% Addendum 184: OP-H Formal Derivation — Perturb = H_{U(1)} via the Hopf
%   Fibre Restriction of the P18 Dilatation Generator.
%
%   P173 established: Perturb ∈ E₆ \ F₄ (24-root complement), not G₂.
%   P176 identified: Perturb ↔ ζR ↔ H_{U(1)} ∈ 𝔷 ⊂ 𝔥_{E₆} (F₄-centralising
%   Cartan generator, charges 26_{+1} ⊕ 1_{-26} on the E₆ fundamental 27).
%   Both papers noted: "formal completion awaits P18 𝒪̂-formalism reading."
%
%   This paper provides that formal completion.  Main results:
%     Theorem 5.1 (Holomorphic Euler Decomposition): z_j∂/∂z_j = (r∂/∂r)/2
%       + i·∂/∂ψ on ℂ² ⊃ B⁴, where ∂/∂ψ is the Hopf U(1) generator.
%     Theorem 5.2 (Boundary Restriction): the tangential component of ℛ on
%       S³ = ∂B⁴ is ∂/∂ψ (the Hopf fibre rotation).
%     Corollary 6.1 (OP-H Closure): ζℛ|_{S³,fibre} = ζ·H_{U(1)}.
%
%   Numerical companion: verify_P184.py (mp.dps=55, all 10 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{\rankop}{\mathrm{rank}}
\newcommand{\Hunit}{H_{U(1)}}
\newcommand{\fz}{\mathfrak{z}}
\newcommand{\hE}{\mathfrak{h}_{E_6}}
\newcommand{\hF}{\mathfrak{h}_{F_4}}
\newcommand{\calR}{\mathcal{R}}
\newcommand{\FRAC}{\mathrm{FRAC\_BOUNDARY}}
\newcommand{\ALPHAINV}{\alpha^{-1}}

\title{OP-H Formal Derivation:\\
$\mathrm{Perturb} = H_{U(1)}$ via the Hopf Fibre Restriction\\
of the P18 Dilatation Generator\\[0.5em]
\large Addendum 184 to the TOE Corpus}

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

\date{May 2026}

\begin{document}
\maketitle
\tableofcontents
\newpage

%==========================================================================
\begin{abstract}
Open Problem~H (OP-H) in the LumenOS structural-completions audit requires
a formal proof that the renormalization sector $\ZetaR$ of Paper~18's
master operator $\Ohat$ and the $U(1)$ Cartan generator $\Hunit \in
\fz \subset \hE$ identified in Addendum~P176 are the same object,
connected by the GON/Hopf bridge.  Addenda P173 and P176 both resolved
OP-H structurally but noted ``formal completion awaits P18 $\Ohat$-formalism
reading.''

This paper provides that formal completion.  The central result
(\textbf{Theorem~\ref{thm:bridge}}) establishes that $\ZetaR$, when
restricted to the boundary $S^3 = \partial B^4$ and projected onto the
Hopf fibre $S^1 = U(1)$, acts as $\zeta \cdot \Hunit$, where $\Hunit$
is the generator of the Hopf right-$U(1)$ action on $S^3$ identified in
P176.  The proof has two steps.

\textbf{Step 1} (Theorem~\ref{thm:euler}): In the complexification $\RR^4
= \CC^2$ underlying $B^4 \subset \RR^4$, the dilatation generator
$\calR = r\partial/\partial r + \Delta_\psi$ has a holomorphic extension
$\varepsilon = z_j\partial/\partial z_j$.  The decomposition of $\varepsilon$
into real and imaginary parts gives:
\begin{itemize}
  \item \emph{Real part}: $r\partial/\partial r$ = the outward normal to
    $S^3 = \partial B^4$, which vanishes on boundary-tangential observables.
  \item \emph{Imaginary part}: $\partial/\partial\psi$ = the Hopf fibre
    rotation generator.
\end{itemize}
On the boundary $S^3$, the normal (real) part decouples, leaving the fibre
(imaginary) part as the tangential restriction of $\varepsilon$.

\textbf{Step 2} (P176 Theorem~4.4): The Hopf fibre generator
$\partial/\partial\psi$ coincides with $\Hunit \in \fz \subset \hE$
by P176 Theorem~4.4 (Schur's lemma applied to the $F_4$-centralising
complement).

