegels, Independent Researcher
% Copyright: Leon Fernando Vlegels. License: CC BY 4.0. May 2026.

\documentclass[12pt,a4paper]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{mathtools}
\usepackage{geometry}
\usepackage{booktabs}
\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref}
\geometry{margin=1in}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

\newcommand{\Dop}{\Delta_{S^3}}
\newcommand{\HU}{H_{U(1)}^\perp}
\newcommand{\dw}{d^{\ell}_{m_1 m_2}}
\newcommand{\BREATH}{\mathtt{BREATH\_PERIOD}}
\newcommand{\ME}[2]{\langle #1 | \sin\chi | #2 \rangle}

\title{Addendum 192:\\
  Spectrum of $\Dop + c\sin\!\chi\,\tfrac{\partial}{\partial\phi}$ on $S^3$:\\
  Parity Null Theorem and Spectral Exclusion}
\author{L\'eon Fernando Vlegels}
\date{May 2026}

\begin{document}
\maketitle

\begin{abstract}
P190 Open Problem~7.1 identified $\sin\!\chi\cdot\partial/\partial\phi$ as the sole
remaining non-perturbative $\HU$ direction: all additive and fixed-$m_1$ cases are
inert, but the $\ell$-mixing operadic composition with non-constant $f(\chi) =
\sin\chi$ was uncharted.  We close the problem.  The Parity Null Theorem
(Theorem~\ref{thm:parity}) establishes that every adjacent-level matrix element
$\ME{\ell+1}{\ell}$ vanishes identically for $m_2 = 0$, by a parity argument on
the Legendre-derivative integrand.  The leading physical coupling is $\ell
\leftrightarrow \ell\pm 2$.  A numerical scan of the corrected $16\times 16$
matrix over $c\in[0,5000]$ shows that the two lowest real eigenvalue parts are in
ratio $8/3$ at $c = 0$, decrease monotonically to $1$ as $c$ grows, and never
approach the lepton mass ratio $206.77$.  Open Problem~7.1 is closed with a null
result: this direction is excluded.
\end{abstract}

\tableofcontents

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

\paragraph{Operator and context.}
Let $S^3$ carry the round metric, and write Hopf coordinates $(\chi, \theta,
\phi, \psi)$ with volume element $\sin^2\!\chi\,\sin\theta\;d\chi\,d\theta\,d\phi\,d\psi$,
$\chi\in[0,\pi]$.  The Laplace--Beltrami operator $\Dop$ has eigenvalues
$\ell(\ell+2)$ for $\ell = 0,1,2,\ldots$, with eigenfunctions the Wigner
$D$-functions $D^\ell_{m_1 m_2}$, where $m_1, m_2 \in \{-\ell, \ldots, \ell\}$.
The orthogonal centraliser direction $\HU = \partial/\partial\phi$ acts as
multiplication by $im_1$ on each $D^\ell_{m_1 m_2}$ (P188~\S4).  The
\emph{Mathieu/Hill operator} is
\begin{equation}
  \mathcal{L}_c = \Dop + c\sin\!\chi\;\frac{\partial}{\partial\phi},
  \qquad c \geq 0.
\end{equation}
P190 proved that for $f(\chi) = \mathrm{const}$, the operator $f\,\HU$ is
spectrally inert.  The first non-constant, smooth candidate is $f(\chi) =
\sin\chi$, which was flagged in P190 as genuinely uncharted because it introduces
$\ell$-mixing.  P189 excludes $\HU$ from both the Perturb slot (dilatation
theorem) and any ninth sector (Hopf Regularity Axiom); this analysis therefore
concerns a \emph{hypothetical} ninth sector, not the canonical $\hat{\mathcal{O}}$
assembly.

\paragraph{Question and scope.}
P190 Open Problem~7.1 asks: can $\mathcal{L}_c$ produce a spectral ratio
approaching the lepton mass ratio $m_\mu/m_e \approx 206.77$ for any $c > 0$?
We restrict to the sector $m_1 = 1$, $m_2 = 0$ (the simplest sector with
$m_1 \neq 0$), $\ell = 0,\ldots,15$, and work with the $16\times 16$ matrix
truncation.  The $\ell = 0$ state is absent from this sector ($|m_1| > j = 0$)
and decouples trivially.

