% Addendum 190: Non-perturbative H_{U(1)}^⊥ and the Mass Formula
%   (Resolution of Open Problem 7.1 from Addendum P188)
%
%   P188 proved that adding ε·H_{U(1)}^⊥ as a 9th sector in first-order
%   perturbation theory requires a physically implausible coupling to close the
%   lepton mass formula gap (m_μ/m_e from ~52.8 to ~206.8). Open Problem 7.1
%   asks whether any non-perturbative inclusion of H_{U(1)}^⊥ — via its square
%   (H_{U(1)}^⊥)², via higher polynomials, or via operadic compositions — can
%   improve the lepton mass ratio.
%
%   This addendum delivers three structural theorems:
%
%   Theorem 3.1 (Second-order correction is zero): The second-order perturbation
%     correction from ε·H_{U(1)}^⊥ vanishes identically. H_{U(1)}^⊥ = ∂/∂φ is
%     diagonal in the Wigner D-function basis, so all off-diagonal matrix elements
%     vanish and the second-order sum is empty.
%
%   Theorem 4.1 (Spectral inertness for any f(H_{U(1)}^⊥)): Any function f of
%     H_{U(1)}^⊥ — including (H_{U(1)}^⊥)² and all higher polynomials — is
%     diagonal with eigenvalue f(im₁) on D^ℓ_{m₁,m₂}. For any transition in
%     which the electron and muon carry the SAME azimuthal quantum number m₁,
%     the correction f(im₁) cancels exactly in the spectral gap, leaving the
%     mass ratio unchanged. This is spectrally inert by the same mechanism
%     identified in P187 §7 for V_self.
%
%   Theorem 5.1 (Cross-m₁ case: naturally-sized coefficient, no selection rule):
%     For transitions in which the electron (ℓ=1) and muon (ℓ=2) carry different
%     azimuthal quantum numbers, adding c·(H_{U(1)}^⊥)² modifies the spectral
%     gap by c·(m₁²_electron − m₁²_muon). The required coefficient to reach
%     m_μ/m_e = 206.8 is |c| ≈ 1.72 (naturally-sized, ≈ √3 to 0.7%). But no
%     physical principle within R1--R8 or the TOE geometry selects a specific
%     cross-m₁ state identification, so this case does not constitute a
%     predictive assembly.
%
%   Operadic composition (§6): replacing ζ·H_{U(1)} with ζ·(H_{U(1)} + f·H_{U(1)}^⊥)
%     for constant f remains diagonal and inert. For non-constant f(χ) on S³,
%     ℓ-mixing is introduced and the spectrum is qualitatively changed; this case
%     is genuinely OPEN and requires further analysis.
%
%   Verdict: OPEN — the non-perturbative H_{U(1)}^⊥ case is structurally reduced
%   but not closed. All additive non-perturbative inclusions (second-order,
%   polynomial, fixed-m₁) are proved inert. The cross-m₁ polynomial case finds
%   a naturally-sized coefficient but lacks a selection rule. The operadic
%   composition with non-constant f(χ) on S³ remains uncharted.
%
%   Numerical companion: verify_P190.py (mp.dps=55; 83/83 checks pass).
%
% L. F. Vlegels, Independent Researcher
% Copyright: Léon 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{enumitem}
\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref}
\geometry{margin=1in}

\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{openproblem}[theorem]{Open Problem}
\newtheorem{observation}[theorem]{Observation}

% TOE shorthand
\newcommand{\Ohat}{\hat{O}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\Hunit}{H_{U(1)}}
\newcommand{\Hperp}{H_{U(1)}^{\perp}}
\newcommand{\ALPHAINV}{\alpha^{-1}}
\newcommand{\Kill}{\mathrm{K}}
\newcommand{\fz}{\mathfrak{z}}
\newcommand{\hE}{\mathfrak{h}_{E_6}}
\newcommand{\calR}{\mathcal{R}}
\newcommand{\PertOp}{\mathrm{Perturb}}

\title{Non-perturbative $H_{U(1)}^{\perp}$ and the Lepton Mass Formula:\\
  Resolution of Open Problem~7.1 from Addendum~P188\\[0.5em]
  \large Addendum 190 to the TOE Corpus}

\author{L.\ F.\ Vlegels\\
Independent Researcher\\
\texttt{axiomofchoicepuzzle@gmail.com}}

\date{May 2026}

\begin{document}
\maketitle
\tableofcontents
\newpage

%==========================================================================
\begin{abstract}
Addendum~P188 characterised $\Hperp = \partial/\partial\phi \in \fz \subset
\hE$ — the Killing-orthogonal complement of $\Hunit$ in the two-dimensional
$F_4$-centralising subalgebra — and showed that adding $\varepsilon \cdot
\Hperp$ as a 9th sector of $\hat{O}$ in first-order perturbation theory
requires a coupling $\kappa \approx 728$ (physical convention: $|c| \approx
1.72$) to close the lepton mass formula gap $(m_\mu/m_e \colon 52.8
\to 206.8)$.  P188 Open Problem~7.1 asked whether any \emph{non-perturbative}
inclusion of $\Hperp$ — via $(\Hperp)^2$, higher polynomials $f(\Hperp)$, or
operadic compositions — can do better.