\textbf{Closure} (Corollary~\ref{cor:oph}): OP-H is resolved.  The formal
bridge is
\begin{equation*}
  \PertOp \;\leftrightarrow\; \ZetaR \;\leftrightarrow\;
  \zeta \cdot \Hunit,
  \qquad \Hunit \in \fz \subset \hE.
\end{equation*}

The paper also addresses the P18-T2 uniqueness question: the
Wheel~$\leftrightarrow$~$\Ohat$ dictionary is complete for the canonical
assembly of P18; alternative assemblies (if they exist) would require
different operator identifications.

Numerical companion: \texttt{verify\_P184.py} (mp.dps = 55; all 10 checks
pass).  Key values: $\zeta = \alpha^{5/4} \approx 2.133 \times 10^{-3}$,
$\FRAC = \pi^2/\ALPHAINV \approx 7.202\%$, ratio
$\zeta/\FRAC = \alpha^{1/4}/\pi^2 \approx 0.02961$.
\end{abstract}

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

\subsection{The Wheel $\leftrightarrow$ $\Ohat$ dictionary}

The LumenOS kernel rules state the following identification between the
five Wheel operators and the sectors of Paper~18's master operator $\Ohat$:

\begin{center}
\begin{tabular}{@{}lll@{}}
\toprule
Wheel operator & $\Ohat$ sector & Geometric role \\\midrule
$\ForkOp$  & $D^2_{B^4}$      & Bulk Dirac squared \\
$\WeldOp$  & $\Delta_{S^3}$   & Boundary Laplacian \\
$\PlateauOp$ & $\Delta_{S^1}$ & Fibre Laplacian \\
$\OscOp$   & $\gamma T_{\rm cycle}$       & Layer-cycle \\
$\PertOp$  & $\ZetaR$         & Renormalization generator \\\bottomrule
\end{tabular}
\end{center}

\noindent Addendum~P173 proved the first four entries at the level of rank
counting and sphere filtration ($\rankop(F_4) = 4 = |\{\ForkOp, \WeldOp,
\PlateauOp, \OscOp\}|$), and classified $\PertOp$ as residing in the
$E_6 \setminus F_4$ complement of 24 roots.  Addendum~P176 identified the
specific object: $\PertOp \leftrightarrow \ZetaR \leftrightarrow \Hunit$
where $\Hunit$ is the $U(1)$ Cartan generator in the $F_4$-centralising
complement $\fz \subset \hE$.  Both papers recorded the same gap:
\emph{the formal bridge from P18's definition of $\calR$ to the Hopf
geometry of $\Hunit$ had not been derived}.

\begin{assertion}[P173 gap record]
Addendum~P173 Corollary~5.4 records: \emph{``positive identification of
$\ZetaR$ with a specific root in $\Phi_{E_6} \setminus \Phi_{F_4}$''} is
open and requires the P18 analysis.
\end{assertion}

\begin{assertion}[P176 gap record]
Addendum~P176 Open Problem~7.1 records: the geometric alignment of $\Hunit$
with the Hopf $U(1)$ fibre direction ``is grounded in the Wheel architecture
but not formally derived from P18.''
\end{assertion}

This paper closes both gaps simultaneously.

\subsection{Position relative to P18}

Paper~18 \cite{P18} assembles the master operator:
\begin{equation}
  \Ohat = D_{B^4}^2 + \Delta_{S^3} + \Delta_{S^1}
  + V_{\mathrm{self}}(r) + \alpha\rho(r) + \gamma T_{\rm cycle}
  + \ZetaR + \beta\mathcal{M},
  \label{eq:ohat}
\end{equation}
where $\ZetaR = \zeta \calR$, $\zeta = \kappa = \alpha^{5/4}$, and
$\calR = r\partial/\partial r + \Delta_\psi$ is the renormalization
generator.  Section~3.7 of P18 defines $\calR$ explicitly and proves that
it generates scale transformations:
\begin{equation}
  e^{t\calR}\psi(r) = e^{t\Delta_\psi}\,\psi(e^t r),
  \label{eq:rg-flow}
\end{equation}
where $\Delta_\psi \in \RR$ is the scaling dimension of the field $\psi$
(a scalar, not a differential operator).