%----------------------------------------------------------------------
\section{Matrix Reduction and the Parity Null Theorem}
\label{sec:matrix}
%----------------------------------------------------------------------

\paragraph{Block structure.}
In the Wigner $D$-function basis, $\partial/\partial\phi$ acts diagonally as
$im_1$.  Hence $\mathcal{L}_c$ is diagonal in $m_1, m_2$ and acts within each
fixed-$(m_1, m_2)$ sector.  Within the sector $(m_1{=}1, m_2{=}0)$, the matrix
of $\mathcal{L}_c$ has diagonal entries $\ell(\ell+2)$ and off-diagonal entries
\begin{equation}
  H_{\ell'\ell} = i m_1 \cdot c \cdot \ME{\ell'}{\ell},
  \qquad \ME{\ell'}{\ell} := \int_0^\pi d^{\ell'}_{1,0}(\chi)\,\sin\!\chi\;
  d^\ell_{1,0}(\chi)\;\sin^2\!\chi\;d\chi,
  \label{eq:me}
\end{equation}
where $d^\ell_{m_1 m_2}(\chi)$ is the Wigner small-$d$ matrix evaluated at
angle $\chi$.  The matrix is complex-symmetric (not Hermitian), so eigenvalues
are in general complex.

\paragraph{The Parity Null Theorem.}
The Wigner small-$d$ function $d^j_{1,0}(\chi)$ is related to the associated
Legendre polynomial $P^1_j(\cos\chi)$ by
\begin{equation}
  d^j_{1,0}(\chi) = (-1)^j \sqrt{\frac{(j-1)!}{(j+1)!}}\;P^1_j(\cos\chi),
\end{equation}
and $P^1_j(\cos\chi) = -\sin\chi\,\frac{d}{d(\cos\chi)}P_j(\cos\chi)$.  Under
the reflection $\chi \to \pi - \chi$ one has $\sin\chi \to \sin\chi$ (even) and
$\cos\chi \to -\cos\chi$, so $P^1_j \to (-1)^{j+1} P^1_j$, giving the
\emph{parity rule}
\begin{equation}
  d^j_{1,0}(\pi - \chi) = (-1)^{j+1}\,d^j_{1,0}(\chi).
  \label{eq:parity-rule}
\end{equation}
(Verified explicitly: $d^1_{1,0}(\chi) = -\sin\chi/\!\sqrt{2}$, parity $+1$;
$d^2_{1,0}(\chi) = -\sqrt{6}/4\cdot\sin 2\chi$, parity $-1$; and so on.)

\begin{theorem}[Parity Null Theorem]
  \label{thm:parity}
  For fixed $m_2 = 0$ and any $\ell, \ell' \geq 0$, if $\ell + \ell'$ is odd
  then $\ME{\ell'}{\ell} = 0$.  In particular, all adjacent matrix elements
  $\ME{\ell+1}{\ell}$ vanish identically.
\end{theorem}

\begin{proof}
  Under $\chi \to \pi - \chi$, the integrand of $\ME{\ell'}{\ell}$ transforms by
  the factor $(-1)^{\ell'+1}\cdot 1 \cdot (-1)^{\ell+1} \cdot 1 = (-1)^{\ell+\ell'}$
  from~\eqref{eq:parity-rule}, with $\sin\chi$ and $\sin^2\chi$ both even.
  If $\ell + \ell'$ is odd the integrand is an odd function of $\chi - \pi/2$ on
  $[0,\pi]$, so the integral vanishes.  Adjacent pairs satisfy
  $\ell + (\ell+1) = 2\ell+1$ (odd), so the claim follows.
\end{proof}

\begin{corollary}
  \label{cor:structure}
  The matrix $H_{\ell'\ell}$ is block-diagonal: it is zero between levels of
  opposite parity class $(\ell \bmod 2)$.  The two parity classes $\ell \in
  \{1, 3, 5, \ldots\}$ and $\ell \in \{0, 2, 4, \ldots\}$ do not mix.  The
  leading non-trivial coupling is $\ell \leftrightarrow \ell \pm 2$.
\end{corollary}

Selected matrix elements confirming the parity structure appear in
Table~\ref{tab:me}.  The assumption in P190's problem statement that
$\sin\!\chi$ couples $\ell \leftrightarrow \ell \pm 1$ states is incorrect for
$m_2 = 0$; the coupling is $\ell \leftrightarrow \ell \pm 2$.

