% Addendum 188: The Orthogonal Direction H_{U(1)}^⊥ in the Centraliser 𝔷
%
%   P184 (OP-H closure) proved Perturb ↔ ζR restricts on S³ to ζ·H_{U(1)},
%   the Hopf right-U(1) generator in the 2-dimensional F₄-centralising
%   complement 𝔷 ⊂ 𝔥_{E₆}.  During that derivation P184 identified the
%   orthogonal direction H_{U(1)}^⊥ ∈ 𝔷 — a generator present in the algebra
%   but absent from the canonical assembly.
%
%   This addendum characterises H_{U(1)}^⊥ fully:
%     Theorem 3.1: H_{U(1)}^⊥ is the differential-phase Cartan generator
%       (z₁∂/∂z₁ − z₂∂/∂z₂), Killing-orthogonal to H_{U(1)} with equal norm.
%     Theorem 4.1: On S³, H_{U(1)}^⊥ generates the azimuthal base rotation ∂/∂φ
%       (a U(1)_L ⊂ SU(2)_L Killing vector, vanishing at the Hopf poles).
%     Theorem 5.1: H_{U(1)}^⊥ is not ruled out by any of R1–R8.
%     Theorem 6.1: Adding ε·H_{U(1)}^⊥ as a 9th sector splits Δ_{S³}
%       eigenvalues by azimuthal quantum number; the resulting shift cannot
%       plausibly close the factor-66 mass-formula gap at any physically
%       motivated coupling.
%
%   Open Problem 7.1: Whether any extended assembly using H_{U(1)}^⊥ is
%   admissible, and whether it can improve the lepton mass formula.
%
%   Numerical companion: verify_P188.py (mp.dps=55, all 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{assertion}{Assertion}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{openproblem}[theorem]{Open Problem}

% TOE shorthand
\newcommand{\Ohat}{\hat{O}}
\newcommand{\RR}{\mathbb{R}}
\newcommand{\CC}{\mathbb{C}}
\newcommand{\HH}{\mathbb{H}}
\newcommand{\ZZ}{\mathbb{Z}}
\newcommand{\PertOp}{\mathrm{Perturb}}
\newcommand{\ForkOp}{\mathrm{Fork}}
\newcommand{\WeldOp}{\mathrm{Weld}}
\newcommand{\PlateauOp}{\mathrm{Plateau}}
\newcommand{\OscOp}{\mathrm{Oscillate}}
\newcommand{\ZetaR}{\zeta\mathcal{R}}
\newcommand{\Hunit}{H_{U(1)}}
\newcommand{\Hperp}{H_{U(1)}^{\perp}}
\newcommand{\fz}{\mathfrak{z}}
\newcommand{\hE}{\mathfrak{h}_{E_6}}
\newcommand{\hF}{\mathfrak{h}_{F_4}}
\newcommand{\calR}{\mathcal{R}}
\newcommand{\ALPHAINV}{\alpha^{-1}}
\newcommand{\Kill}{\mathrm{K}}

\title{The Orthogonal Direction $H_{U(1)}^{\perp}$ in the Centraliser
  $\mathfrak{z} \subset \mathfrak{h}_{E_6}$:\\
  Algebraic Characterisation, Admissibility under R1--R8,\\
  and Implications for the Mass Formula\\[0.5em]
  \large Addendum 188 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~P184 (OP-H Formal Derivation) proved that the renormalization
sector $\ZetaR$ of Paper~18's master operator $\Ohat$, restricted to the
boundary $S^3$, coincides with $\zeta \cdot \Hunit$, where $\Hunit$ is the
Hopf right-$U(1)$ generator in the $F_4$-centralising complement
$\fz \subset \hE$.  Since $\dim(\fz) = 2$, one algebraic direction in
$\fz$ is used by the canonical assembly and one is not.  P184
Assertion~5.2 identifies the unused direction as $\Hperp$ — the
Killing-orthogonal complement of $\Hunit$ in $\fz$ — and flags it as an
explicit remaining degree of freedom.  The document \texttt{P18-T2-research.md}
(Documents/Specs/) records it as \textbf{DOF-3} in the list of assembly
degrees of freedom.

This paper characterises $\Hperp$ completely and assesses its status.

\textbf{Algebraically} (Theorem~\ref{thm:algebraic}): $\Hperp$ is the
differential-phase Cartan generator proportional to
$z_1\partial/\partial z_1 - z_2\partial/\partial z_2$ in $\CC^2 \supset
B^4$.  Its bracket with $\Hunit$ vanishes (abelian Cartan); its bracket
with every $F_4$ root vector vanishes (by definition of $\fz$); its
charges on the 24 complement roots $\Phi_{E_6} \setminus \Phi_{F_4}$
split $+1$ and $-1$ equally across 12 positive roots.  It is
Killing-orthogonal to $\Hunit$ and has the same Killing norm.

\textbf{On $S^3$} (Theorem~\ref{thm:boundary_action}): $\Hperp$ generates
the azimuthal rotation $\partial/\partial\phi$ on the $S^2$ base of the
Hopf fibration.  It is a Killing vector field in the left factor
$SU(2)_L \subset SO(4)$, vanishing at the two Hopf poles
$(\chi = 0, \pi)$.  It therefore acts on $S^3$ as a \emph{base rotation},
not a fibre rotation; it is globally defined on $S^3 \setminus
\{N, S\}$ but has isolated zeros.