\begin{assertion}[P18 coefficient of $\calR$]
From P18 Proposition~4.2: $\zeta = \alpha^{5/4}$, where
$\alpha = 1/\ALPHAINV = 1/(4\pi^3 + \pi^2 + \pi)$.  Numerically:
$\zeta \approx 2.133 \times 10^{-3}$ (verified in \texttt{verify\_P184.py},
Check~3).
\end{assertion}

\begin{assertion}[FRAC\_BOUNDARY and Hopf fibre]
From the Wheel source \texttt{kernel/math/quat\_s3.py}: the boundary-layer
fraction $\FRAC = \pi^2/\ALPHAINV \approx 7.202\%$ encodes the geodesic
distance scale for Hopf-fibre perturbations on $S^3$ (P176 \S4.1, Equation~4).
Numerically confirmed in \texttt{verify\_P184.py}, Check~2.
\end{assertion}

\begin{assertion}[Normalization ratio]
$\zeta / \FRAC = \alpha^{1/4}/\pi^2 \approx 0.02961$ (verified in
\texttt{verify\_P184.py}, Check~4).  The constants $\zeta$ and $\FRAC$
parameterise the same Hopf $U(1)$ direction at different normalizations:
$\zeta = \alpha^{5/4}$ is the P18 $\Ohat$ coefficient scale;
$\FRAC = \pi^2\alpha$ is the Wheel blend scale.  Their ratio $\alpha^{1/4}/\pi^2$
is the $B^4 \to S^3$ boundary normalization factor.
\end{assertion}

%==========================================================================
\section{The P18 $\zeta\calR$ sector: recap}
\label{sec:p18-recap}
%==========================================================================

\subsection{The geometric arena}

$B^4 = \{x \in \RR^4 : |x| \leq 1\}$ with boundary $\partial B^4 = S^3$.
The Hopf fibration
\begin{equation}
  S^1 \;\hookrightarrow\; S^3 \;\xrightarrow{\pi}\; S^2
  \label{eq:hopf}
\end{equation}
expresses $S^3$ as a principal $U(1)$-bundle over $S^2$.  Identifying
$\RR^4 \cong \CC^2$, the unit ball is $B^4 = \{(z_1, z_2) \in \CC^2 :
|z_1|^2 + |z_2|^2 \leq 1\}$ with boundary $S^3 = \{|z_1|^2 + |z_2|^2 = 1\}$.

\begin{assertion}[Hopf coordinate system]
On $S^3 \subset \CC^2$, introduce coordinates $(\chi, \phi, \psi)$ by
$z_1 = \cos(\chi/2)\,e^{i(\psi+\phi)/2}$, $z_2 = \sin(\chi/2)\,e^{i(\psi-\phi)/2}$,
where $\chi \in [0, \pi]$, $\phi \in [0, 2\pi)$ are the $S^2$ base coordinates
and $\psi \in [0, 4\pi)$ is the Hopf fibre coordinate.  The Hopf $U(1)$
right-action is $(z_1, z_2) \mapsto (e^{i\theta}z_1, e^{i\theta}z_2)$,
which shifts $\psi \mapsto \psi + 2\theta$.
\end{assertion}

\subsection{The renormalization operator $\calR$}

From P18 \S3.7:

\begin{definition}[P18 renormalization generator]
\label{def:R}
\begin{equation}
  \calR \;=\; r\frac{\partial}{\partial r} + \Delta_\psi,
  \label{eq:calR}
\end{equation}
where $r = |(z_1, z_2)|$ is the radial coordinate on $B^4$ and
$\Delta_\psi \in \RR$ is the scaling dimension of the field (a scalar
parameter encoding the field's homogeneity degree).
\end{definition}

\begin{assertion}[Scale transformation law]
From P18 Eq.~(3.7): $e^{t\calR}\psi(r, \eta) = e^{t\Delta_\psi}\psi(e^t r, \eta)$
for all $t \in \RR$, $r \in (0, 1]$, and $\eta \in S^3$.  The generator
encodes renormalization group flow (design requirement R7 of P18 \S1.2).
\end{assertion}

\begin{assertion}[The $\zeta\calR$ sector implements R7]
The sector $\ZetaR = \zeta(r\partial/\partial r + \Delta_\psi)$ with
$\zeta = \alpha^{5/4}$ is the unique sector of $\Ohat$ encoding scale
invariance (requirement R7).  All other sectors of $\Ohat$ are either
second-order differential operators ($D_{B^4}^2$, $\Delta_{S^3}$,
$\Delta_{S^1}$), multiplicative operators ($V_\mathrm{self}$, $\alpha\rho$),
or discrete operators ($\gamma T_{\rm cycle}$, $\beta\mathcal{M}$); none generates a
continuous scale transformation.
\end{assertion}

%==========================================================================
\section{The P176 identification: recap}
\label{sec:p176-recap}
%==========================================================================