This addendum proves three structural results that reduce Open Problem~7.1.

\textbf{Theorem~\ref{thm:second_order_zero}}: The second-order perturbation
correction from $\varepsilon \cdot \Hperp$ vanishes identically.  Since
$\Hperp = \partial/\partial\phi$ is already diagonal in the Wigner $D$-function
basis — the same basis that diagonalises $\Delta_{S^3}$ — all off-diagonal
matrix elements $\langle m | \Hperp | n \rangle$ with $m \neq n$ are zero, and
the standard second-order sum is empty.

\textbf{Theorem~\ref{thm:polynomial_inert}}: For \emph{any} function $f$
applied to $\Hperp$, the operator $f(\Hperp)$ remains diagonal in the
$D$-function basis with eigenvalue $f(im_1)$ on $D^\ell_{m_1,m_2}$.  For any
electron--muon identification in which both states carry the same azimuthal
quantum number $m_1$, the correction $f(im_1)$ cancels exactly in the spectral
gap $\Delta E = E_\mu - E_e$, leaving the mass ratio unchanged regardless of
the coefficient.  This is the same spectral-inertness mechanism identified for
$V_\mathrm{self}$ in P187.

\textbf{Theorem~\ref{thm:cross_m1}}: For cross-$m_1$ identifications (electron
and muon at different $m_1$ values), the correction is $c(m_{1,e}^2 -
m_{1,\mu}^2)$ from $c \cdot (\Hperp)^2$.  For the natural identification
(electron at $(\ell{=}1, m_1{=}1)$, muon at $(\ell{=}2, m_1{=}0)$), the
required $c \approx 1.720$ is \emph{naturally sized} — within $0.7\%$ of
$\sqrt{3}$ — but no physical principle within $R1$--$R8$ or the TOE geometry
selects a specific $m_1$ for either lepton.  The case is therefore
\emph{structurally possible but not predictive}.

The operadic composition replacing $\zeta \Hunit$ with $\zeta(\Hunit + f \cdot
\Hperp)$ for non-constant $f = f(\chi)$ introduces $\ell$-mixing and is
genuinely open; the smooth candidate $f(\chi) = \sin\chi$ is identified as the
natural next target.

\textbf{Verdict: OPEN} — the non-perturbative $\Hperp$ question is structurally
reduced but not closed.

Numerical companion: \texttt{verify\_P190.py} ($83/83$ checks pass,
\texttt{mp.dps}~$=55$).
\end{abstract}

%==========================================================================
\section{Background and Open Problem~7.1}
\label{sec:background}
%==========================================================================

\subsection{The P188 result recalled}

The two-dimensional $F_4$-centralising Cartan complement $\fz \subset \hE$
(P176, P184) has an orthonormal basis
\begin{equation}
  \Hunit = \tfrac{1}{2}(z_1\partial_{z_1} + z_2\partial_{z_2})
  \;\text{ and }\;
  \Hperp = \tfrac{1}{2}(z_1\partial_{z_1} - z_2\partial_{z_2}),
\end{equation}
with $\Kill(\Hunit,\Hunit) = \Kill(\Hperp,\Hperp) = 24$ and $\Kill(\Hunit,\Hperp) = 0$
(P188 Theorem~3.1).  On $S^3$ in Hopf coordinates $(\chi,\phi,\psi)$:
$\Hunit = \partial/\partial\psi$ (Hopf fibre generator, nowhere-vanishing)
and $\Hperp = \partial/\partial\phi$ (azimuthal base generator, vanishing at the
Hopf poles $\chi = 0,\pi$; P188 Theorem~4.1).

P188 Theorem~6.1 showed that adding $\varepsilon \cdot \Hperp$ as a 9th sector
in first-order perturbation theory splits the $\ell$-th eigenspace of
$\Delta_{S^3}$ by azimuthal quantum number $m_1$:
\begin{equation}
  E_{\ell,m_1,m_2}^{(1)} = \ell(\ell+2) + \kappa m_1,
  \quad \kappa \in \mathbb{R} \text{ (with } \varepsilon = -i\kappa\text{)}.
  \label{eq:first_order}
\end{equation}
Closing the mass-formula gap (P187 convention: $m_\mu/m_e \colon 52.8 \to 206.8$
using $\mu_1/\mu_0 \approx 0.7933$) requires $\kappa \approx 1.72$ for the
identified P187 transitions and $\kappa \approx 728$ in the P18 eigenvalue
convention ($\mu_1/\mu_0 = \alpha$).  P188 declared the latter implausible at
any natural coupling.  Both estimates agree that first-order perturbation theory
alone cannot close the gap with a naturally-normalised coupling.

\subsection{Open Problem~7.1 from P188}

P188 Open Problem~7.1(iii) asks: does $(\Hperp)^2 = (\partial/\partial\phi)^2$
as a second-order operator produce a spectrum with a ratio closer to 207?  Parts
(i) and (ii) ask more broadly whether any assembly incorporating $\Hperp$ —
additively or via operadic/Kleisli composition — is admissible and improves the
mass formula.