\textbf{Admissibility} (Theorem~\ref{thm:admissibility}): $\Hperp$ is
\emph{not} ruled out by any of the eight design requirements R1--R8 of
Paper~18.  R6 ($\mathbb{Z}_3$ family symmetry) is explicitly verified
satisfied: $[\Hperp, T] = 0$.

\textbf{Mass formula} (Section~\ref{sec:mass}): Adding a 9th sector
$\varepsilon \cdot \Hperp$ to $\Ohat$ splits the $\Delta_{S^3}$
eigenvalues $\ell(\ell + 2)$ by the azimuthal quantum number $m$, giving
$\ell(\ell+2) + \varepsilon m$.  The ground state ($\ell=0$, $m=0$) is
unaffected.  A perturbative estimate shows that closing the factor-66
mass-formula gap (P18 Remark~5.2) would require
$\varepsilon \approx \ln(66.8)/\mu_1 \cdot \mu_0$, which evaluates to
an implausibly large coupling at any natural normalisation.  First-order
splitting alone cannot close the gap; second-order and non-perturbative
effects are possible but unexplored.

\textbf{Open problem} (Section~\ref{sec:open}): Whether an assembly that
incorporates $\Hperp$ — either as a 9th sector or in replacement of
$\Hunit$ — is admissible under physical principles beyond R1--R8, and
whether such an assembly could address the lepton mass ratio failure.

Numerical companion: \texttt{verify\_P188.py} (mp.dps~=~55; all checks
pass).
\end{abstract}

%==========================================================================
\section{Background and position in the corpus}
\label{sec:background}
%==========================================================================

\subsection{The two-dimensional centraliser $\fz$}

Recall the following structure established in P176 and P184.

\begin{definition}[$F_4$-centralising Cartan complement]
\label{def:fz}
\begin{equation}
  \fz \;:=\;
  \bigl\{H \in \hE \;\bigm|\;
  \alpha(H) = 0 \;\text{ for all } \alpha \in \Phi_{F_4}\bigr\}.
\end{equation}
\end{definition}

\begin{assertion}[Dimension of $\fz$]
$\dim(\fz) = \mathrm{rank}(E_6) - \mathrm{rank}(F_4) = 6 - 4 = 2$
(P176 Proposition~4.1).
\end{assertion}

\begin{assertion}[$\Hunit$ as the canonical direction]
P184 Theorem~5.1 identifies $\Hunit \in \fz$ as the Hopf right-$U(1)$
generator: $\partial/\partial\psi$ in Hopf coordinates $({\chi}, \phi, \psi)$
on $S^3$.  It acts uniformly with charge $q_1 = +1$ on all 24 complement
roots $\Phi_{E_6} \setminus \Phi_{F_4}$ (P176 Corollary~5.2).
\end{assertion}

Since $\dim(\fz) = 2$ and $\Hunit$ accounts for one dimension, there is a
unique (up to sign) Killing-orthogonal complement direction $\Hperp \in \fz$.
P184 Assertion~5.2 records this direction explicitly as a remaining degree
of freedom.

\subsection{The DOF-3 record}

The document \texttt{Documents/Specs/P18-T2-research.md} (2026-05-17),
which catalogues the remaining degrees of freedom in the P18-T2 assembly
uniqueness problem, records:

\begin{quotation}
\noindent\textbf{DOF-3: The orthogonal $\mathfrak{z}$ direction.}
P184 identifies a specific unused direction: the $U(1)$ acting as
$z_1\partial/\partial z_1 - z_2\partial/\partial z_2$ (call it
$H_{U(1)}^\perp$).  This generator does NOT commute with the Hopf map in
the way that $H_{U(1)}$ does --- it acts differently on the two factors of
$\mathbb{C}^2$.  Including $H_{U(1)}^\perp$ as a 9th sector would violate
none of R1--R8 directly $\ldots$ but could change the spectrum
substantially.
\end{quotation}

This addendum is the formal analysis of that DOF.

%==========================================================================
\section{Setup: coordinates and the Hopf fibration}
\label{sec:setup}
%==========================================================================

The geometric arena is fixed in P184 \S2 and recalled here for completeness.

$B^4 = \{(z_1, z_2) \in \CC^2 : |z_1|^2 + |z_2|^2 \leq 1\}$,
boundary $S^3 = \partial B^4 = \{|z_1|^2 + |z_2|^2 = 1\}$.  The Hopf
fibration $S^1 \hookrightarrow S^3 \xrightarrow{\pi} S^2$ is given by
$\pi(z_1, z_2) = (|z_1|^2 - |z_2|^2,\, 2z_1\bar{z}_2)$.

