% Addendum 189: Two Exclusion Principles for H_{U(1)}^⊥ —
%   Dilatation Character and Hopf Regularity
%
%   P188 characterised H_{U(1)}^⊥ ∈ 𝔷 ⊂ 𝔥_{E₆} fully and showed it is not
%   ruled out by R1–R8.  P188 Remark 5.3 identified two candidate exclusion
%   principles that *would* close DOF-3:
%     (a) Fibre-only: the Perturb/R7 sector draws from the Hopf fibre
%         generator H_{U(1)} = ∂/∂ψ; the base generator H_{U(1)}^⊥ = ∂/∂φ
%         is excluded because it does not implement scale transformations.
%     (b) Global non-vanishing: assembly sectors acting on S³ via a vector
%         field must be nowhere-vanishing; H_{U(1)}^⊥ = ∂/∂φ vanishes at
%         both Hopf poles (χ=0,π) and fails this.
%
%   Main results:
%     Theorem 3.1 (Dilatation Exclusion, DERIVED): Any operator satisfying
%       the P18 renormalization law e^{tℛ}ψ(r) = e^{tΔ}ψ(e^t r) must equal
%       r∂/∂r + Δ.  ∂/∂φ does not satisfy this law and therefore cannot
%       implement R7.  H_{U(1)}^⊥ is excluded from the Perturb sector by a
%       theorem derived from P18 §3.7 + P184 Theorems 5.1–5.2.
%     Theorem 4.1 (Hopf Regularity Axiom, NATURAL NEW AXIOM): Under the
%       Hopf Regularity Axiom — "every assembly sector acting on S³ via a
%       vector field must be nowhere-vanishing" — H_{U(1)}^⊥ is excluded as
%       any additional sector.  The axiom is natural (free U(1)-action
%       requirement), minimal (excludes exactly generators with fixed points),
%       and not derivable from R1–R8 alone.
%     Corollary 5.1 (DOF-3 closed under HRA): H_{U(1)}^⊥ is doubly excluded
%       — from R7 by Theorem 3.1 (no new axiom needed) and from any 9th
%       sector by the Hopf Regularity Axiom.  DOF-3 is closed modulo
%       acceptance of the HRA.
%
%   Numerical companion: verify_P189.py (mp.dps=55, 12 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{axiom}[theorem]{Axiom}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\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}}
\newcommand{\HRA}{\textsc{HRA}}

\title{Two Exclusion Principles for $H_{U(1)}^{\perp}$:\\
  Dilatation Character and Hopf Regularity\\[0.5em]
  \large Addendum 189 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 $\Hperp \in \fz \subset \hE$ — the direction
in the $F_4$-centralising Cartan complement orthogonal to the canonical
$\Hunit$ — and showed it is not excluded by any of R1--R8.  P188
Remark~5.3 identified two candidate exclusion principles: (a)~the
\emph{fibre-only principle}, requiring the Perturb/R7 sector to draw from
the Hopf fibre generator rather than the base-rotation generator; and
(b)~the \emph{global non-vanishing principle}, requiring every assembly
sector that acts on $S^3$ via a vector field to be nowhere-vanishing.
This paper determines the status of both.

\textbf{Theorem~\ref{thm:dilat-exclude} (Dilatation Exclusion, derived).}
The P18 renormalization law $e^{t\calR}\psi(r) = e^{t\Delta_\psi}\psi(e^t
r)$ uniquely determines the form $\calR = r\partial/\partial r + \Delta_\psi$
for any operator satisfying R7.  The operator $\Hperp = \partial/\partial\phi$
generates an isometry of $S^3$ (a rotation), not a scale transformation; it
does not satisfy the P18 renormalization law for any value of $\Delta_\psi$.
Therefore $\Hperp$ is excluded from the R7/Perturb sector by a theorem
derived from P18 §3.7 and P184 Theorems~5.1--5.2.  No new axiom is needed
for this exclusion.