\subsection{Key tool: Wigner $D$-functions on $S^3$}

The Wigner $D$-functions $D^\ell_{m_1,m_2}(\phi,\chi,\psi)$ form an
orthonormal basis of $L^2(S^3)$ (Gel'fand--Tsetlin basis for $SU(2)$).  In
Hopf coordinates:
\begin{equation}
  D^\ell_{m_1,m_2}(\phi,\chi,\psi)
  = e^{im_1\phi}\,d^\ell_{m_1,m_2}(\chi)\,e^{im_2\psi},
  \label{eq:Dfunc}
\end{equation}
where $d^\ell_{m_1,m_2}(\chi)$ are the Wigner small-$d$ matrices and $\ell \in
\{0, \tfrac{1}{2}, 1, \tfrac{3}{2}, \ldots\}$, $m_1, m_2 \in \{-\ell, \ldots,
\ell\}$.  The $D$-functions are simultaneous eigenfunctions:
\begin{equation}
  \Delta_{S^3}\,D^\ell_{m_1,m_2} = \ell(\ell+2)\,D^\ell_{m_1,m_2}, \quad
  \frac{\partial}{\partial\psi}\,D^\ell_{m_1,m_2} = im_2\,D^\ell_{m_1,m_2}, \quad
  \frac{\partial}{\partial\phi}\,D^\ell_{m_1,m_2} = im_1\,D^\ell_{m_1,m_2}.
  \label{eq:eigenvalues}
\end{equation}
The third identity follows directly from~\eqref{eq:Dfunc}: differentiating
$e^{im_1\phi}$ with respect to $\phi$ gives $im_1 e^{im_1\phi}$, while
$d^\ell_{m_1,m_2}(\chi)$ and $e^{im_2\psi}$ are unaffected.  This means
$\Hperp = \partial/\partial\phi$ is \emph{already diagonal} in the $D$-function basis.

%==========================================================================
\section{The P18 Mass Formula and the Spectral Gap}
\label{sec:massformula}
%==========================================================================

The P18 mass formula (Conjecture~5.1) is:
\begin{equation}
  \frac{m_n}{m_e} = \exp\!\left(\frac{\mu_1}{\mu_0}\,\Delta E_n\right),
  \label{eq:mass-formula}
\end{equation}
where $\mu_k = \int_0^1 r^k \rho(r)\,dr$ are the radial moments of the
geometric density $\rho(r) = 16\pi^3r^3 + 3\pi^2r^2 + 2\pi r$ (P03):
\begin{equation}
  \mu_0 = 4\pi^3 + \pi^2 + \pi \approx 137.036 = \alpha^{-1},
  \quad
  \mu_1 = \frac{16\pi^3}{5} + \frac{3\pi^2}{4} + \frac{2\pi}{3} \approx 108.717,
  \quad
  \frac{\mu_1}{\mu_0} \approx 0.7933.
  \label{eq:moments}
\end{equation}

P187 (following P18 Remark~5.2) computes: with $\Delta_{S^3}$ alone, the
electron--muon spectral gap is $\Delta E = \ell(\ell{+}2)|_{\ell=2} -
\ell(\ell{+}2)|_{\ell=1} = 8 - 3 = 5$, giving:
\begin{equation}
  \frac{m_\mu}{m_e}\bigg|_{\text{unperturbed}}
  = \exp(0.7933 \times 5) = e^{3.967} \approx 52.8,
  \label{eq:base_ratio}
\end{equation}
versus the experimental value $m_\mu/m_e \approx 206.77$.  The required
spectral gap to match observation is:
\begin{equation}
  \Delta E^* = \frac{\mu_0}{\mu_1}\ln(206.77)
  \approx \frac{\ln(206.77)}{0.7933}
  \approx 6.720,
  \label{eq:required_gap}
\end{equation}
so the canonical assembly falls short by $\delta E = 6.720 - 5 = 1.720$.

\begin{remark}[Conventions for $\mu_1/\mu_0$]
\label{rem:convention}
P188 Theorem~6.1 uses $\mu_1/\mu_0 = \alpha \approx 1/137$ (the P18 eigenvalue
convention, where the mass formula exponent is normalised to the fine-structure
constant).  Under this convention, the required eigenvalue at $\ell = 1$ is
$3 + \kappa = (\mu_0/\mu_1)\ln(206.8) \approx 730.9$, giving $\kappa \approx
728$.  The present paper uses the physical moment ratio $\mu_1/\mu_0 \approx
0.7933$ throughout, which gives the smaller shortfall $\delta E \approx 1.720$.
Both conventions agree on the qualitative conclusion: first-order $\varepsilon
\cdot \Hperp$ requires a coupling of the same order as the target shift
itself.
\end{remark}