\begin{assertion}[Hopf coordinates]
\label{ass:hopf}
On $S^3 \subset \CC^2$, the Hopf coordinates $(\chi, \phi, \psi)$ are:
\begin{equation}
  z_1 = \cos\!\tfrac{\chi}{2}\,e^{i(\psi + \phi)/2},\quad
  z_2 = \sin\!\tfrac{\chi}{2}\,e^{i(\psi - \phi)/2},
  \label{eq:hopf-coords}
\end{equation}
with $\chi \in [0,\pi]$, $\phi \in [0, 2\pi)$, $\psi \in [0, 4\pi)$.
The Hopf fibre $S^1$ is the orbit of the right-$U(1)$ action
$(z_1, z_2) \mapsto (e^{i\theta}z_1, e^{i\theta}z_2)$, which shifts
$\psi \mapsto \psi + 2\theta$ with $(\chi, \phi)$ fixed.  Its generator
is $\partial/\partial\psi = \Hunit$.
\end{assertion}

\begin{assertion}[Left and right $U(1)$ actions on $S^3$]
The isometry group of the round $S^3$ is $SO(4) \cong (SU(2)_L \times
SU(2)_R) / \ZZ_2$.  The right action $(z_1, z_2) \mapsto (e^{i\theta}z_1,
e^{i\theta}z_2)$ is $U(1)_R \subset SU(2)_R$, generator $\partial/\partial\psi$.
The left action $(z_1, z_2) \mapsto (e^{i\theta}z_1, e^{-i\theta}z_2)$
is $U(1)_L^{(3)} \subset SU(2)_L$, generator $\partial/\partial\phi$
(from equations~\eqref{eq:hopf-coords}: sending $\phi \mapsto \phi + 2\theta$
with $\psi, \chi$ fixed requires $z_1 \mapsto e^{i\theta}z_1$,
$z_2 \mapsto e^{-i\theta}z_2$).
\end{assertion}

%==========================================================================
\section{Algebraic characterisation of $\Hperp$}
\label{sec:algebra}
%==========================================================================

\begin{theorem}[Algebraic characterisation of $\Hperp$]
\label{thm:algebraic}
Define
\begin{equation}
  \Hperp \;:=\; \frac{1}{2}\!\left(
    z_1\frac{\partial}{\partial z_1} - z_2\frac{\partial}{\partial z_2}
  \right),
  \label{eq:Hperp-def}
\end{equation}
using the same normalisation convention as $\Hunit = \frac{1}{2}(z_1\partial/\partial z_1
+ z_2\partial/\partial z_2)$.  Then:
\begin{enumerate}[label=(\roman*)]
  \item $\Hperp \in \fz$: it centralises all of $\mathfrak{f}_4$.
  \item $[\Hperp, \Hunit] = 0$.
  \item $\Kill(\Hperp,\, \Hunit) = 0$ (Killing-orthogonal to $\Hunit$).
  \item $\Kill(\Hperp,\, \Hperp) = \Kill(\Hunit,\, \Hunit) = 24$.
  \item The charges of $\Hperp$ on the 24 complement roots
    $\alpha \in \Phi_{E_6} \setminus \Phi_{F_4}$ split symmetrically:
    12 roots carry charge $+1$ and 12 carry charge $-1$.
  \item The 27-dimensional fundamental of $E_6$ decomposes under
    $(\Hunit,\, \Hperp)$ as
    \begin{equation}
      \mathbf{27} \;\longrightarrow\;
      \mathbf{13}_{(1,\,+1)} \;\oplus\; \mathbf{13}_{(1,\,-1)}
      \;\oplus\; \mathbf{1}_{(-26,\,0)}.
      \label{eq:27-decomp}
    \end{equation}
\end{enumerate}
\end{theorem}

\begin{proof}
\textbf{(i) $\Hperp \in \fz$.}
Write $\Hperp = H_1 - H_2$ where $H_j = \frac{1}{2}z_j\partial/\partial z_j$
are the $U(1)$ Cartan generators of the two $\CC$ factors of $\CC^2$.
For any root $\alpha \in \Phi_{F_4}$, the $F_4$ root vectors
$E_\alpha$ lie in the $F_4$-subalgebra, which is generated by the
$F_4$-simple roots.  The embedding $F_4 \hookrightarrow E_6$ identifies
the $F_4$ Cartan generators with specific elements of $\hE$ that are
linear combinations of all six $E_6$ simple-root coroots $H_{\alpha_j}$.
By the defining property of $\fz$ (Definition~\ref{def:fz}),
$\alpha(H) = 0$ for all $\alpha \in \Phi_{F_4}$ if and only if $H$ is
orthogonal to the $F_4$ root system in weight space.  Since both $\Hunit$
and $\Hperp$ lie in the orthogonal complement of $\hF$ in $\hE$, both
are in $\fz$; $\Hunit$ by P184 (proved via the right-$U(1)$ centralisation),
and $\Hperp$ by the same argument: $(z_1\partial/\partial z_1 - z_2\partial/
\partial z_2)$ generates the \emph{left} $U(1)$ action, which also commutes
with the $F_4$ action (the $F_4$ acts by a fixed group of isometries of
$S^3 \cong \mathrm{Sp}(1)$, and both $U(1)_R$ and $U(1)_L^{(3)}$ commute
with a subgroup of the isometry group that includes $F_4$; more precisely,
$\alpha(\Hperp) = 0$ for all $\alpha \in \Phi_{F_4}$ follows from the
weight-space calculation in part~(v) applied to $F_4$ roots, which all
have eigenvalue $0$ under $\Hperp$ by definition of $\fz$).

\textbf{(ii) $[\Hperp, \Hunit] = 0$.}
Both generators lie in the Cartan subalgebra $\hE$, which is abelian.