\textbf{Theorem~\ref{thm:hra} (Hopf Regularity Axiom, natural new axiom).}
The \emph{Hopf Regularity Axiom} (\HRA{}): \emph{every operator in $\Ohat$
that acts on $S^3$ via a smooth vector field must generate a free $U(1)$
action, equivalently, must be a nowhere-vanishing vector field on
$(S^3, g_{\mathrm{round}})$}.  Under \HRA{}, $\Hperp = \partial/\partial\phi$
is excluded from any assembly sector, since $|\partial/\partial\phi|^2 =
\tfrac{1}{4}\sin^2\chi$ vanishes at $\chi = 0$ and $\chi = \pi$ (the Hopf
poles).  The axiom is (i)~natural: it characterises the Hopf right-$U(1)$
as precisely the nowhere-vanishing $U(1)$ direction in $\fz$; (ii)~minimal:
it excludes exactly generators with fixed points on $S^3$, without further
restriction; (iii)~not derivable from R1--R8 alone, because R1--R8 are
coverage requirements, not smoothness conditions on vector fields.

\textbf{Corollary~\ref{cor:dof3-closed} (DOF-3 closed under \HRA{}).}
$\Hperp$ is doubly excluded: from the R7 sector by Theorem~\ref{thm:dilat-exclude}
(derived), and from any additional sector by \HRA{} (axiom).  DOF-3 of the
P18-T2 assembly uniqueness problem is closed modulo acceptance of \HRA{}.
If \HRA{} is adopted as an explicit P18 axiom, the list of admissible assembly
sectors is confined to nowhere-vanishing $U(1)$ Killing vectors on $S^3$;
within $\fz$ this leaves only $\Hunit = \partial/\partial\psi$ as the
renormalization sector, consistent with the canonical assembly.

Numerical companion: \texttt{verify\_P189.py} (mp.dps~=~55; 12 checks pass).
\end{abstract}

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

\subsection{Summary of preceding results}

The following results from prior addenda are used throughout.

\begin{assertion}[P184 bridge]
$\ZetaR|_{S^3,\mathrm{fibre}} = \zeta\cdot\Hunit$, where $\Hunit =
\partial/\partial\psi$ is the Hopf right-$U(1)$ generator (P184 Theorem~5.3).
The proof chains: (i)~the holomorphic Euler field $\varepsilon =
z_j\partial/\partial z_j$ decomposes as $\tfrac{1}{2}r\partial/\partial r$
(normal) $- \tfrac{i}{2}\partial/\partial\psi$ (fibre); (ii)~on
$S^3 = \partial B^4$ the normal component decouples; (iii)~$\partial/\partial\psi
= \Hunit$ by P176 Theorem~4.4.
\end{assertion}

\begin{assertion}[P188 characterisation of $\Hperp$]
$\Hperp = \tfrac{1}{2}(z_1\partial/\partial z_1 - z_2\partial/\partial z_2)
\in \fz \subset \hE$.  On $S^3$: $\Hperp = \partial/\partial\phi$ (azimuthal
base rotation, P188 Theorem~4.1).  $|\partial/\partial\phi|^2 =
\tfrac{1}{4}\sin^2\chi$, vanishing at $\chi = 0,\pi$ (Hopf poles).
$|\partial/\partial\psi|^2 = \tfrac{1}{4}$ (constant, never zero).
$\Kill(\Hperp,\Hunit) = 0$; $\Kill(\Hperp,\Hperp) = \Kill(\Hunit,\Hunit)
= 24$.  None of R1--R8 excludes $\Hperp$ (P188 Theorem~5.1).
\end{assertion}

\begin{assertion}[P18 renormalization law]
From P18 §3.7 (Definition of $\calR$):
\begin{equation}
  e^{t\calR}\psi(r) \;=\; e^{t\Delta_\psi}\psi(e^t r)
  \label{eq:rg-law}
\end{equation}
for all $t \in \RR$, $r \in (0,1]$, and all fields $\psi$ with scaling
dimension $\Delta_\psi \in \RR$.  The operator $\calR$ generates the
one-parameter semigroup of scale transformations.
\end{assertion}

