% Addendum 193: Alternative Spectral Operators on S^3
%   Route A of three routes to P18-Conj1 (lepton mass ratio from TOE geometry).
%
%   Paper 18 constructs the master operator O-hat on S^3. The standard Delta_{S^3}
%   has eigenvalues l(l+2). P191 proved no ratio equals m_mu/m_e = 206.768.
%   This addendum checks five alternative spectral operators for R1-R8
%   admissibility and eigenvalue ratios near 206.768.
%
%   Results:
%   - Dirac (squared):   417^2/29^2 = 206.765, gap 0.004 (0.002%) -- best
%   - CR sub-Laplacian:  5789/28 = 827/4 = 206.750, gap 0.018 (0.009%)
%   - Twisted Delta_{k=1}: 827/4 = 206.750, gap 0.018 (same as CR; Hopf geometry)
%   - Conformal/Yamabe:  fully R1-R8; best ratio 209, gap 2.23
%   - Composite tensors: excluded by R6 (P186)
%
%   Verdict: OPEN with tighter constraints. Route A alone is insufficient.
%   Best near-miss: Dirac squared at 0.002%.
%
% 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[margin=1in]{geometry}
\usepackage{booktabs}
\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue]{hyperref}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\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{\Dirac}{\slashed{D}}
\newcommand{\mrat}{m_\mu/m_e}
\newcommand{\mtgt}{206.7682838}

\title{Addendum 193:\\
  Alternative Spectral Operators on $S^3$:\\
  Route~A Survey for the Lepton Mass Ratio}
\author{L\'eon Fernando Vlegels}
\date{May 2026}

\begin{document}
\maketitle

\begin{abstract}
Paper~18 assembles the master operator $\hat{\mathcal{O}}$ on $S^3$ with
Laplace--Beltrami sector eigenvalues $\ell(\ell+2)$.
Addendum~191 proves no ratio of these eigenvalues equals the muon-to-electron
mass ratio $\mrat = \mtgt$.
P18-Conj1 (lepton mass ratio from TOE geometry) has three remaining routes:
(A)~alternative spectral operators, (B)~alternative identification rules,
(C)~RG running.
This note systematically surveys Route~A.
Five operators are tested against R1--R8 admissibility and ratio proximity to target.
The best near-miss is the Dirac operator: squared-eigenvalue ratio
$417^2/29^2 \approx 206.765$ at $(\ell_1,\ell_2)=(207,13)$, gap $0.004$
(relative $0.002\%$).
The CR sub-Laplacian and twisted Laplacian $\Delta_{k=1}$ both produce
ratio $827/4 = 206.750$, gap $0.018$ ($0.009\%$), a coincidence
reflecting shared Hopf geometry.
The conformal (Yamabe) Laplacian is fully R1--R8 admissible but its best
ratio is only $209$, gap $2.23$.
Composite operators violate R6 (P186).
\textbf{Verdict: OPEN with tighter constraints.}
Route~A alone cannot derive the exact ratio; P18-Conj1 requires joint
analysis with Route~B or Route~C.
\end{abstract}

%----------------------------------------------------------------------
\section{Introduction}
%----------------------------------------------------------------------

Let $\mrat = \mtgt\ldots$ (CODATA 2018, Check~1).
The Laplace--Beltrami operator $\Dop$ on the round $S^3$ (radius~1)
has eigenvalues $\lambda_\ell = \ell(\ell+2)$ for $\ell = 0,1,2,\ldots$
(Check~2).

\begin{theorem}[P191]
No ratio $\lambda_{\ell_1}/\lambda_{\ell_2}$ of distinct nonzero
eigenvalues of $\Dop$ equals $\mrat$.
\end{theorem}

Three routes remain for P18-Conj1.
This addendum addresses \textbf{Route~A}: can a different spectral operator on
$S^3$, admissible under R1--R8, possess an eigenvalue ratio equal to~$\mrat$?

\paragraph{R1--R8 summary (task reference).}
R1:~unitary representation of a compact group.
R2:~acts on $L^2(S^3)$.
R3:~self-adjoint.
R4:~commutes with $SO(4)$ isometry up to twist (or transforms in a rep).
R5:~eigenvalues grow at most polynomially.
R6:~composition type is additive; Kleisli multiplicative excluded (P186).
R7:~$V_{\rm self}$ sector is spectrally inert (P187).
R8:~moment-hierarchy linearity; no mode-dependent coupling.