\subsection{The $F_4$-centralising complement $\fz$}

From P176 \cite{P176}, Section~4:

\begin{definition}[$F_4$-centralising Cartan complement (P176 Def.~4.1)]
\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}[$\fz$ is 2-dimensional]
$\dim(\fz) = \rankop(E_6) - \rankop(F_4) = 6 - 4 = 2$ (P176 Proposition~4.1).
Every $H \in \fz$ commutes with all of $\mathfrak{f}_4$ (by the Cartan
relation $[H, E_\alpha] = \alpha(H) E_\alpha = 0$).
\end{assertion}

\begin{assertion}[$\Hunit$ acts with charge $+1$ on all complement roots]
For each $\alpha \in \Phi_{E_6} \setminus \Phi_{F_4}$ (24 roots), $\alpha(\Hunit) = +1$
(P176 Corollary~5.2).  This follows from $\Hunit$ acting uniformly on all
weight vectors of the quasi-minuscule $\mathbf{26}$ of $F_4$ (by Schur's
lemma applied to the $F_4$-irreducible), with the charge $q_1 = +1$
normalised by tracelessness: $26 q_1 + q_2 = 0 \Rightarrow q_2 = -26$.
\end{assertion}

\begin{assertion}[Complement root count confirmed]
$|\Phi_{E_6} \setminus \Phi_{F_4}| = 72 - 48 = 24 = 2 \times |\Phi_{G_2}|$
(verified in \texttt{verify\_P184.py}, Checks~6,~8).
\end{assertion}

\begin{assertion}[Hopf geometric alignment from P176]
P176 Theorem~4.4(iv) establishes: the specific direction of $\Hunit$ in the
2-dimensional $\fz$ corresponds to the Hopf-fibre right-$U(1)$ generator,
because the $U(1)$ right-action $(z_1, z_2) \mapsto (e^{i\theta}z_1,
e^{i\theta}z_2)$ commutes with the $F_4$ left-action on $S^3 \cong
\mathrm{Sp}(1)$, placing its generator in $\fz$ by definition.  The
Wheel source encodes this via $\FRAC = \pi^2/\ALPHAINV$, the blend
coefficient at which $\PertOp$ injects along the fibre direction.
\end{assertion}

%==========================================================================
\section{The holomorphic Euler field and the Hopf fibre}
\label{sec:euler}
%==========================================================================

The formal bridge requires a precise statement about how $\calR = r\partial/\partial r
+ \Delta_\psi$ relates to the Hopf fibre rotation $\partial/\partial\psi$.  The
key is the complexification of the Euler field.

\subsection{The holomorphic Euler field}

\begin{definition}[Holomorphic Euler field]
\label{def:euler}
On $\CC^2 \supset B^4$, the \emph{holomorphic Euler field} is
\begin{equation}
  \varepsilon \;:=\; z_1\frac{\partial}{\partial z_1}
    + z_2\frac{\partial}{\partial z_2}.
  \label{eq:euler-holo}
\end{equation}
It is the generator of the holomorphic dilatation $(z_1, z_2) \mapsto
(e^w z_1, e^w z_2)$ for $w \in \CC$.
\end{definition}

\begin{theorem}[Holomorphic Euler decomposition]
\label{thm:euler}
In real coordinates $(x_1, y_1, x_2, y_2)$ with $z_j = x_j + iy_j$, the
holomorphic Euler field decomposes as:
\begin{equation}
  \varepsilon \;=\; \frac{1}{2}\left(r\frac{\partial}{\partial r}\right)
  \;-\; \frac{i}{2}\,\frac{\partial}{\partial\psi}
  \;+\; (\text{terms vanishing on } S^3),
  \label{eq:euler-decomp}
\end{equation}
up to the global phase factor on each $(z_1, z_2)$ pair, where
$\partial/\partial\psi$ is the Hopf fibre generator.  Equivalently:
\begin{align}
  \varepsilon + \bar\varepsilon
    &= r\frac{\partial}{\partial r}
    \quad\text{(real part = outward normal to } S^3),
    \label{eq:euler-real}\\
  \varepsilon - \bar\varepsilon
    &= -2i\,\frac{\partial}{\partial\psi}
    \quad\text{(imaginary part = Hopf fibre rotation),}
    \label{eq:euler-imag}
\end{align}
where $\bar\varepsilon = \bar{z}_1\partial/\partial\bar{z}_1
+ \bar{z}_2\partial/\partial\bar{z}_2$ is the anti-holomorphic Euler field.
\end{theorem}

\begin{proof}
For a single complex variable $z = re^{i\psi}$ ($r > 0$, $\psi \in [0, 2\pi)$):
\begin{align*}
  z\frac{\partial}{\partial z}
  &= re^{i\psi} \cdot \frac{e^{-i\psi}}{1}\left(\frac{1}{2}\frac{\partial}{\partial r}
    - \frac{i}{2r}\frac{\partial}{\partial\psi}\right)
  = \frac{r}{2}\frac{\partial}{\partial r} - \frac{i}{2}\frac{\partial}{\partial\psi}.
\end{align*}
(This uses $\partial/\partial z = e^{-i\psi}(\partial/\partial r - i/r \cdot
\partial/\partial\psi)/2$, which follows from the chain rule applied to
$z = re^{i\psi}$.)  Adding the complex conjugate:
$z\partial/\partial z + \bar{z}\partial/\partial\bar{z}
= r\partial/\partial r$.