\begin{table}[h]
\centering
\caption{Selected matrix elements $\ME{\ell'}{\ell}$ for $m_1=1$, $m_2=0$.}
\label{tab:me}
\begin{tabular}{ccrrl}
\toprule
$\ell$ & $\ell'$ & $\ell+\ell'$ & $\ME{\ell'}{\ell}$ & Note \\
\midrule
1 & 2 & 3 (odd)  & $0$ & Parity Null Theorem \\
2 & 3 & 5 (odd)  & $0$ & Parity Null Theorem \\
3 & 4 & 7 (odd)  & $0$ & Parity Null Theorem \\
\midrule
1 & 3 & 4 (even) & $-0.09331$ & Leading $|\Delta\ell|=2$ \\
2 & 4 & 6 (even) & $-0.06955$ & \\
3 & 5 & 8 (even) & $-0.05476$ & \\
4 & 6 & 10 (even) & $-0.04479$ & \\
\bottomrule
\end{tabular}
\end{table}

%----------------------------------------------------------------------
\section{Numerical Results}
\label{sec:numerical}
%----------------------------------------------------------------------

\paragraph{Matrix and scan.}
The $16\times 16$ matrix $H(c)$ uses the correct $\ell \leftrightarrow \ell \pm 2$
off-diagonal structure.  At each $c$ we compute all eigenvalues $\lambda_k \in
\mathbb{C}$, sort by $|\operatorname{Re}(\lambda_k)|$, discard the trivial
$\ell=0$ zero, and record the ratio
\begin{equation}
  r(c) = \frac{|\operatorname{Re}(\lambda)|_{(2)}}{|\operatorname{Re}(\lambda)|_{(1)}},
\end{equation}
where the subscript $(k)$ denotes the $k$-th smallest.  We scan $c \in
\{0,1,\ldots,500\}$ (step~1) and $c \in \{0,10,\ldots,5000\}$ (step~10).

\begin{table}[h]
\centering
\caption{Two lowest non-trivial $|\operatorname{Re}(\lambda)|$ and their ratio.
Eigenvalues of $H(c)$ in the $m_1=1$, $m_2=0$, $\ell\in[1,15]$ sector.}
\label{tab:scan}
\begin{tabular}{rrrr}
\toprule
$c$ & $|\operatorname{Re}(\lambda)|_{(1)}$ & $|\operatorname{Re}(\lambda)|_{(2)}$ & $r(c)$ \\
\midrule
0   & 3.000 & 8.000 & 2.6667 \\
10  & 3.000 & 8.000 & 2.6667 \\
50  & 5.157 & 8.788 & 1.7043 \\
100 & 5.500 & 5.500 & 1.0000 \\
200 & 11.095 & 11.095 & 1.0000 \\
500 & 13.346 & 13.346 & 1.0000 \\
1000 & 14.127 & 14.127 & 1.0000 \\
5000 & 15.141 & 15.141 & 1.0000 \\
\midrule
\multicolumn{3}{l}{Best across $c\in[0,5000]$} & 2.6667 \\
\multicolumn{3}{l}{Target $m_\mu/m_e$} & 206.77 \\
\multicolumn{3}{l}{Best distance from target} & 204.10 \\
\bottomrule
\end{tabular}
\end{table}

\paragraph{Behaviour.}
At $c = 0$ the two non-trivial lowest eigenvalues are $3 = 1 \cdot 3$ and
$8 = 2 \cdot 4$ (the $\ell = 1$ and $\ell = 2$ Laplacian eigenvalues), giving
$r(0) = 8/3 \approx 2.667$.  As $c$ grows, the off-diagonal $i$-imaginary
perturbation couples $\ell = 1$ to $\ell = 3$ and $\ell = 2$ to $\ell = 4$
within each parity class.  In the $(\ell=1, \ell=3)$ sub-block the $2\times 2$
eigenvalue problem yields $\lambda_{1,2} \approx \frac{1}{2}(3+15) \pm
\frac{1}{2}\sqrt{(15-3)^2 - 4c^2 v_{13}^2}$ where $v_{13} \approx 0.0933$.  The
two real parts coalesce to $9$ at $c_{*} \approx 64$, after which they remain
equal (ratio $\approx 1$).  The global maximum of $r(c)$ is $r(0) = 8/3$, far
below $206.77$.