%==========================================================================
\section{Second-Order Perturbation from $\varepsilon \cdot \Hperp$}
\label{sec:second_order}
%==========================================================================

\begin{theorem}[Second-order correction vanishes]
\label{thm:second_order_zero}
The second-order perturbation-theory correction to any eigenvalue of $\Delta_{S^3}$
from the perturbation $V = \varepsilon \cdot \Hperp$ is identically zero for all
$\varepsilon \in \mathbb{C}$.
\end{theorem}

\begin{proof}
The second-order correction to the eigenvalue of state $|n\rangle = D^\ell_{m_1,m_2}$
is (degenerate second-order perturbation theory):
\begin{equation}
  E_n^{(2)} = \sum_{|m\rangle \notin \mathcal{H}_\ell}
  \frac{|\langle m | V | n \rangle|^2}{E_n^{(0)} - E_m^{(0)}},
  \label{eq:second_order}
\end{equation}
where the sum is over states $|m\rangle = D^{\ell'}_{m_1',m_2'}$ with $\ell'
\neq \ell$ (outside the degenerate subspace $\mathcal{H}_\ell$ of $\Delta_{S^3}$
at eigenvalue $\ell(\ell+2)$).

From~\eqref{eq:eigenvalues}, $V = \varepsilon \cdot \partial/\partial\phi$ acts
on $D^\ell_{m_1,m_2}$ as:
\[
  V\,D^\ell_{m_1,m_2} = \varepsilon\,im_1\,D^\ell_{m_1,m_2}.
\]
Therefore:
\begin{equation}
  \langle D^{\ell'}_{m_1',m_2'} | V | D^\ell_{m_1,m_2} \rangle
  = \varepsilon\,im_1 \langle D^{\ell'}_{m_1',m_2'} | D^\ell_{m_1,m_2} \rangle
  = \varepsilon\,im_1 \cdot \delta_{\ell'\ell}\,\delta_{m_1'm_1}\,\delta_{m_2'm_2}.
  \label{eq:matrix_element}
\end{equation}
For $\ell' \neq \ell$, the delta $\delta_{\ell'\ell} = 0$ kills every
off-diagonal matrix element in the sum~\eqref{eq:second_order}.  Hence
$E_n^{(2)} = 0$.

The argument is independent of $\varepsilon$ and of the specific state $n$:
$V$ maps every $D$-function to a scalar multiple of itself, leaving all other
states orthogonal.  No degenerate mixing within the same $\ell$-eigenspace
contributes to the inter-space sum~\eqref{eq:second_order} because the
$\Delta_{S^3}$ degenerate subspaces at different $\ell$ are orthogonal.
\end{proof}

\begin{corollary}[Perturbation from $\varepsilon\Hperp$ is exactly first-order]
\label{cor:exact_first_order}
The full perturbative expansion of the eigenvalue of $\Delta_{S^3} + \varepsilon\Hperp$
terminates at first order.  The exact eigenvalue on $D^\ell_{m_1,m_2}$ is:
\begin{equation}
  E_{\ell,m_1,m_2}^{\varepsilon} = \ell(\ell+2) + \varepsilon\,im_1.
\end{equation}
This is not a perturbative approximation: since $\Hperp$ is diagonal in the
$D$-function basis, the sum $\Delta_{S^3} + \varepsilon\Hperp$ is simultaneously
diagonalised by the same basis, and the eigenvalues are exact.
\end{corollary}

%==========================================================================
\section{Non-perturbative Polynomial $f(\Hperp)$}
\label{sec:polynomial}
%==========================================================================

\begin{theorem}[Any $f(\Hperp)$ is spectrally inert for fixed-$m_1$ transitions]
\label{thm:polynomial_inert}
Let $f : \mathbb{C} \to \mathbb{C}$ be any function with $f(im_1) \in \mathbb{R}$
for all integers $m_1$ (so that the modified eigenvalues are real).  Define the
modified operator $\hat{O}' = \hat{O} + c \cdot f(\Hperp)$ for $c \in \mathbb{R}$.
Then:
\begin{enumerate}[label=(\roman*)]
  \item $f(\Hperp)$ is diagonal in the $D$-function basis with eigenvalue $f(im_1)$:
    \begin{equation}
      f(\Hperp)\,D^\ell_{m_1,m_2} = f(im_1)\,D^\ell_{m_1,m_2}.
      \label{eq:f_diag}
    \end{equation}
  \item The eigenvalue of $\hat{O}'$ on $D^\ell_{m_1,m_2}$ is:
    \begin{equation}
      E_{\ell,m_1,m_2}' = \ell(\ell+2) + c\,f(im_1)
      \quad (+\text{contributions from other sectors}).
      \label{eq:modified_eigenvalue}
    \end{equation}
  \item For any electron state at $(\ell_e, m_1)$ and muon state at $(\ell_\mu, m_1)$
    with the \emph{same} $m_1$:
    \begin{equation}
      \Delta E' = E_{\ell_\mu, m_1, \cdot}' - E_{\ell_e, m_1, \cdot}'
      = \ell_\mu(\ell_\mu+2) - \ell_e(\ell_e+2),
      \label{eq:cancellation}
    \end{equation}
    independent of $c$ and $f$.  The mass ratio $m_\mu/m_e$ is unchanged.
\end{enumerate}
\end{theorem}

\begin{proof}
\textbf{(i).} By the spectral mapping theorem: since $\Hperp$ is diagonalised by
the $D$-functions with eigenvalue $im_1$, any function $f(\Hperp)$ is
diagonalised by the same basis with eigenvalue $f(im_1)$.  This is exact (not
perturbative); it follows from the fact that $\Hperp$ is a normal operator on
$L^2(S^3)$ and its spectral decomposition is the $D$-function expansion.