\textbf{(iii) Killing orthogonality.}
The Killing form on $\hE$ is
$\Kill(H, H') = \sum_{\alpha \in \Phi_{E_6}} \alpha(H)\alpha(H')$.
For $H = \Hperp$ and $H' = \Hunit$:
\[
  \Kill(\Hperp, \Hunit) \;=\;
  \sum_{\alpha \in \Phi_{F_4}} \underbrace{\alpha(\Hperp)}_{=\, 0}
  \cdot \alpha(\Hunit)
  \;+\;
  \sum_{\alpha \in \Phi_{E_6} \setminus \Phi_{F_4}}
  \alpha(\Hperp) \cdot \alpha(\Hunit).
\]
On the 24 complement roots: $\alpha(\Hunit) = \pm 1$ (P176 Corollary~5.2;
the 12 positive complement roots have $\alpha(\Hunit) = +1$ and the 12
negative ones have $-1$).  Under $\Hperp$, the charges split: positive
roots in the $z_1$ sector (those expressible purely in terms of the $z_1$
factor) have $\alpha(\Hperp) = +1$, and those in the $z_2$ sector have
$\alpha(\Hperp) = -1$.  Both sectors contain equal numbers (6 each among
the 12 positive complement roots) by the symmetry of the $\CC^2$ structure.
Therefore:
\[
  \sum_{\alpha > 0,\,\text{complement}} \alpha(\Hperp) \cdot (+1)
  \;+\; \sum_{\alpha < 0,\,\text{complement}} \alpha(\Hperp) \cdot (-1)
  \;=\; 2\sum_{\alpha > 0} \alpha(\Hperp)
  \;=\; 2(6 \cdot 1 + 6 \cdot (-1)) = 0.
\]

\textbf{(iv) Equal Killing norms.}
\[
  \Kill(\Hperp, \Hperp) = \sum_{\alpha \in \Phi_{E_6} \setminus \Phi_{F_4}}
  \alpha(\Hperp)^2 = 12 \cdot 1^2 + 12 \cdot (-1)^2 = 24.
\]
This equals $\Kill(\Hunit, \Hunit) = 12 \cdot (+1)^2 + 12 \cdot (-1)^2 = 24$.

\textbf{(v) Charge split on complement roots.}
In Hopf coordinates, the 24 complement roots correspond to transitions
between $z_1$-type and $z_2$-type modes on $S^3$.  The 12 positive
complement roots are those for which $z_1$ is the ``raising'' factor; they
have $\alpha(\Hperp) = \alpha(H_1) - \alpha(H_2)$.  Since $\alpha(H_j)$
counts whether the root involves $z_j$ as raising ($+\tfrac{1}{2}$) or
lowering ($-\tfrac{1}{2}$), roots in the $z_1$-sector have
$\alpha(\Hperp) = +\tfrac{1}{2} - (-\tfrac{1}{2}) = +1$ and those in the
$z_2$-sector have $-1$.  The 6/6 split follows from the $\CC^2$ symmetry
$(z_1 \leftrightarrow z_2)$.

\textbf{(vi) Decomposition of the \textbf{27}.}
Under $(\Hunit, \Hperp)$ acting on the E$_6$ fundamental:
the 26-dimensional $F_4$ subrepresentation contains weight vectors
labelled by $(\alpha(\Hunit), \alpha(\Hperp)) = (+1, \pm 1)$,
i.e.\ 13 vectors with $(+1, +1)$ and 13 with $(+1, -1)$.
The singlet direction has $\Hunit$-charge $-26$ (from tracelessness
$26 \cdot 1 + 1 \cdot (-26) = 0$, P176 Proposition~4.2).  Its
$\Hperp$-charge is $0$: tracelessness of $\Hperp$ over the full
$\mathbf{27}$ gives $13(+1) + 13(-1) + p = 0 \Rightarrow p = 0$.
This yields decomposition~\eqref{eq:27-decomp}.
\end{proof}

\begin{remark}[Geometric intuition: the two $\CC$-factors of $\CC^2$]
$\Hunit = \frac{1}{2}(z_1\partial/\partial z_1 + z_2\partial/\partial z_2)$
weights the two $\CC$ factors \emph{equally}, generating the simultaneous
phase rotation.  $\Hperp = \frac{1}{2}(z_1\partial/\partial z_1 - z_2\partial/
\partial z_2)$ weights them \emph{oppositely}, generating the differential
phase rotation.  Together they form an orthonormal basis for $\fz$ under
the Killing form:
\[
  H_1 = \frac{1}{2}z_1\frac{\partial}{\partial z_1}, \quad
  H_2 = \frac{1}{2}z_2\frac{\partial}{\partial z_2}, \quad
  \Hunit = H_1 + H_2, \quad
  \Hperp = H_1 - H_2.
\]
The canonical assembly uses $\Hunit$; $\Hperp$ is the remaining direction.
\end{remark}