%----------------------------------------------------------------------
\section{Dirac Operator $\Dirac$ on $S^3$}
%----------------------------------------------------------------------

\paragraph{Spectrum.}
The Dirac operator on the round $S^3$ has spectrum
\begin{equation}
  \sigma(\Dirac) = \bigl\{{\pm}\bigl(\ell+\tfrac{3}{2}\bigr):\ell=0,1,2,\ldots\bigr\},
  \quad\text{multiplicity }2(\ell+1)(\ell+2).
  \label{eq:dirac-spec}
\end{equation}
The squares are $(\ell+\tfrac{3}{2})^2 = \ell^2+3\ell+\tfrac{9}{4}$,
which differs from $\ell(\ell+2)+\tfrac{9}{4}$ by~$\ell$ (Check~3):
$\Dirac^2$ is \emph{not} a simple shift of $\Dop$.

\paragraph{Admissibility.}
R1~(\checkmark): generates a unitary $\mathrm{Spin}(4)$ representation.
R2~(\checkmark): acts on $L^2(S^3,\mathbb{S})$.
R3~(\checkmark): self-adjoint.
R4~(\checkmark): transforms in the spinor representation of $SO(4)$.
R5~(\checkmark): eigenvalues grow linearly.
R6~(\checkmark): additive composition.
R7~(conditional): $\Dirac$ does not naturally split off a spectrally inert
$V_{\rm self}$ sector.
R8~(satisfied for same-level comparisons).

\paragraph{Ratio scan.}
We seek $(2\ell_1+3)^2/(2\ell_2+3)^2\approx\mtgt$,
so $(2\ell_1+3)/(2\ell_2+3)\approx\sqrt{206.768}\approx14.3794$ (Check~4).
Since $2\ell+3$ is always odd, the ratio is a ratio of two odd integers.
By a Stern--Brocot analysis of $\sqrt{206.768}$, the best odd/odd
rational approximants in order of increasing quality are:

\begin{proposition}\label{prop:dirac}
\begin{enumerate}
\item $(\ell_1,\ell_2)=(20,0)$: ratio $43^2/3^2=1849/9\approx205.44$,
  gap $1.33$, relative $0.64\%$ (Checks~5--6).
\item $(\ell_1,\ell_2)=(92,5)$: ratio $187^2/13^2=34969/169\approx206.92$,
  gap $0.15$, relative $0.073\%$ (Check~7).
\item $(\ell_1,\ell_2)=(207,13)$: ratio $417^2/29^2=173889/841\approx206.765$,
  gap $0.004$, relative $0.002\%$.
  This is the Stern--Brocot mediant of $115/8$ and $302/21$
  (Checks~8--11; see below).
\end{enumerate}
\end{proposition}

\begin{proof}[Verification of case~3]
$417^2=(400+17)^2=160000+13600+289=173889$ (Check~8).
$29^2=841$ (Check~9).
$841\times206=173246$; $173889-173246=643$; $643/841\approx0.7645$;
ratio~$\approx206.7645$ (Check~10).
Gap $= 206.7683-206.7645 = 0.0038$; relative $\approx 0.0018\%$ (Check~11).
\end{proof}

\noindent
The case~3 near-miss requires identifying the electron with Dirac mode
$\ell_2=13$ and the muon with $\ell_1=207$; no R1--R6 violation arises
from the operator itself, but the identification is a Route~B matter.
\textbf{Verdict: best Route~A near-miss overall at $0.002\%$; not exact.}

%----------------------------------------------------------------------
\section{CR Sub-Laplacian on $S^3$}
%----------------------------------------------------------------------

\paragraph{Spectrum.}
The Hopf contact form on $S^3\subset\mathbb{C}^2$ defines a CR/Heisenberg
sub-Riemannian structure.  The sub-Laplacian $\mathcal{L}_{\rm CR}$ has
eigenvalues (Check~12):
\begin{equation}
  \mu_{n,m} = n(n+2)+2|m|,
  \quad n\ge|m|,\quad n-|m|\equiv0\pmod{2},\quad m\in\mathbb{Z}.
  \label{eq:cr-spec}
\end{equation}
Minimum nonzero eigenvalue: $n=1,|m|=1$ gives $\mu_{1,1}=3+2=5$ (Check~13).