\textbf{(ii).} Direct from (i) and the linearity of the eigenvalue.

\textbf{(iii).} In~\eqref{eq:modified_eigenvalue}, the correction $c f(im_1)$ is
the same for the electron (state $D^{\ell_e}_{m_1,\cdot}$) and the muon (state
$D^{\ell_\mu}_{m_1,\cdot}$) since both share the same $m_1$.  It cancels
exactly in the difference~\eqref{eq:cancellation}.
\end{proof}

\begin{corollary}[P187 analogy: spectral inertness]
The mechanism of Theorem~\ref{thm:polynomial_inert}(iii) is structurally
identical to the spectral inertness of $V_\mathrm{self}$ proved in P187
Corollary~7.3: in both cases, a perturbation that acts the same way on both
lepton states cancels in the mass ratio.  For $V_\mathrm{self}$, the perturbation
was radial (mode-uniform at leading order); for $f(\Hperp)$, it is $m_1$-uniform
within any fixed-$m_1$ sector.
\end{corollary}

\begin{remark}[$(\Hperp)^2$ spectrum explicitly]
\label{rem:Hperp_sq_spectrum}
Taking $f(z) = -z^2$ (so $f(\Hperp) = (\partial/\partial\phi)^2$):
\[
  (\Hperp)^2\,D^\ell_{m_1,m_2}
  = \left(\frac{\partial}{\partial\phi}\right)^2 D^\ell_{m_1,m_2}
  = (im_1)^2 D^\ell_{m_1,m_2}
  = -m_1^2\,D^\ell_{m_1,m_2}.
\]
The spectrum of $c \cdot (\Hperp)^2$ on the $\ell$-th eigenspace is
$\{-c m_1^2 : m_1 = -\ell, \ldots, \ell\}$.  For $\ell = 1$: $\{0, -c\}$
(non-degenerate); for $\ell = 2$: $\{0, -c, -4c\}$.

\medskip
For $c > 0$: the highest-energy states within each $\ell$-sector are those
with $m_1 = 0$; the lowest are $m_1 = \pm\ell$.  For $c < 0$: the ordering
is reversed.  In either case, for the \emph{same} $m_1$ at $\ell = 1$ and
$\ell = 2$, the correction cancels and $\Delta E = 5$ is unchanged.
\end{remark}

%==========================================================================
\section{The Cross-$m_1$ Case}
\label{sec:cross_m1}
%==========================================================================

The fixed-$m_1$ cancellation (Theorem~\ref{thm:polynomial_inert}) forces us
to ask whether a cross-$m_1$ identification — electron and muon at different
$m_1$ values — can break the inertness.

\begin{theorem}[Cross-$m_1$ gap and required coefficient]
\label{thm:cross_m1}
For an operator $c\cdot(\Hperp)^2$ added to $\hat{O}$, the spectral gap between
an electron state at $(\ell_e, m_{1,e})$ and a muon state at $(\ell_\mu, m_{1,\mu})$
with $m_{1,e} \neq m_{1,\mu}$ is:
\begin{equation}
  \Delta E = \ell_\mu(\ell_\mu+2) - \ell_e(\ell_e+2)
  + c\bigl(m_{1,e}^2 - m_{1,\mu}^2\bigr).
  \label{eq:cross_m1_gap}
\end{equation}
Setting $\ell_e = 1$ and $\ell_\mu = 2$ (the P187 assignment):
\begin{equation}
  \Delta E = 5 + c\bigl(m_{1,e}^2 - m_{1,\mu}^2\bigr).
\end{equation}
The required $\Delta E^* \approx 6.720$ (from Section~\ref{sec:massformula})
is achieved for the coefficient:
\begin{equation}
  c^* = \frac{\delta E}{m_{1,e}^2 - m_{1,\mu}^2},
  \quad \delta E = \Delta E^* - 5 \approx 1.720,
  \label{eq:c_star}
\end{equation}
provided $m_{1,e}^2 \neq m_{1,\mu}^2$.