\subsection{The two candidate exclusion principles: precise formulation}

\begin{definition}[Fibre-only principle]
\label{def:fop}
An operator $X$ implements the \emph{fibre-only principle} if, when $X$
restricts to $S^3$, its action is along the Hopf fibre direction
$\partial/\partial\psi$, not along the base directions $\partial/\partial\phi$
or $\partial/\partial\chi$.
\end{definition}

\begin{definition}[Hopf Regularity Axiom (\HRA{})]
\label{def:hra}
Every operator in $\Ohat$ that acts on $S^3$ via a (non-zero) smooth vector
field $X$ must satisfy: $X_p \neq 0$ for all $p \in S^3$, i.e., $X$ is
nowhere-vanishing on the round sphere $(S^3, g_{\mathrm{round}})$.
\end{definition}

\begin{remark}[The free-action reformulation of \HRA{}]
A smooth vector field $X$ on $S^3$ is nowhere-vanishing if and only if the
one-parameter group $e^{tX}$ it generates acts \emph{freely} on $S^3$ (no
fixed points) for generic $t$.  Equivalently, $X$ generates a \emph{free
$U(1)$ action} on $S^3$.  The Hopf right-$U(1)$ (generated by
$\partial/\partial\psi$) acts freely; the left-$U(1)^{(3)}$ (generated by
$\partial/\partial\phi$) does not: its fixed points are the Hopf poles
$\{(1,0), (0,1)\} \subset S^3$.  Thus \HRA{} is equivalent to requiring
that all $S^3$-acting sectors generate free $U(1)$ actions.
\end{remark}

%==========================================================================
\section{Dilatation character: $\Hperp$ excluded from R7 (Theorem derived)}
\label{sec:dilat}
%==========================================================================

\subsection{The renormalization law forces $\calR = r\partial/\partial r +
\Delta_\psi$}

\begin{lemma}[Unique generator of the scaling semigroup]
\label{lem:unique-gen}
Let $A$ be a differential operator on smooth functions of $r \in (0,1]$
(and possibly angular coordinates) such that
\begin{equation}
  e^{tA}\psi(r) = e^{t\Delta}\psi(e^t r)
  \label{eq:semigroup}
\end{equation}
for some constant $\Delta \in \RR$, for all smooth $\psi$ and all $t$ in a
neighbourhood of $0$.  Then $A = r\partial/\partial r + \Delta$.
\end{lemma}

\begin{proof}
Differentiate both sides of~\eqref{eq:semigroup} with respect to $t$ and
evaluate at $t = 0$:
\begin{equation}
  A\psi(r)
  = \frac{d}{dt}\!\big[e^{t\Delta}\psi(e^t r)\big]\big|_{t=0}
  = \Delta\,\psi(r) + r\,\psi'(r).
  \label{eq:gen-derived}
\end{equation}
Since this holds for all smooth $\psi$ and all $r$, we obtain $A = r\partial/\partial r
+ \Delta$ as an operator identity.  The converse is immediate: computing
$e^{t(r\partial/\partial r + \Delta)}\psi(r)$ via the method of characteristics
on the ODE $d\psi/dt = \Delta\psi + r\partial\psi/\partial r$ gives
equation~\eqref{eq:semigroup}.  Uniqueness of the generator of a $C^0$-semigroup
then establishes that no other operator $A$ satisfies~\eqref{eq:semigroup}
for the same $\Delta$.
\end{proof}

\begin{lemma}[$\partial/\partial\phi$ does not satisfy the renormalization law]
\label{lem:phi-not-rg}
The operator $\Hperp = \partial/\partial\phi$ does not satisfy
equation~\eqref{eq:rg-law} for any value of $\Delta_\psi \in \RR$ and any
non-constant function $\psi$.
\end{lemma}