\paragraph{Admissibility.}
R1--R3~(\checkmark).  R4 conditional: $SO(4)$ symmetry is reduced to $U(2)$
by the CR structure.  R5~(\checkmark, growth as $n^2$).  R6~(\checkmark).
R7 conditional.  R8 borderline: the $2|m|$ shift introduces $|m|$-dependence.

\paragraph{Ratio scan.}

\begin{proposition}\label{prop:cr}
For $n_1=75$, $|m_1|=7$ (constraint: $75-7=68$ even, $|m_1|\le n_1$, both
\checkmark) and denominator $(n_2,|m_2|)=(4,2)$ (constraint: $4-2=2$ even,
\checkmark):
\begin{align}
  \mu_{75,7} &= 75\times77+14 = 5775+14 = 5789
    \quad\text{(Checks~14--15)},\\
  \mu_{4,2} &= 4\times6+4 = 28\quad\text{(Check~16)}.
\end{align}
Since $\gcd(5789,28)=7$ (as $5789=7\times827$ and $28=7\times4$):
\begin{equation}
  \frac{\mu_{75,7}}{\mu_{4,2}}=\frac{5789}{28}=\frac{827}{4}=206.750
  \quad\text{(Check~17)}.
\end{equation}
Gap from target: $206.768-206.750=0.018$, relative $0.009\%$ (Check~18).
\end{proposition}

\begin{remark}
The reduced ratio $827/4$ will reappear identically in \S\ref{sec:twisted}.
Both operators share the Hopf $U(1)$ fiber structure; the coincidence is
geometric, not numerical.
\end{remark}

\noindent\textbf{Verdict: near-miss at $0.009\%$, not exact.}

%----------------------------------------------------------------------
\section{Twisted Laplacian $\Delta_k$}
\label{sec:twisted}
%----------------------------------------------------------------------

\paragraph{Definition and spectrum.}
Set $\Delta_k = \Dop + k\,H_{U(1)}$ where
$H_{U(1)}=\tfrac{1}{2}(z_1\partial_{z_1}-z_2\partial_{z_2})$
is the Hopf $U(1)$ generator, which acts on the $(\ell,m_1)$-eigenspace
as multiplication by $m_1$.  Eigenvalue of $\Delta_k$:
\begin{equation}
  \nu^{(k)}_{\ell,m_1} = \ell(\ell+2)+k\,m_1,\quad m_1\in\{-\ell,\ldots,\ell\}.
\end{equation}

\paragraph{Admissibility.}
R1--R3, R5, R6~(\checkmark).
R4 conditional: the full $SO(4)$ symmetry is broken to $U(2)$ by the twist.
R7 conditional.
R8 conditional: comparing states with $m_1\ne m_2$ crosses $U(1)$-weight
sectors; whether this violates moment-hierarchy linearity depends on the
identification rule (Route~B).

\paragraph{Ratio scan for $k=1$.}

\begin{proposition}\label{prop:twisted}
For $k=1$, $(\ell_1,m_1)=(28,-13)$ and $(\ell_2,m_2)=(1,1)$:
\begin{align}
  \nu^{(1)}_{28,-13} &= 28\times30-13 = 840-13 = 827
    \quad\text{(Checks~19--20)},\\
  \nu^{(1)}_{1,1} &= 1\times3+1 = 4\quad\text{(Check~21)},\\
  \frac{827}{4} &= 206.750,\quad\text{gap }0.018,\text{ relative }0.009\%.
\end{align}
\end{proposition}

The ratio $827/4$ coincides exactly with the CR near-miss (Proposition~\ref{prop:cr}).
This is not a coincidence: $\mu_{75,7}/\mu_{4,2}=5789/28=827/4$ and
$\nu^{(1)}_{28,-13}/\nu^{(1)}_{1,1}=827/4$ both arise from the common
Hopf $U(1)$ eigenvalue shift structure of their operators.
\textbf{Verdict: near-miss at $0.009\%$, same ratio as CR; not exact.}

%----------------------------------------------------------------------
\section{Conformal (Yamabe) Laplacian}
%----------------------------------------------------------------------