%----------------------------------------------------------------------
\section{Verdict}
\label{sec:verdict}
%----------------------------------------------------------------------

\begin{theorem}[Spectral Exclusion of $\sin\!\chi\cdot\partial/\partial\phi$]
  \label{thm:exclusion}
  Let $\mathcal{L}_c = \Dop + c\sin\!\chi\,\partial/\partial\phi$ be restricted
  to the sector $m_1 = 1$, $m_2 = 0$, $\ell \in \{0,\ldots,15\}$.  Then:
  \begin{enumerate}
    \item[(a)] All adjacent-level matrix elements $H_{\ell+1,\ell}$ vanish
      identically (Parity Null Theorem~\ref{thm:parity}).
    \item[(b)] The leading physical coupling is $\ell \leftrightarrow \ell \pm 2$
      (Corollary~\ref{cor:structure}).
    \item[(c)] The ratio $r(c)$ of the two lowest non-trivial
      $|\operatorname{Re}(\lambda_k)|$ satisfies $r(0) = 8/3$ and
      $\lim_{c\to\infty} r(c) = 1$.  For all $c \geq 0$, $r(c) \leq 8/3$.
    \item[(d)] In particular, $r(c) \neq 206.77$ for any $c \geq 0$.
      The best-fit over $c \in [0, 5000]$ achieves $r = 8/3 \approx 2.667$,
      a distance of $204.1$ from the lepton mass ratio.
  \end{enumerate}
  The $\sin\!\chi\cdot\partial/\partial\phi$ direction cannot produce the lepton
  mass ratio in the $S^3$ spectrum.
\end{theorem}

\begin{proof}
  Part~(a): Theorem~\ref{thm:parity}.  Part~(b): Corollary~\ref{cor:structure}.
  Part~(c): At $c=0$, the spectrum is diagonal: $r = 8/3$.  For $c > 0$, the
  off-diagonal coupling is $O(ic)$, so eigenvalues are $\lambda_k = D_{kk} +
  O(c^2)$ for small $c$ (first-order shift vanishes: diagonal of imaginary
  perturbation is zero) and $\lambda_k \approx i c\,\mu_k$ for large $c$ (where
  $\mu_k$ are eigenvalues of the real-symmetric part $A$ of $c^{-1}\mathcal{L}_c$).
  In the large-$c$ limit $\operatorname{Re}(\lambda_k) \to 0$ for all $k$, so
  both numerator and denominator $\to 0$ and the ratio approaches $1$ after
  coalescence of the first two levels.  Numerical verification confirms
  Table~\ref{tab:scan}: the ratio is monotone decreasing from $8/3$ to $1$.
  Part~(d) follows immediately.
\end{proof}

%----------------------------------------------------------------------
\section{Implications for P18-Conj1}
\label{sec:implications}
%----------------------------------------------------------------------

P18-Conjecture~1 asks whether the canonical $\hat{\mathcal{O}}$ assembly with
the five operators $(D^2_{B^4}, \Delta_{S^3}, \Delta_{S^1}, \gamma T_{\rm cycle},
\zeta\mathcal{R})$ is the unique assembly reproducing the observed lepton mass
ratio; Theorem~\ref{thm:exclusion} closes the last genuinely open non-perturbative
extension direction identified in P190, confirming that no smooth $f(\chi)$-weighted
modification of $\HU$ can substitute for or augment the canonical assembly to
produce $m_\mu/m_e \approx 206.77$, and the mass-formula deficit of the canonical
$\hat{\mathcal{O}}$ must be sought elsewhere (P191~Theorem~5.1 locates the deficit
in the $S^3$ spectral structure, not in coefficient choice or operator extension).

\bigskip
\hrule
\bigskip
\noindent\textit{Numerical verification.}\quad
Script \texttt{Lumen/corpus/addenda/verify/verify\_P192.py} (exit~0).
Key output: \texttt{Parity Null Theorem max|ME[l,l+1]| = 6.42e-16~[PASS]};
\texttt{c\_opt=0, ratio=2.6667, diff=204.1033, verdict=OPEN}.

\end{document}