\begin{enumerate}[label=(\alph*)]
  \item \textbf{Case A} ($m_{1,e} = 1$, $m_{1,\mu} = 0$):
    $c^* = 1.720/1 \approx 1.720$.
  \item \textbf{Case B} ($m_{1,e} = 0$, $m_{1,\mu} = 2$):
    $c^* = 1.720/(-4) \approx -0.430$.
  \item \textbf{Case C} ($m_{1,e} = 1$, $m_{1,\mu} = 2$):
    $c^* = 1.720/(-3) \approx -0.573$.
\end{enumerate}
\end{theorem}

\begin{proof}
Direct substitution into~\eqref{eq:modified_eigenvalue} with $f(im_1) = -m_1^2$:
$E_{\ell,m_1}' = \ell(\ell+2) - c m_1^2$.  Then
$\Delta E = E_{\ell_\mu,m_{1,\mu}}' - E_{\ell_e, m_{1,e}}' =
(\ell_\mu(\ell_\mu+2) - c m_{1,\mu}^2) - (\ell_e(\ell_e+2) - c m_{1,e}^2)
= 5 + c(m_{1,e}^2 - m_{1,\mu}^2)$.
\end{proof}

\begin{remark}[Case A coefficient and $\sqrt{3}$]
\label{rem:sqrt3}
Case A gives $c^* \approx 1.720$.  Numerically $\sqrt{3} \approx 1.732$, and
$|c^* - \sqrt{3}|/\sqrt{3} \approx 0.67\%$.  The value $\sqrt{3}$ appears
naturally in the TOE root system geometry (e.g., the long-root norm ratio in
$A_2 \subset G_2 \subset F_4$) and as the length of the non-zero weights of
the adjoint representation of $G_2$.  This numerical proximity is noted as a
potentially significant coincidence but is not proved here: no derivation of
$c = \sqrt{3}$ from the TOE algebra currently exists.

Verification: $\exp(0.7933 \times (5 + 1.720)) = \exp(0.7933 \times 6.720)
\approx 206.8$ (see \texttt{verify\_P190.py} Check~8).
\end{remark}

\begin{remark}[Physical interpretation of cross-$m_1$]
\label{rem:cross_m1_physics}
The identification of the physical electron with a specific $m_1$ value is
not motivated by any principle within R1--R8 or the current TOE.  The quantum
number $m_1$ is the eigenvalue of $\partial/\partial\phi = \Hperp$ (the
azimuthal base rotation); in the canonical assembly $\Hperp$ does not appear
as an operator at all, so $m_1$ is unobservable and all states with the same
$\ell$ but different $m_1$ are physically degenerate.  Introducing $c \cdot
(\Hperp)^2$ breaks this degeneracy, but without a further principle
specifying which $m_1$-eigenstate corresponds to which lepton, the cross-$m_1$
identification is ambiguous.  The three cases A, B, C in
Theorem~\ref{thm:cross_m1} produce the correct mass ratio for different and
mutually exclusive state identifications.

This contrasts with the $\mathbb{Z}_3$ family symmetry (R6), which does
provide a selection rule for the three-fold degeneracy in $m_2$ (the Hopf
fibre quantum number): the $T$-eigenvalues $e^{i m_2 \cdot 4\pi/3}$ label
the three lepton families.  No analogue of R6 exists for $m_1$.
\end{remark}

%==========================================================================
\section{Operadic Composition with $\Hperp$}
\label{sec:operadic}
%==========================================================================

P188 Open Problem~7.1(ii) also raises operadic (non-additive) composition.
The canonical Perturb sector is $\PertOp \leftrightarrow \zeta \Hunit$ (P184
Theorem~5.3).  We consider replacing this with
$\zeta(\Hunit + f \cdot \Hperp)$ for a function $f$ on $S^3$.

\subsection{Constant $f$: remains diagonal and inert}

If $f$ is a constant, the combined generator
$\partial/\partial\psi + f \cdot \partial/\partial\phi$ has eigenvalue
$i(m_2 + f m_1)$ on $D^\ell_{m_1,m_2}$.  This is again diagonal; for the
Perturb sector, the eigenvalue contribution is $\zeta\cdot i(m_2 + f m_1)$.
The dominant mass-ratio correction is still from $\Delta_{S^3}$, which is
unaffected.  Constant $f$ does not help.

\subsection{Non-constant $f = f(\chi)$: $\ell$-mixing, genuinely open}

For $f = f(\chi)$ a smooth non-constant function of the polar angle $\chi$ on
$S^2$ (the Hopf base), the operator $f(\chi) \cdot \partial/\partial\phi$ is
\emph{not} diagonal in the $D$-function basis.  The matrix element
\begin{equation}
  \bigl\langle D^{\ell'}_{m_1',m_2'} \bigm| f(\chi) \cdot \partial/\partial\phi
  \bigm| D^\ell_{m_1,m_2} \bigr\rangle
  = im_1 \cdot \delta_{m_1'm_1}\,\delta_{m_2'm_2} \cdot
  \int_0^\pi f(\chi)\,d^{\ell'}_{m_1,m_2}(\chi)\,d^\ell_{m_1,m_2}(\chi)\,\sin\chi\,d\chi
  \label{eq:chi_matrix}
\end{equation}
connects eigenspaces at different $\ell'$ and $\ell$ (the $\chi$-integral is
generically non-zero for $\ell' \neq \ell$ when $f$ is non-constant), creating
$\ell$-mixing and a qualitatively different spectrum.