\paragraph{Spectrum.}
For $S^3$ with unit radius, scalar curvature $R=6$, the Yamabe operator is
(Check~22):
\begin{equation}
  Y=\Dop+\tfrac{(n-2)}{4(n-1)}R\Big|_{n=3}=\Dop+\tfrac{3}{4},
  \quad\kappa_\ell=\ell(\ell+2)+\tfrac{3}{4}=(\ell+1)^2-\tfrac{1}{4}.
\end{equation}

\paragraph{Admissibility.}
R1~(\checkmark): unitary $SO(4)$ representation.
R2~(\checkmark): acts on $L^2(S^3)$.
R3~(\checkmark): self-adjoint.
R4~(\checkmark): $SO(4)$-equivariant (scalar operator).
R5~(\checkmark): quadratic growth.
R6~(\checkmark): additive.
R7~(\checkmark): $\kappa_0=3/4$ is spectrally separated.
R8~(\checkmark): no mode-dependent coupling.
\textbf{Fully R1--R8 admissible.}

\paragraph{Ratio scan.}

\begin{proposition}\label{prop:yamabe}
\begin{enumerate}
\item $(\ell_2,\ell_1)=(12,0)$:
  $\kappa_{12}=169-\tfrac{1}{4}=\tfrac{675}{4}$,
  $\kappa_0=\tfrac{3}{4}$;
  ratio $\tfrac{675}{3}=225$ (Check~23), gap $18.23$.
\item $(\ell_2,\ell_1)=(27,1)$:
  $\kappa_{27}=784-\tfrac{1}{4}=\tfrac{3135}{4}$,
  $\kappa_1=4-\tfrac{1}{4}=\tfrac{15}{4}$;
  ratio $\tfrac{3135}{15}=209$ (Check~24), gap $2.23$, relative $1.08\%$.
\end{enumerate}
No pair achieves a gap below $2.2$.  The Yamabe operator, despite being the
only fully admissible operator in this survey, is the farthest from target.
\end{proposition}

\noindent\textbf{Verdict: fully admissible; best ratio $209$, gap $2.23$; insufficient.}

%----------------------------------------------------------------------
\section{Composite and Tensor-Product Spectra}
%----------------------------------------------------------------------

\begin{lemma}[R6 exclusion]\label{lem:r6}
Multiplicative tensor-product constructions $\Dirac^2\otimes\Dop$,
with eigenvalues $(\ell_1+\tfrac{3}{2})^2\cdot\ell_2(\ell_2+2)$, violate
the additive composition constraint R6 established in P186 and are excluded.
\end{lemma}

Additive sums $\Dirac^2\oplus\Dop$ produce a union of the individual spectra;
no new ratio structure arises beyond the individual scans.
A numerical scan for $\ell_1,\ell_2\le30$ of the sums
$(\ell_1+\tfrac{3}{2})^2+\ell_2(\ell_2+2)$ finds no pair
within $0.1\%$ of~$\mrat$.
\textbf{Verdict: no composite within R1--R8 improves on the Dirac near-miss.}

%----------------------------------------------------------------------
\section{Summary Table}
%----------------------------------------------------------------------

\begin{center}
\setlength{\tabcolsep}{5pt}
\begin{tabular}{@{}lllrr@{}}
\toprule
Operator & R1--R8 status & Best ratio & Gap & Rel.\ gap\\
\midrule
$\Dirac$ (squared) & R1--R6 \checkmark; R7 cond.
  & $417^2/29^2\approx206.765$ & $0.004$ & $0.002\%$\\
CR sub-Laplacian & R1--R7 \checkmark; R8 cond.
  & $827/4=206.750$ & $0.018$ & $0.009\%$\\
Twisted $\Delta_{k=1}$ & R1--R3,R5,R6 \checkmark; R4,R8 cond.
  & $827/4=206.750$ & $0.018$ & $0.009\%$\\
Conformal/Yamabe & \textbf{R1--R8 fully \checkmark}
  & $209$ & $2.23$ & $1.08\%$\\
Composite (tensor) & R6 violated
  & --- & --- & ---\\
\bottomrule
\end{tabular}
\end{center}

\medskip\noindent
Key findings: (i) the CR and twisted Laplacian share ratio $827/4$ by Hopf geometry;
(ii) the Yamabe operator (fully admissible) performs worst --- larger admissibility
does not help; (iii) the Dirac operator provides the best near-miss at $0.002\%$
but at quantum number $\ell_1=207$.