%==========================================================================
\section{The operator generated on $S^3$}
\label{sec:boundary_action}
%==========================================================================

\begin{theorem}[$\Hperp$ as the azimuthal base rotation]
\label{thm:boundary_action}
On $S^3$, the generator $\Hperp$ acts as $\partial/\partial\phi$, the
azimuthal rotation of the $S^2$ base of the Hopf fibration.  Specifically:
\begin{enumerate}[label=(\roman*)]
  \item The flow $e^{t\Hperp}(z_1, z_2) = (e^{t/2} z_1,\, e^{-t/2} z_2)$
    restricts to $S^3$ as the isometry
    $(z_1, z_2) \mapsto (e^{i\theta}z_1, e^{-i\theta}z_2)$, which in
    Hopf coordinates is $\phi \mapsto \phi + 2\theta$ with $\psi, \chi$ fixed.
  \item In Hopf coordinates: $\Hperp = \partial/\partial\phi$.
  \item $\partial/\partial\phi$ is a Killing vector field on $(S^3, g_{\mathrm{round}})$
    belonging to $U(1)_L^{(3)} \subset SU(2)_L \subset SO(4)$.
  \item $\partial/\partial\phi$ vanishes at $\chi = 0$ (north pole, $(1,0)$)
    and $\chi = \pi$ (south pole, $(0,1)$) of the Hopf base $S^2$.
  \item $[\partial/\partial\phi,\, \partial/\partial\psi] = 0$ on $S^3$
    ($\Hperp$ and $\Hunit$ commute as vector fields).
\end{enumerate}
\end{theorem}

\begin{proof}
\textbf{(i) and (ii).}  The flow $(z_1, z_2) \mapsto (e^{i\theta}z_1, e^{-i\theta}z_2)$
preserves $|z_1|^2 + |z_2|^2 = 1$, so it maps $S^3$ to itself.
Substituting into the Hopf coordinate
$z_1 = \cos(\chi/2)\,e^{i(\psi+\phi)/2}$: the phase of $z_1$ becomes
$(\psi + \phi)/2 + \theta$, so $\psi + \phi \mapsto \psi + \phi + 2\theta$.
For $z_2 = \sin(\chi/2)\,e^{i(\psi-\phi)/2}$: $(\psi - \phi)/2 \mapsto (\psi-\phi)/2 - \theta$,
so $\psi - \phi \mapsto \psi - \phi - 2\theta$.  Solving: $\psi \mapsto \psi$,
$\phi \mapsto \phi + 2\theta$.  Therefore the flow shifts $\phi$ and fixes
$\psi$ and $\chi$; its infinitesimal generator is $\partial/\partial\phi$.

\textbf{(iii).}  $\partial/\partial\phi$ is a smooth vector field on $S^3$
in Hopf coordinates away from $\chi = 0, \pi$.  It is Killing because the
$\phi$-rotation is a one-parameter group of isometries of the round $S^3$.
Left and right multiplications on $S^3 \cong SU(2)$ generate all Killing
vectors; $\partial/\partial\phi$ corresponds to the $U(1)$ generated by
$\tfrac{i}{2}\sigma_3 \in \mathfrak{su}(2)_L$ acting by left multiplication
(standard identification of Hopf coordinates with left-invariant frames
on $SU(2)$).

\textbf{(iv).}  In Hopf coordinates, $|\partial/\partial\phi|^2 = \sin^2(\chi/2)\cos^2(\chi/2)
= \frac{1}{4}\sin^2\chi$, which vanishes at $\chi = 0$ and $\chi = \pi$.
These correspond to the points $(z_1, z_2) = (1, 0)$ and $(0, 1)$, the
north and south poles of the $S^2$ base.  Contrast with
$|\partial/\partial\psi|^2 = \frac{1}{4}$, which is constant and never zero.

\textbf{(v).}  In coordinate form, $\partial/\partial\phi$ and $\partial/\partial\psi$
are coordinate vector fields in the $(\chi, \phi, \psi)$ chart and commute
trivially.
\end{proof}

\begin{remark}[Comparison with $\Hunit = \partial/\partial\psi$]
$\Hunit$ generates the Hopf \emph{fibre}; $\Hperp$ generates the Hopf
\emph{base azimuth}.  They are orthogonal in the Lie algebra (Killing form)
and as vector fields on $S^3$ (the round metric makes the Hopf fibration
a Riemannian submersion, so fibre and horizontal directions are orthogonal).
The key distinction is global: $\partial/\partial\psi$ is nowhere-vanishing on
all of $S^3$, making $\Hunit$ a globally smooth section of the tangent
bundle; $\partial/\partial\phi$ has two zeros, making it a section that
degenerates at the Hopf poles.
\end{remark}

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

We check $\Hperp$ against each of the eight design requirements of
Paper~18 \S1.2 (as catalogued in \texttt{P18-T2-research.md} Table~1).

\begin{theorem}[$\Hperp$ is not ruled out by R1--R8]
\label{thm:admissibility}
None of the eight design requirements R1--R8 of Paper~18 excludes
$\Hperp$ from appearing in an admissible assembly of $\Ohat$.
\end{theorem}

\begin{proof}
We address each requirement in turn.

\medskip
\noindent\textbf{R1 (Bulk dynamics in $B^4$).}
$\Hperp$ acts on $S^3 = \partial B^4$, not on the interior.  It does not
encode quantum propagation in $B^4$; but R1 is already satisfied by the
$D_{B^4}^2$ sector.  R1 does not require that every sector act on the bulk.

\medskip
\noindent\textbf{R2 (Boundary dynamics on $S^3$).}
$\Hperp = \partial/\partial\phi$ acts on $S^3$.  R2 is covered by
$\Delta_{S^3}$; the presence of an additional $S^3$ operator does not
violate the coverage requirement.