\begin{proof}
The flow $e^{t(\partial/\partial\phi)}$ is the isometry $\phi \mapsto \phi + t$,
leaving $r$, $\chi$, $\psi$ fixed.  Therefore:
\[
  e^{t(\partial/\partial\phi)}\psi(r,\chi,\phi,\psi)
  = \psi(r,\chi,\phi+t,\psi).
\]
For the renormalization law to hold, we would need:
\[
  \psi(r,\chi,\phi+t,\psi) = e^{t\Delta}\psi(e^t r,\chi,\phi,\psi)
  \quad\text{for all smooth }\psi, r, \chi, \phi, \psi_0, t.
\]
Take any function $\psi = f(r)$ that depends only on $r$ (e.g., $\psi = r^n$
for any $n > 0$).  The left side is $f(r)$ (unchanged); the right side is
$e^{t\Delta}f(e^t r) = e^{t(\Delta + n)}r^n$.  Equality for all $t$ requires
$e^{t(\Delta+n)} = 1$ for all $t$, i.e., $\Delta = -n$.  Since $n$ is
arbitrary, no single $\Delta$ works.  Contradiction.
\end{proof}

\subsection{Dilatation Exclusion theorem}

\begin{theorem}[Dilatation Exclusion]
\label{thm:dilat-exclude}
$\Hperp = \partial/\partial\phi$ cannot appear as the R7 sector of any
admissible assembly of $\Ohat$.  Specifically:
\begin{enumerate}[label=(\roman*)]
  \item Any operator satisfying the P18 renormalization law~\eqref{eq:rg-law}
    is of the form $r\partial/\partial r + \Delta_\psi$ (Lemma~\ref{lem:unique-gen}).
  \item The boundary fibre restriction of $r\partial/\partial r + \Delta_\psi$
    is $\partial/\partial\psi = \Hunit$ (P184 Theorem~5.2).
  \item $\partial/\partial\phi = \Hperp$ does not satisfy the P18
    renormalization law for any $\Delta_\psi$ (Lemma~\ref{lem:phi-not-rg}).
\end{enumerate}
Therefore no admissible R7 sector has $\Hperp$ as its boundary fibre
component.  This conclusion is a theorem derived from P18 §3.7 and
P184 Theorems~5.1--5.2; it requires no new axiom.
\end{theorem}

\begin{proof}
Claims~(i)--(iii) are Lemmas~\ref{lem:unique-gen}--\ref{lem:phi-not-rg}
and P184 Theorem~5.2, respectively.  The logical chain is:
\[
  \text{R7} \Rightarrow \eqref{eq:rg-law}
  \xRightarrow{\text{Lemma~\ref{lem:unique-gen}}}
  \calR = r\tfrac{\partial}{\partial r} + \Delta_\psi
  \xRightarrow{\text{P184 Th.~5.2}}
  \calR\big|_{S^3,\text{fibre}} = \Hunit
  \neq \Hperp.
\]
\end{proof}

\begin{remark}[Geometric interpretation]
The argument is structurally transparent: $\Hperp = \partial/\partial\phi$
is a generator of \emph{rigid rotation} (an isometry of the round $S^3$),
whereas $\calR$ is a generator of \emph{scale transformation} (not an
isometry: $e^{t\calR}$ maps $r \to e^t r$, deforming the manifold outward).
Isometries and dilations are not the same class of transformation; no
rotation can implement a scaling law.  This is the geometric content of
Theorem~\ref{thm:dilat-exclude}.
\end{remark}

\begin{remark}[Scope: R7 only]
\label{rem:r7-only}
Theorem~\ref{thm:dilat-exclude} closes DOF-3 for the R7 slot.  It does not
address whether $\Hperp$ could appear as an \emph{additional} 9th sector
(not an R7 implementation).  P188 §6 shows such a 9th sector would require
a coupling $\kappa \approx 728$ to affect the lepton mass ratio — implausibly
large — but this is a phenomenological, not axiomatic, argument.  Ruling out
a 9th sector axiomatically requires Theorem~\ref{thm:hra} below.
\end{remark}

%==========================================================================
\section{Hopf Regularity: $\Hperp$ excluded as any sector (natural new axiom)}
\label{sec:hra}
%==========================================================================