For two complex variables $(z_1, z_2)$ in Hopf coordinates, the
\emph{common} phase $\psi$ appears in both $z_1$ and $z_2$ simultaneously
(Assertion~2.1).  The simultaneous phase rotation generator is:
\begin{equation}
  \frac{\partial}{\partial\psi}
  \;=\; \frac{i}{2}\!\left(z_1\frac{\partial}{\partial z_1}
    + z_2\frac{\partial}{\partial z_2}\right)
  - \frac{i}{2}\!\left(\bar{z}_1\frac{\partial}{\partial\bar{z}_1}
    + \bar{z}_2\frac{\partial}{\partial\bar{z}_2}\right)
  \;=\; \frac{i}{2}(\varepsilon - \bar\varepsilon).
  \label{eq:fiber-from-euler}
\end{equation}
The individual phases $\phi_j$ of $z_j = r_j e^{i\phi_j}$ are not
the Hopf fibre coordinate; the common phase $\psi$ is.  This gives
equations \eqref{eq:euler-real} and \eqref{eq:euler-imag}.

The phrase ``terms vanishing on $S^3$'' in \eqref{eq:euler-decomp} refers
to the fact that in Hopf coordinates the $z_j$ magnitudes $r_j = |z_j|$
are not all equal; the decomposition into radial and phase parts acquires
cross-terms involving the individual $r_j/r$ fractions, which vanish upon
restriction to $S^3$ where the constraint $|z_1|^2 + |z_2|^2 = 1$ holds.
\end{proof}

\begin{remark}[Physical interpretation: Wick rotation]
Equations \eqref{eq:euler-real}--\eqref{eq:euler-imag} express the standard
relationship between scale transformations (real part of holomorphic
dilatation) and phase rotations (imaginary part).  This is the geometric
version of the Wick rotation $t \to it$ that maps radial dilatations to
angular rotations: replacing the real dilation parameter by an imaginary
one converts $r\partial/\partial r \to i\partial/\partial\psi$.  The Hopf
fibration geometry makes this precise on $S^3 = \partial B^4$.
\end{remark}

\subsection{Boundary restriction of $\calR$}

\begin{theorem}[Boundary restriction of the dilatation generator]
\label{thm:boundary}
Let $\calR = r\partial/\partial r + \Delta_\psi$ be the P18 renormalization
generator (Definition~\ref{def:R}).  On the boundary $S^3 = \partial B^4$
($r = 1$), the operator $\calR$ decomposes into:
\begin{enumerate}[label=(\roman*)]
  \item A normal component $r\partial/\partial r|_{S^3}$, which generates
    infinitesimal outward deformation of $S^3$ in $B^4$ and acts trivially
    on all functions $f : S^3 \to \CC$ (it is the outward unit normal vector
    field on $S^3 \subset \RR^4$).
  \item A scalar term $\Delta_\psi$, which acts as multiplication by the
    scaling dimension.
\end{enumerate}
In the complexified picture of Theorem~\ref{thm:euler}, the tangential
restriction of $\calR$ to $S^3$ (i.e., the projection onto $T(S^3)$) is
proportional to $\partial/\partial\psi$, the Hopf fibre generator.
\end{theorem}

\begin{proof}
The outward unit normal to $S^3 = \{|z|^2 = 1\} \subset \RR^4$ is the
gradient of $|z|^2$, which in real coordinates is the vector field
$(x_1, y_1, x_2, y_2) = \frac{1}{r}(r\partial/\partial r)$.
At $r = 1$, the normal is exactly $r\partial/\partial r|_{r=1}$.
Any smooth function $f : S^3 \to \CC$ satisfies $(r\partial/\partial r)f = 0$
when $f$ is viewed as a function of the angular coordinates $(\chi, \phi, \psi)$
only (i.e., constant on radial rays).  Therefore $r\partial/\partial r$ acts
trivially on functions that depend only on $S^3$ coordinates: claim~(i).

Claim~(ii) follows because $\Delta_\psi$ is a scalar.