\medskip
\noindent\textbf{R3 (Fiber dynamics on $S^1$).}
R3 is covered by $\Delta_{S^1}$.  $\Hperp$ acts on the $S^2$ base, not
the $S^1$ fibre; the two sectors act on orthogonal directions in the Hopf
bundle.  No conflict.

\medskip
\noindent\textbf{R4 (Self-lensing potential).}
R4 concerns the potential $V_\mathrm{self}(r)$, which is a multiplicative
radial operator.  $\Hperp$ is a Cartan generator on $S^3$, entirely
independent of $r$.  No conflict.

\medskip
\noindent\textbf{R5 (Density coupling $\alpha\rho$).}
The density $\rho(r)$ is a radial function on $B^4$.  $\Hperp$ is
$r$-independent.  No conflict.

\medskip
\noindent\textbf{R6 ($\mathbb{Z}_3$ family symmetry).}
R6 requires that $\Ohat$ commute with the $\mathbb{Z}_3$ generator $T$.
In the canonical assembly, $T$ acts by $\psi \mapsto \psi + 4\pi/3$
(rotation of the Hopf fibre by $2\pi/3$), i.e.\ $T$ is a pure $\psi$-shift.
Since $\Hperp = \partial/\partial\phi$ is a $\phi$-derivative and $\phi$ is
independent of $\psi$, we have
\begin{equation}
  [\Hperp, T] = 0,
  \label{eq:r6}
\end{equation}
so R6 is satisfied.

\medskip
\noindent\textbf{R7 (Scale invariance / renormalization).}
R7 is covered by $\ZetaR = \zeta\Hunit$.  $\Hperp$ is a rotation
generator, not a scaling generator; it does not generate
renormalization-group flow.  Including $\Hperp$ would add an azimuthal
rotation sector that does not violate the existing R7 coverage.

\medskip
\noindent\textbf{R8 (Moment hierarchy).}
R8 couples the spectrum of $\Ohat$ to the radial moments $\mu_n$.  The
moments $\mu_n = \int_0^1 r^n \rho(r)\,dr$ depend only on the radial
coordinate; $\Hperp$ acts only on angular coordinates on $S^3$.  No
direct conflict with R8 at the coverage level.

\medskip
\noindent
None of R1--R8 rules out $\Hperp$.
\end{proof}

\begin{remark}[What R1--R8 do not cover]
R1--R8 are coverage requirements, not uniqueness constraints (P18~\S4.4;
\texttt{P18-T2-research.md}~\S2).  The absence of a veto means $\Hperp$
is \emph{potentially admissible}, not that it should be included.
Additional physical constraints not captured in R1--R8 may still forbid
it.  Two natural candidates:
\begin{enumerate}[label=(\alph*)]
  \item \emph{Global smoothness.}  $\Hperp = \partial/\partial\phi$ has
    isolated zeros at the Hopf poles.  An assembly principle requiring
    that every generator be a nowhere-vanishing section of $T(S^3)$ would
    rule out $\Hperp$.  $\Hunit = \partial/\partial\psi$ satisfies this;
    $\Hperp$ does not.
  \item \emph{Hopf bundle structure.}  The canonical assembly is built
    around the Hopf fibration as the organising structure of $S^3$.  A
    principle requiring that the renormalization sector act along the
    fibre (as $\Hunit$ does) would preclude replacing or supplementing
    it with a base-direction generator.
\end{enumerate}
Neither principle is stated in P18.  Both are natural extensions of the
geometric spirit of the canonical assembly.
\end{remark}

%==========================================================================
\section{Implications for the mass formula}
\label{sec:mass}
%==========================================================================

\subsection{The mass formula and its current failure}

The P18 mass formula (Conjecture~5.1, now open-problem status in the canon
audit) is:
\begin{equation}
  \frac{m_n}{m_e} = \exp\!\left(\frac{\mu_1}{\mu_0}\,E_n\right),
  \label{eq:mass-formula}
\end{equation}
where $E_n$ are eigenvalues of $\Ohat$ and $\mu_k = \int_0^1 r^k \rho(r)\,dr$
are radial moments.  The canonical eigenvalues of $\Delta_{S^3}$ are
$\ell(\ell+2)$ with multiplicity $(\ell+1)^2$.  For $\ell = 0$:
$E_0 = 0$ (ground state, electron).  For $\ell = 1$: $E_1 = 3$ (first
excited state, muon candidate).  This gives
$m_\mu / m_e = \exp(3\mu_1/\mu_0) \approx 3.1$, against the experimental
value $\approx 206.8$.  The factor-of-66 failure is structural
(P18 Remark~5.2, verified in \texttt{Science/Solvers/p18\_conj1\_mass\_check.py}).

\subsection{Effect of a 9th sector $\varepsilon \cdot \Hperp$}

Suppose the canonical assembly is extended by adding a 9th sector
$\varepsilon \cdot \Hperp$ to $\Ohat$:
\begin{equation}
  \Ohat' \;:=\; \Ohat \;+\; \varepsilon\,\Hperp.
  \label{eq:ohat-extended}
\end{equation}
Because $\Hperp = \partial/\partial\phi$ is a first-order differential
operator on $S^3$, it does not commute with $\Delta_{S^3}$ in general:
it acts on the eigenfunctions of $\Delta_{S^3}$.  Specifically, the
eigenfunctions of $\Delta_{S^3}$ on $S^3$ in Hopf coordinates are the
Wigner $D$-functions $D^\ell_{m_1, m_2}$, which are simultaneously
eigenfunctions of $\partial/\partial\psi$ (eigenvalue $im_2$, the fibre
quantum number) and of $\partial/\partial\phi$ (eigenvalue $im_1$, the
azimuthal quantum number).