\begin{observation}[The smooth candidate $f(\chi) = \sin\chi$]
\label{obs:smooth}
The operator $\sin\chi \cdot \partial/\partial\phi$ is globally smooth on $S^3$:
the factor $\sin\chi \to 0$ as $\chi \to 0, \pi$ cancels the zeros of
$\partial/\partial\phi$ at the Hopf poles.  Contrast with $\Hperp = \partial/\partial\phi$
alone, which has isolated zeros.  If the additional smoothness requirement
``every generator in $\hat{O}$ that acts on $S^3$ must be a globally smooth
(nowhere-vanishing) vector field'' were adopted as a geometric principle (P188
Remark~6.3), it would exclude $\Hperp$ alone but admit $\sin\chi \cdot
\partial/\partial\phi$ only if the smoothness criterion is satisfied by the
\emph{operator} rather than the underlying vector field.

In the sense of operators on $L^2(S^3)$: $\sin\chi \cdot \partial/\partial\phi$
is a well-defined bounded operator (the multiplication by $\sin\chi$ is bounded,
and $\partial/\partial\phi$ is only unbounded due to the gradient-type structure).
The full operator $\Delta_{S^3} + \zeta\sin\chi \cdot \partial/\partial\phi$ is
not diagonal in the $D$-function basis, and its spectrum is not known in
closed form.  The $\ell$-mixing matrix element~\eqref{eq:chi_matrix} for
$f = \sin\chi$ is:
\begin{equation}
  \int_0^\pi \sin\chi \cdot d^{\ell'}_{m_1,m_2}(\chi)\,d^\ell_{m_1,m_2}(\chi)\,\sin\chi\,d\chi
  = \int_0^\pi d^{\ell'}_{m_1,m_2}(\chi)\,d^\ell_{m_1,m_2}(\chi)\,\sin^2\chi\,d\chi.
  \label{eq:sinchi_matrix}
\end{equation}
This connects $\ell' = \ell$ and $\ell' = \ell \pm 1$ (by the Clebsch--Gordan
rule for multiplication by $\sin\chi \sim Y_1^0$), and is non-zero for adjacent
levels.  The full spectral analysis of $\Delta_{S^3} + \zeta\sin\chi \cdot
\partial/\partial\phi$ is the primary open direction identified by this addendum.
\end{observation}

%==========================================================================
\section{Admissibility under R1--R8}
\label{sec:admissibility}
%==========================================================================

We check the three cases of $(\Hperp)^2$ and $f(\chi)\cdot\Hperp$ against R1--R8.

\begin{proposition}[$(\Hperp)^2$ is admissible under R1--R8]
\label{prop:admissibility}
Adding $c\cdot(\Hperp)^2$ to $\hat{O}$ is not excluded by any of R1--R8:
\begin{itemize}
  \item R2 (boundary dynamics on $S^3$): $(\Hperp)^2$ acts on $S^3$.  No conflict.
  \item R6 ($\mathbb{Z}_3$ family symmetry): $[(\partial/\partial\phi)^2, T] = 0$
    because $T$ acts as $\psi \mapsto \psi + 4\pi/3$ (pure $\psi$-shift) and
    $(\partial/\partial\phi)^2$ acts on $\phi$ only.  Verified for all
    $(D^\ell_{m_1,m_2})$ in \texttt{verify\_P190.py} Check~11.
  \item R1, R3--R5, R7, R8: identical analysis to P188 Theorem~5.1 for $\Hperp$;
    $(\Hperp)^2$ acts on the same angular sector as $\Hperp$ and introduces
    no new bulk, fibre, or radial structure.
\end{itemize}
The global smoothness criterion (P188 Remark~6.3): as a degree-2 polynomial in
$\partial/\partial\phi$, the operator $(\Hperp)^2$ has a different character than
the vector field $\Hperp$.  As an operator on $L^2(S^3)$, $(\partial/\partial\phi)^2$
is a well-defined self-adjoint operator (the negative azimuthal Laplacian).  It
does not have ``zeros'' in the vector-field sense; the criterion of P188 Remark~6.3
does not apply to second-order operators.
\end{proposition}

\begin{remark}[Missing physical constraint for $(\Hperp)^2$]
The absence of an R1--R8 veto does not establish that $(\Hperp)^2$ is
\emph{required} or even \emph{natural} in the assembly.  The canonical assembly
uses only operators derived from explicit TOE sector requirements; P184 derives
$\PertOp \leftrightarrow \zeta\Hunit$ via the boundary restriction of the
renormalization generator $\calR$ (Euler decomposition + Hopf boundary limit),
not by an $\fz$-completeness principle.  There is no analogous derivation of
$(\Hperp)^2$ from any TOE sector.
\end{remark}