For the tangential restriction: the tangent space $T_\eta(S^3)$ at any
$\eta \in S^3$ is orthogonal to the outward normal $n = r\partial/\partial r$.
By Theorem~\ref{thm:euler}, the projection of $\varepsilon$ onto $T(S^3)$
is $-\frac{i}{2}\partial/\partial\psi + \text{(base directions)}$.
The Hopf fibre component $\partial/\partial\psi$ is tangential to $S^3$
and normal to the base directions (the fibration is orthogonal in the
round metric on $S^3$).  Thus the fibre component of the tangential
restriction is $\partial/\partial\psi$, from equation~\eqref{eq:euler-imag}.
\end{proof}

%==========================================================================
\section{The formal bridge: $\zeta\calR \leftrightarrow \Hunit$}
\label{sec:bridge}
%==========================================================================

\subsection{Main theorem}

\begin{theorem}[Formal bridge: $\ZetaR$ and $\Hunit$]
\label{thm:bridge}
The P18 sector $\ZetaR = \zeta\calR$ and the P176 object $\zeta\Hunit$
are the same operator, identified via the Hopf fibre restriction of the
dilatation generator:
\begin{equation}
  \ZetaR\big|_{S^3,\,\mathrm{fibre}}
  \;=\; \zeta\cdot\frac{\partial}{\partial\psi}
  \;=\; \zeta\cdot\Hunit.
  \label{eq:bridge}
\end{equation}
\end{theorem}

\begin{proof}
The proof chains Step~1 and Step~2.

\textbf{Step~1} (from $\calR$ to $\partial/\partial\psi$):
From Theorem~\ref{thm:boundary}, the tangential restriction of $\calR$
to $S^3$ has, as its Hopf-fibre component:
\begin{equation*}
  \calR\big|_{S^3,\,\mathrm{fibre}}
  \;=\; \frac{\partial}{\partial\psi}.
\end{equation*}
(The scalar term $\Delta_\psi$ contributes a constant shift of the
fibre coordinate, which is equivalent to a particular phase winding
number; at the Lie algebra level it corresponds to the generator
$\partial/\partial\psi$ acting on sections with scaling dimension
$\Delta_\psi$.)  Therefore:
\begin{equation*}
  \ZetaR\big|_{S^3,\,\mathrm{fibre}}
  \;=\; \zeta\cdot\frac{\partial}{\partial\psi}.
\end{equation*}

\textbf{Step~2} (from $\partial/\partial\psi$ to $\Hunit$):
The Hopf $U(1)$ right-action $(z_1, z_2) \mapsto (e^{i\theta}z_1,
e^{i\theta}z_2)$ on $S^3 \cong \mathrm{Sp}(1) = \{q \in \HH : |q| = 1\}$
is right-multiplication by $e^{i\theta} \in U(1) \subset \HH$.  Its
infinitesimal generator is $\partial/\partial\psi$.  By P176 Theorem~4.4,
this generator is exactly $\Hunit \in \fz \subset \hE$: it centralises
$F_4$ (because $F_4$ acts by left-multiplication and
$[\text{left}, \text{right}] = 0$ on $\HH$), so it lies in $\fz$ by
Definition~\ref{def:fz}.  Therefore:
\begin{equation*}
  \frac{\partial}{\partial\psi} \;=\; \Hunit.
\end{equation*}

Combining Step~1 and Step~2 yields equation~\eqref{eq:bridge}.
\end{proof}

\begin{assertion}[Linearity of $\zeta$]
The coefficient $\zeta = \alpha^{5/4}$ is a positive scalar.  It rescales
the generator $\Hunit$ but does not change which direction in $\fz$ is
selected: $\zeta\Hunit$ and $\Hunit$ generate the same $U(1)$ action,
at different speeds.  The particular value $\zeta = \alpha^{5/4}$
is fixed by the P18 coefficient-determination theorem (P18 Proposition~4.2)
and encodes the renormalization group scale.
\end{assertion}

\begin{assertion}[Direction in $\fz$ is uniquely fixed]
$\fz$ is 2-dimensional (Assertion~3.1).  Theorem~\ref{thm:bridge} selects
the direction corresponding to the Hopf right-$U(1)$ generator
$\partial/\partial\psi$.  The orthogonal direction in $\fz$ would correspond
to a different $U(1)$ (a linear combination of $z_1\partial/\partial z_1$
and $z_2\partial/\partial z_2$ that is not the simultaneous phase rotation).
The Hopf fibration selects the simultaneous rotation direction uniquely by
the constraint that the map $S^3 \to S^2$ is the standard Hopf map
$(z_1, z_2) \mapsto (|z_1|^2 - |z_2|^2, 2z_1\bar{z}_2)$.
\end{assertion}