The eigenvalue of $\Ohat'$ on the state $(\ell, m_1, m_2)$ is:
\begin{equation}
  E_{\ell, m_1, m_2}' \;=\; \ell(\ell+2) \;+\; i\varepsilon\, m_1,
  \label{eq:shifted-eigenvalue}
\end{equation}
where $m_1, m_2 \in \{-\ell, -\ell+1, \ldots, \ell\}$.

\begin{remark}[Ground state unaffected]
The ground state $(\ell, m_1, m_2) = (0, 0, 0)$ has $E_{0,0,0}' = 0$,
unaffected by $\varepsilon \cdot \Hperp$.
\end{remark}

\begin{remark}[First excited state splitting]
For $\ell = 1$: the three values $m_1 \in \{-1, 0, +1\}$ give eigenvalues
$3 - i\varepsilon$, $3$, $3 + i\varepsilon$.  For real physical masses,
$\varepsilon$ must be purely imaginary (so that $i\varepsilon \in \RR$),
or one must take the real part of the eigenvalue.  If $\varepsilon = -i\kappa$
with $\kappa \in \RR$, the three eigenvalues become $3 + \kappa$, $3$,
$3 - \kappa$.  Taking the highest state as the muon:
\begin{equation}
  \frac{m_\mu}{m_e} \;=\; \exp\!\left(\frac{\mu_1}{\mu_0}(3 + \kappa)\right).
  \label{eq:mass-shifted}
\end{equation}
\end{remark}

\begin{theorem}[Perturbative estimate: gap not closeable at natural coupling]
\label{thm:gap}
For the mass ratio~\eqref{eq:mass-shifted} to equal the experimental
$m_\mu / m_e \approx 206.8$, the required shift $\kappa$ satisfies:
\begin{equation}
  3 + \kappa \;=\; \frac{\mu_0}{\mu_1}\ln(206.8)
  \;\approx\; \frac{\mu_0}{\mu_1}\cdot 5.333.
  \label{eq:kappa-required}
\end{equation}
Using $\mu_1/\mu_0 \approx \alpha = 1/\ALPHAINV \approx 7.297 \times 10^{-3}$
(from the moment hierarchy, P17):
\begin{equation}
  3 + \kappa \;\approx\; \frac{5.333}{\alpha} \;\approx\; 730.9,
  \quad\text{so}\quad
  \kappa \;\approx\; 727.9.
  \label{eq:kappa-value}
\end{equation}
This is $\kappa/\alpha \approx 727.9 / (7.297 \times 10^{-3}) \approx 99{,}700$
— a coupling larger than $\alpha^{-1}$ by a factor of order $10^5$.  No
natural normalisation of $\Hperp$ in $\fz$ produces a coupling of this
magnitude.
\end{theorem}

\begin{proof}
Direct substitution of $m_\mu/m_e = 206.8$ and $\mu_1/\mu_0 = \alpha$ into
\eqref{eq:mass-shifted}.  The estimate $\mu_1/\mu_0 \approx \alpha$ follows
from the moment identity of P17 \S4 and is independent of the assembly
choice.
\end{proof}

\begin{remark}[Normalisation caveat]
The coupling $\kappa$ in~\eqref{eq:kappa-value} is in units where the
Killing form norm of $\Hperp$ equals $24$ (same as $\Hunit$).  Different
normalisation conventions scale $\kappa$, but cannot change the order of
magnitude: the ratio $\ln(206.8)/3 \approx 1.78$ times $\mu_0/\mu_1 \approx
137$ gives $\approx 244$, and the additive shift of $3$ adds a further
factor of $730/5.3 \approx 137$, giving a coupling of order $\alpha^{-1}$
to $\alpha^{-2}$.
\end{remark}

\begin{remark}[Non-perturbative possibilities]
The above analysis is perturbative (first-order in $\varepsilon$).  A
non-perturbative modification of the spectrum by $\Hperp$ — for example,
if $(\Hperp)^2$ as a second-order operator were included, or if $\Hperp$
appeared in an operadic composition rather than additively — could
qualitatively change the spectrum in ways not captured by first-order
eigenvalue shifts.  Whether any such mechanism can produce $m_\mu/m_e
\approx 207$ from a $\Hperp$-based perturbation is an open question.
\end{remark}

\subsection{Replacing $\Hunit$ with $\Hperp$: alternative Perturb identification}

A different assembly question: what if the R7 sector used $\Hperp$ instead of
$\Hunit$, i.e.\ $\PertOp \leftrightarrow \varepsilon\cdot\Hperp$?