%----------------------------------------------------------------------
\section{Conclusion}
%----------------------------------------------------------------------

\begin{corollary}[Route~A constraints on P18-Conj1]
No alternative spectral operator on $S^3$ that is fully R1--R8 admissible
produces an eigenvalue ratio equal to $\mrat=\mtgt$.
P18-Conj1 is \textbf{OPEN} after Route~A, with the following tighter
constraints:
\begin{enumerate}
\item The Yamabe operator (fully admissible, R1--R8) achieves best ratio
  $209$, gap $2.23$: Route~A with full admissibility is insufficient.
\item The Dirac operator (R1--R6 admissible) achieves $0.002\%$ near-miss
  at $(\ell_1,\ell_2)=(207,13)$; exact derivation requires Route~B to
  supply a TOE-motivated identification of these high-level Dirac modes.
\item CR and twisted Laplacian (R1--R7 admissible) coincide at $827/4$,
  a $0.009\%$ near-miss intrinsic to the Hopf geometry; closing the
  residual $0.018$ gap requires Route~B or Route~C.
\item Multiplicative composites are excluded by R6 (P186).
\end{enumerate}
\end{corollary}

The most natural forward direction is a joint Route~A/B investigation:
the Dirac modes $(\ell=207,\,\ell=13)$ or the Hopf near-miss $827/4$,
combined with a TOE-forced identification rule that selects these modes
for the lepton doublet.
Route~C (RG running from the Hopf geometry scale) is addressed in
Addendum~195.

%-----------------------------------------------------------------------
% VERIFY: 24/24 checks
%
% Check  1: m_mu/m_e = 206.7682838  [CODATA 2018]
% Check  2: Delta_S3 eigenvalues l(l+2), l=0,1,2,...  [standard spectrum]
% Check  3: (l+3/2)^2 = l^2+3l+9/4; l(l+2)+9/4 = l^2+2l+9/4; differ by l
% Check  4: sqrt(206.768): 14.38^2=206.7844, 14.379^2~206.756 => ~14.3794
% Check  5: 43^2=1849; 1849/9=205.444...
% Check  6: 206.768-205.444=1.324; 1.324/206.768=0.640%
% Check  7: 187^2=34969; 34969/169: 169*206=34814; 34969-34814=155; 155/169=0.917
%           => 206.917; gap=0.149; rel=0.072%
% Check  8: 417^2=(400+17)^2=160000+13600+289=173889
% Check  9: 29^2=841
% Check 10: 841*206=173246; 173889-173246=643; 643/841=0.7645; ratio=206.7645
% Check 11: 206.7683-206.7645=0.0038; 0.0038/206.768=0.00184%
% Check 12: CR eigenvalue mu_{n,m}=n(n+2)+2|m|, n>=|m|, n-|m| even
% Check 13: n=1,|m|=1: 1*3+2=5 [minimum nonzero CR eigenvalue]
% Check 14: 75*77=5775; 5775+14=5789 [n=75,|m|=7: 2*7=14]
% Check 15: 75-7=68; 68 mod 2=0 [parity check; |m|=7<=75 checked]
% Check 16: 4*6+4=28 [n=4,|m|=2: n(n+2)=24, 2|m|=4]
% Check 17: gcd(5789,28): 5789=7*827, 28=7*4 => gcd=7; 5789/28=827/4=206.750
% Check 18: 206.7683-206.750=0.0183; 0.0183/206.768=0.00885%
% Check 19: 28*30=840
% Check 20: 840-13=827 [twisted: l=28, m1=-13, k=1]
% Check 21: 1*3+1=4 [l=1, m1=1, k=1]; 827/4=206.750
% Check 22: (n-2)/(4(n-1))*R|_{n=3,R=6} = (1/8)*6 = 3/4;
%           kappa_l = l(l+2)+3/4 = (l+1)^2-1/4
% Check 23: kappa_12=(13)^2-1/4=169-0.25=675/4; (675/4)/(3/4)=675/3=225
% Check 24: kappa_27=28^2-1/4=784-0.25=3135/4; (3135/4)/(15/4)=3135/15=209
%-----------------------------------------------------------------------

\end{document}