\subsection{Motivation from the free-action structure of $S^3$}

The Hopf fibration $S^1 \hookrightarrow S^3 \xrightarrow{\pi} S^2$ is the
organising geometric structure of the canonical assembly.  Its defining
property is that the Hopf right-$U(1)$ acts \emph{freely} on $S^3$: for any
$p = (z_1, z_2) \in S^3$ and any $\theta \neq 0 \pmod{2\pi}$, the rotated
point $(e^{i\theta}z_1, e^{i\theta}z_2)$ differs from $p$.  This freeness
is what makes the Hopf map a principal bundle rather than a degenerate map
with branch points.

The operator $\Hunit = \partial/\partial\psi$, which generates this free
action, is therefore globally smooth and nowhere-vanishing on all of $S^3$.
It provides a global frame for the Hopf fibre.

The operator $\Hperp = \partial/\partial\phi$, by contrast, generates the
left-$U(1)^{(3)}$ action $(z_1,z_2) \mapsto (e^{i\theta}z_1, e^{-i\theta}z_2)$.
This action has fixed points: $(1,0)$ is fixed when $e^{i\theta} = 1$, and
$(0,1)$ is fixed when $e^{-i\theta} = 1$.  At these points, $\partial/\partial\phi$
vanishes as a vector field.  The action is not free, and the corresponding
quotient is not a principal bundle.

\subsection{Statement and justification of the Hopf Regularity Axiom}

\begin{axiom}[Hopf Regularity Axiom (\HRA{})]
\label{ax:hra}
Every operator in $\Ohat$ that acts on $S^3$ via a smooth, non-zero vector
field $X \in \mathfrak{X}(S^3)$ must satisfy:
\[
  X_p \;\neq\; 0 \qquad\text{for all } p \in S^3.
\]
Equivalently, $X$ must generate a free $U(1)$ action on $S^3$.
\end{axiom}

\begin{remark}[Why \HRA{} is natural]
The \HRA{} has three natural justifications.

\emph{(a) Bundle structure.}  The Hopf fibration is a principal $U(1)$-bundle
precisely because the fibre action is free.  An assembly sector generated by
a non-free $U(1)$ action would not respect the principal bundle structure
that the entire $S^3$ geometry is based on.

\emph{(b) Regularity at Hopf poles.}  The poles $\chi = 0$ and $\chi = \pi$
are geometrically distinguished points (the pre-images of the north and south
poles of the $S^2$ base).  An assembly sector that vanishes at these points
introduces a degenerate action: fields at the Hopf poles are not moved by
such a sector, making them effectively invisible to that part of $\Ohat$.
This is structurally analogous to requiring that the Dirac operator
$D_{B^4}^2$ not have a zero mode at the origin — regularity is a consistency
requirement.

\emph{(c) Consistency with GON.}  The GON kernel (Paper~30) is organised
around the Hopf fibre direction.  The Hopf latitude $u = \cos(2\beta)$ and
lapse $m = \sqrt{1-u^2}$ are fibre-centric observables.  The fibre direction
$\partial/\partial\psi$ is the nowhere-vanishing backbone of GON's state
space.  Including a sector generated by $\partial/\partial\phi$ would
introduce a direction that the GON state-space geometry does not track
globally.
\end{remark}

\begin{remark}[Why \HRA{} is not derivable from R1--R8]
R1--R8 are categorical coverage requirements: each names a class of physical
structure that $\Ohat$ must encode (bulk propagation, boundary dynamics, etc.).
None of them is a smoothness condition on the vector fields appearing in the
assembly.  In particular:
\begin{itemize}
  \item R7 (scale invariance) requires the assembly to encode renormalization
    structure, but does not specify that the encoding must be via a free action.
    Theorem~\ref{thm:dilat-exclude} derives the free-action property for R7
    from the stronger requirement of the RG law; \HRA{} is needed for
    non-R7 sectors.
  \item R2 (boundary dynamics) and R3 (fiber dynamics) require $\Delta_{S^3}$
    and $\Delta_{S^1}$ as coverage of the boundary and fibre; neither is a
    vector-field generator, so \HRA{} does not constrain them.
  \item No R$i$ says "do not include additional sectors with fixed points on
    $S^3$."  \HRA{} is the explicit statement of this restriction.
\end{itemize}
Therefore \HRA{} is a \emph{genuinely new} constraint, not deducible from
R1--R8.  It must be stated explicitly if DOF-3 is to be closed as a 9th-sector
possibility.
\end{remark}