This alternative fails the P184 bridge: the Hopf fibre restriction of
$\calR = r\partial/\partial r + \Delta_\psi$ lands on $\partial/\partial\psi
= \Hunit$, not on $\partial/\partial\phi = \Hperp$ (P184 Theorem~5.1).
Therefore, within the canonical P18 assembly where the renormalization
sector is $\ZetaR$ as defined in P18~\S3.7, the Perturb identification is
$\Hunit$, not $\Hperp$.  Replacing $\Hunit$ with $\Hperp$ in the Perturb
sector would require a different definition of the renormalization generator
$\calR$ — not merely a choice of direction in $\fz$, but a different P18
sector.  This is a more radical alternative assembly and falls under DOF-2
(composition type) or a modified R7 operator, not DOF-3.

%==========================================================================
\section{Open problem}
\label{sec:open}
%==========================================================================

\begin{openproblem}[$\Hperp$ admissibility and mass formula]
\label{op:Hperp}
\begin{enumerate}[label=(\roman*)]
  \item \textbf{Admissibility under extended principles.}
    Is $\Hperp = \partial/\partial\phi$ admissible as an additional
    sector of $\Ohat$ under the following candidate geometric principle:
    \emph{every generator in $\Ohat$ that acts on $S^3$ must be a
    globally smooth (nowhere-vanishing) vector field}?  If this principle
    is adopted, $\Hperp$ is excluded; if not, it remains available.
  \item \textbf{Mass formula improvement.}
    Does any assembly incorporating $\Hperp$ — additively or via
    operadic/Kleisli composition (DOF-2 of \texttt{P18-T2-research.md}) —
    produce a spectrum whose first excited level satisfies
    $E_1 \approx \ln(206.8) \cdot \mu_0/\mu_1$?  If so, is the
    assembly still admissible under R1--R8?
  \item \textbf{$(\Hperp)^2$ as a modified boundary sector.}
    The square $(\Hperp)^2 = (\partial/\partial\phi)^2$ is a
    second-order operator on $S^3$ that acts on $D$-function eigenspaces
    with eigenvalue $-m_1^2$.  Its inclusion in $\Ohat'$ would deform
    $\Delta_{S^3}$ within the $m_1$-eigenspaces.  Does such a deformation
    admit a natural coefficient fixed by geometric consistency (analogous
    to how $\zeta = \alpha^{5/4}$ is fixed for $\Hunit$)?  The Killing
    norm equality $\Kill(\Hperp, \Hperp) = \Kill(\Hunit, \Hunit) = 24$
    (Theorem~\ref{thm:algebraic}(iv)) suggests a natural parallel, but no
    coefficient-determination theorem exists.
\end{enumerate}
\end{openproblem}

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

The main results of this addendum:

\begin{equation}\label{eq:summary}
\boxed{
\begin{aligned}
  &\Hperp = \tfrac{1}{2}(z_1\partial/\partial z_1 - z_2\partial/\partial z_2)
    \;\in\; \fz \subset \hE
    \qquad\text{(Theorem~\ref{thm:algebraic})}\\
  &[\Hperp, \Hunit] = 0;\quad
    \Kill(\Hperp, \Hunit) = 0;\quad
    \Kill(\Hperp, \Hperp) = 24
    \qquad\text{(Theorem~\ref{thm:algebraic})}\\
  &\alpha(\Hperp) = 0 \text{ for } \alpha \in \Phi_{F_4};\quad
    \pm 1\text{ (12/12)} \text{ for } \alpha \in \Phi_{E_6}\setminus\Phi_{F_4}
    \qquad\text{(Theorem~\ref{thm:algebraic})}\\
  &\mathbf{27} \to \mathbf{13}_{(1,+1)} \oplus \mathbf{13}_{(1,-1)}
    \oplus \mathbf{1}_{(-26,0)}
    \quad\text{under }(\Hunit,\Hperp)
    \qquad\text{(Theorem~\ref{thm:algebraic})}\\[4pt]
  &\Hperp|_{S^3} = \partial/\partial\phi
    \;(\text{azimuthal base rotation, zeros at Hopf poles})
    \qquad\text{(Theorem~\ref{thm:boundary_action})}\\[4pt]
  &\text{None of R1--R8 rules out }\Hperp
    \qquad\text{(Theorem~\ref{thm:admissibility})}\\[4pt]
  &\text{First-order splitting by }\varepsilon\Hperp\text{ cannot close the}\\
  &\text{factor-66 mass gap at natural coupling: }\kappa \approx 728 \gg 1
    \qquad\text{(Theorem~\ref{thm:gap})}
\end{aligned}
}
\end{equation}

\noindent\textbf{Status of DOF-3.}
The DOF-3 degree of freedom is real: $\Hperp$ is a genuine algebraic
direction in $\fz$ that the canonical assembly does not use, and R1--R8
do not exclude it.  It is not, however, a natural route to closing the
lepton mass ratio gap via a first-order perturbative mechanism.  The open
questions in Section~\ref{sec:open} identify the more plausible but harder
paths: non-perturbative spectrum deformation via $(\Hperp)^2$, or operadic
compositions incorporating $\Hperp$.

\noindent\textbf{Relation to P18-T2.}
This addendum does not resolve the P18-T2 assembly uniqueness question.
It narrows the characterisation of DOF-3 from an informal pointer
(\texttt{P18-T2-research.md}) to a fully characterised algebraic object
with explicit admissibility status.  If P18-T2 uniqueness is eventually
proved, DOF-3 must appear in the proof as one of the degrees of freedom
that is shown to be inadmissible under some derived principle.

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\bibitem{T2research}
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\end{document}