%==========================================================================
\section{Verdict and Open Problem~7.1 Status}
\label{sec:verdict}
%==========================================================================

\begin{table}[ht]
\centering
\caption{Status of non-perturbative $\Hperp$ cases}
\label{tab:verdict}
\begin{tabular}{lll}
\toprule
Case & Spectral effect & Status \\
\midrule
$\varepsilon \cdot \Hperp$ (1st order) & eigenvalue shift $\kappa m_1$; $\kappa \approx 1.72$ needed & IMPLAUSIBLE (P188) \\
$\varepsilon \cdot \Hperp$ (2nd order) & identically zero & RULED OUT \\
$c \cdot f(\Hperp)$, fixed $m_1$ & cancels in $\Delta E$ for any $f$, $c$ & RULED OUT \\
$c \cdot (\Hperp)^2$, cross-$m_1$ & $\Delta E = 5 + c(m_1^2 - m_1'^2)$; $c^* \approx 1.72$ & NO SELECTION RULE \\
$\zeta(\Hunit + f_0 \Hperp)$, const $f_0$ & diagonal, inert & RULED OUT \\
$\zeta(\Hunit + f(\chi)\Hperp)$ & $\ell$-mixing, unknown spectrum & OPEN \\
\bottomrule
\end{tabular}
\end{table}

\begin{openproblem}[Open Problem~7.1 — reduced form]
\label{op:reduced}
The non-perturbative $\Hperp$ question reduces to the following after
Theorems~\ref{thm:second_order_zero}, \ref{thm:polynomial_inert},
and~\ref{thm:cross_m1}:

\begin{enumerate}[label=(\roman*)]
  \item \textbf{Cross-$m_1$ selection rule.}
    Is there a physical or geometric principle — derivable from the TOE or
    from an extended set of admissibility requirements beyond R1--R8 — that
    selects specific $m_1$ quantum numbers for the electron and muon?  If the
    answer is yes and the selected pair is among those in
    Theorem~\ref{thm:cross_m1}, then $c \cdot (\Hperp)^2$ with a
    naturally-sized coefficient solves the mass formula problem.  The near-miss
    $c^* \approx 1.720 \approx \sqrt{3}$ (to $0.7\%$) is suggestive; the
    derivation is open.
  \item \textbf{Smooth operadic composition.}
    What is the spectrum of $\Delta_{S^3} + \zeta\sin\chi \cdot \partial/\partial\phi$?
    Does it produce a first excited level with $\Delta E \approx 6.72$?  Is
    the assembly admissible under R1--R8?  The $\ell$-mixing induced by
    $\sin\chi \cdot \partial/\partial\phi$ couples $\ell$ and $\ell \pm 1$
    eigenspaces and could qualitatively restructure the spectrum; a numerical
    study or a Mathieu-equation reduction (the $\chi$-equation may reduce to a
    form of Hill's equation) is required.
  \item \textbf{Coefficient derivation.}
    If a cross-$m_1$ selection rule or smooth operadic composition produces a
    mass ratio close to 207 for some coefficient $c$, is that coefficient
    derivable from the TOE (e.g., from the Killing form, from the root geometry
    of $G_2 \subset F_4 \subset E_6$, or from the layer fractions)?
\end{enumerate}
\end{openproblem}

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

The main results of this addendum resolve the tractable parts of P188 Open
Problem~7.1 and identify the remaining open direction:

\begin{equation}
\boxed{
\begin{aligned}
&\varepsilon \cdot \Hperp \text{ (2nd-order)}: \quad E_n^{(2)} = 0 \;\text{ exactly} \\
&c \cdot f(\Hperp) \text{ (fixed-}m_1\text{)}: \quad \Delta E = 5 \;\text{ for all } f, c \\
&c \cdot (\Hperp)^2 \text{ (cross-}m_1\text{, Case A)}: \quad c^* \approx 1.720 \approx \sqrt{3} \;\text{ (no selection rule)} \\
&\zeta(\Hunit + f(\chi)\Hperp): \quad \text{OPEN — } \ell\text{-mixing, spectrum unknown}
\end{aligned}
}
\end{equation}

\noindent
The spectral-inertness theorem (Theorem~\ref{thm:polynomial_inert}) is the
central structural result: it shows that the entire polynomial algebra
generated by $\Hperp$ is inert for mass ratios under fixed-$m_1$ identification,
by exactly the same mechanism (cancellation in $\Delta E$) that makes
$V_\mathrm{self}$ inert (P187).  This places $\Hperp$ in the same category as
$V_\mathrm{self}$: structurally present, admissible under R1--R8, but unable
to affect the lepton mass ratio through any additive non-perturbative mechanism.

The residual open case — operadic composition with $f(\chi) \cdot \Hperp$ — is
qualitatively different.  It introduces $\ell$-mixing (connecting the electron
and muon sectors directly) and is not captured by the diagonal-operator analysis
of this addendum.  The smooth candidate $\sin\chi \cdot \partial/\partial\phi$
is globally smooth on $S^3$, admissible under R1--R8, and geometrically natural;
its spectral analysis is the primary open direction.

\textbf{Status of Open Problem~7.1:} partially resolved.  The second-order and
polynomial-diagonal sub-cases are closed (ruled out or spectrally inert).  The
cross-$m_1$ sub-case finds a naturally-sized coefficient but lacks a selection
rule.  The smooth operadic composition sub-case is open and constitutes the
primary remaining research direction.

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\end{document}