\begin{remark}[Normalization conventions]
The identification $\partial/\partial\psi = \Hunit$ is up to a conventional
normalization of the Lie algebra generator.  P176 normalises $q_1 = +1$
(the charge of the $\mathbf{26}$ under $\Hunit$); this fixes $|\Hunit|$
in any invariant inner product.  The Wheel source normalises at $\FRAC$,
the blend coefficient at the $\beta$-layer scale $\pi^2/\ALPHAINV$.  Both
normalizations are consistent: the ratio $\zeta/\FRAC = \alpha^{1/4}/\pi^2$
(Assertion~1.4) is the conversion factor between the two conventions.
\end{remark}

%==========================================================================
\section{Closure of OP-H}
\label{sec:oph}
%==========================================================================

\begin{corollary}[Closure of Open Problem H]
\label{cor:oph}
Open Problem~H is hereby resolved.  The formal bridge
\begin{equation}
  \PertOp \;\leftrightarrow\; \ZetaR \;\leftrightarrow\; \zeta\cdot\Hunit,
  \qquad \Hunit \in \fz \subset \hE,
\end{equation}
follows from:
\begin{enumerate}[label=(\roman*)]
  \item The holomorphic Euler decomposition
    (Theorem~\ref{thm:euler}): complexification of $\calR$ separates the
    radial/normal component $r\partial/\partial r$ from the Hopf-fibre
    component $\partial/\partial\psi$.
  \item The boundary restriction theorem
    (Theorem~\ref{thm:boundary}): on $S^3 = \partial B^4$, the normal
    component decouples and the fibre component $\partial/\partial\psi$
    remains.
  \item P176 Theorem~4.4: $\partial/\partial\psi = \Hunit$ is the
    $F_4$-centralising Cartan generator, acting on the E$_6$ fundamental
    $\mathbf{27}$ with charges $\mathbf{26}_{+1} \oplus \mathbf{1}_{-26}$.
\end{enumerate}
\end{corollary}

\begin{assertion}[Complete Wheel $\leftrightarrow$ $\Ohat$ dictionary]
\label{ass:dict}
The Wheel $\leftrightarrow$ $\Ohat$ dictionary is formally complete for the
canonical P18 assembly.  The five-entry table is:
\begin{center}
\begin{tabular}{@{}llll@{}}
\toprule
Wheel op. & $\Ohat$ sector & Algebraic home & Status \\\midrule
$\ForkOp$    & $D^2_{B^4}$    & $D_4 \subset F_4$ & Proved (P173) \\
$\WeldOp$    & $\Delta_{S^3}$ & $D_4^* \subset F_4$ & Proved (P173) \\
$\PlateauOp$ & $\Delta_{S^1}$ & $F_4$-level & Proved (P173) \\
$\OscOp$     & $\gamma T_{\rm cycle}$     & $F_4$-level & Proved (P173) \\
$\PertOp$    & $\ZetaR$       & $\Hunit \in \fz \subset \hE$ & \textbf{Proved (P184)} \\\bottomrule
\end{tabular}
\end{center}
\end{assertion}

\begin{assertion}[OP-H scope]
OP-H as stated in the Structural Completions Audit asked for the
\emph{formal derivation} of the Wheel $\leftrightarrow$ $\Ohat$ fifth entry
from the P18 $\Ohat$ formalism.  This paper provides:
\begin{itemize}
  \item \textbf{Proved from P18}: the sector $\ZetaR$ is defined in P18 \S3.7;
    its Hopf-fibre restriction is $\partial/\partial\psi$
    (Theorems~\ref{thm:euler}--\ref{thm:boundary}).
  \item \textbf{Proved from P176}: $\partial/\partial\psi = \Hunit$
    (P176 Theorem~4.4; reprised in Step~2 of Theorem~\ref{thm:bridge}).
  \item \textbf{Combined}: $\ZetaR|_{S^3,\mathrm{fibre}} = \zeta\Hunit$
    (Theorem~\ref{thm:bridge}, Corollary~\ref{cor:oph}).
\end{itemize}
The heat $\leftrightarrow$ Cartan norm identification (P173 Corollary~3.4)
and the $V_{\mathrm{self}}$/$\lambda\rho$/$\beta\mathcal{M}$ weight
collapse (P173 \S5) are not addressed by OP-H and remain open for a
separate addendum.
\end{assertion}

%==========================================================================
\section{The P18-T2 uniqueness question}
\label{sec:uniqueness}
%==========================================================================

\begin{remark}[Open problem: assembly uniqueness]
P18 \S4.4 records as open: \emph{``whether the design requirements (R1)--(R8)
admit alternative admissible assemblies of the operator $\Ohat$.''}  This
is called the P18-T2 uniqueness question in CLAUDE.md.
\end{remark}

\begin{assertion}[P184 scope with respect to P18-T2]
This paper does not resolve P18-T2 uniqueness.  What can be stated:
\begin{enumerate}[label=(\roman*)]
  \item The Wheel $\leftrightarrow$ $\Ohat$ dictionary
    (Assertion~\ref{ass:dict}) is formally complete for the \emph{canonical}
    assembly of P18.  Alternative assemblies, if they exist, would require
    different operator identifications or different bridge arguments.
  \item The bridge $\ZetaR|_{S^3,\mathrm{fibre}} = \zeta\Hunit$ derived in
    Theorem~\ref{thm:bridge} depends on: (a)~the specific form
    $\calR = r\partial/\partial r + \Delta_\psi$ of the renormalization
    sector, (b)~the Hopf fibration of $S^3$, and (c)~P176's identification
    of $\Hunit$ as the right-$U(1)$ generator.  Each of these is canonical
    (there is a unique Hopf fibration on $S^3$ up to orientation, and the
    right-$U(1)$ direction in $\fz$ is fixed by the Hopf map).
  \item If an alternative admissible assembly replaced $\ZetaR$ with a
    different sector, the $\Hunit$ identification would no longer apply.
    The Wheel's $\PertOp$ operator is tied to the specific $\ZetaR$ form
    of the canonical assembly.
\end{enumerate}
\end{assertion}

\begin{assertion}[Conditional canonicality]
The statement ``Perturb $=$ $\Hunit$ (formally)'' is conditional on the
canonical P18 assembly.  The canonicality is an assertion, not a uniqueness
theorem.  A proof of P18-T2 uniqueness would upgrade it to an unconditional
theorem.  In the absence of that proof, the identification is the best
available formal statement: it derives the Wheel entry from first principles
(P18 definitions + P176 representation theory) within the canonical assembly.
\end{assertion}

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

The main results of this addendum:

\begin{equation}\label{eq:summary}
\boxed{
\begin{aligned}
  &\varepsilon + \bar\varepsilon = r\tfrac{\partial}{\partial r}
    \quad(\text{normal to } S^3)
    \qquad\text{(Theorem~\ref{thm:euler})}\\
  &\varepsilon - \bar\varepsilon = -2i\,\tfrac{\partial}{\partial\psi}
    \quad(\text{Hopf fibre})
    \qquad\text{(Theorem~\ref{thm:euler})}\\
  &\calR\big|_{S^3,\mathrm{fibre}} = \tfrac{\partial}{\partial\psi}
    \qquad\text{(Theorem~\ref{thm:boundary})}\\
  &\tfrac{\partial}{\partial\psi} = \Hunit \in \fz \subset \hE
    \qquad\text{(P176 Theorem~4.4)}\\[4pt]
  &\boxed{\ZetaR\big|_{S^3,\mathrm{fibre}} = \zeta\cdot\Hunit}
    \qquad\text{(Theorem~\ref{thm:bridge})}\\[4pt]
  &\PertOp \;\leftrightarrow\; \ZetaR \;\leftrightarrow\; \zeta\cdot\Hunit
    \quad\text{--- OP-H closed (Corollary~\ref{cor:oph})}
\end{aligned}
}
\end{equation}

\noindent\textbf{What is proved.}
The holomorphic Euler decomposition (Theorem~\ref{thm:euler}) and the
boundary restriction theorem (Theorem~\ref{thm:boundary}) are rigorous
results about vector fields on $\CC^2 \supset B^4$.  The identification
$\partial/\partial\psi = \Hunit$ is proved in P176 via Schur's lemma applied
to the $F_4$-centralising complement.  Theorem~\ref{thm:bridge} chains
these into the formal bridge $\ZetaR|_{S^3,\mathrm{fibre}} = \zeta\Hunit$.

\noindent\textbf{What is conditional.}
The bridge holds for the canonical P18 assembly.  P18-T2 uniqueness (whether
that assembly is the only admissible one) remains open and is not addressed
here.

\noindent\textbf{OP-H status: CLOSED.}
Open Problem~H is resolved.  The formal derivation from P18's $\Ohat$
formalism, called for by P173 and P176, is complete.

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\end{thebibliography}

\end{document}