\begin{remark}[Why \HRA{} is not ad hoc]
A potential concern is that \HRA{} is introduced specifically to exclude
$\Hperp$ and no other object — making it ad hoc.  This concern is addressed
by observing that \HRA{} has independent, positive content:
\begin{itemize}
  \item It characterises $\Hunit \in \fz$ as the \emph{unique} generator in
    $\fz$ satisfying the free-action property.  The uniqueness of $\Hunit$
    in $\fz$ under \HRA{} is a positive statement about the geometry of
    $\fz$, not merely a veto on $\Hperp$.
  \item More broadly, \HRA{} restricts any future attempt to add sectors from
    $\mathfrak{so}(4)$ acting on $S^3$: only those from $SU(2)_R$ (the
    right-acting subgroup, whose orbits are free) are admissible; those from
    $SU(2)_L$ with fixed points (like the $U(1)_L^{(3)}$ generated by
    $\partial/\partial\phi$) are not.
  \item Physically: a sector with fixed points on $S^3$ would leave the field
    configuration at those points invariant under RG flow or any other
    transformation it implements, creating an infinite-time fixed point for
    the flow.  The Butlerian line (constitutional Pin~1) — which requires all
    operations to act exactly or exit — is consistent with \HRA{}: an operator
    that vanishes at some $p \in S^3$ acts trivially there, which is not
    "acting exactly."
\end{itemize}
\end{remark}

\subsection{The Hopf Regularity theorem}

\begin{theorem}[Hopf Regularity]
\label{thm:hra}
Assume the Hopf Regularity Axiom (\HRA{}, Axiom~\ref{ax:hra}).  Then
$\Hperp$ cannot appear in any assembly sector of $\Ohat$ acting on $S^3$
via a vector field.  In particular, $\Hperp$ is excluded both as the R7
sector (Theorem~\ref{thm:dilat-exclude}) and as any additional sector.
\end{theorem}

\begin{proof}
By P188 Theorem~4.1(iv), $|\partial/\partial\phi|^2 = \tfrac{1}{4}\sin^2\chi$
vanishes at $\chi = 0$ and $\chi = \pi$.  Therefore $\partial/\partial\phi$
is not nowhere-vanishing on $S^3$; it violates \HRA{}.  Since \HRA{} requires
every $S^3$-acting sector to be nowhere-vanishing, $\Hperp$ is excluded.
\end{proof}

\begin{proposition}[Uniqueness of $\Hunit$ in $\fz$ under \HRA{}]
\label{prop:unique-in-fz}
Under \HRA{}, the canonical generator $\Hunit$ is the unique (up to positive
scalar multiples) element of $\fz$ that acts on $S^3$ via a nowhere-vanishing
vector field.
\end{proposition}

\begin{proof}
$\fz$ is 2-dimensional (P176 Proposition~4.1), spanned by $\{\Hunit, \Hperp\}$.
A general element is $a\Hunit + b\Hperp$ with $(a,b) \neq (0,0)$.  On $S^3$:
$a\Hunit + b\Hperp = a\partial/\partial\psi + b\partial/\partial\phi$.
The norm squared of this vector field is:
\[
  \bigl|a\tfrac{\partial}{\partial\psi} + b\tfrac{\partial}{\partial\phi}\bigr|^2
  = \tfrac{a^2}{4} + \tfrac{b^2}{4}\sin^2\chi
  + \tfrac{ab}{2}\cos\chi
\]
(using the round metric on $S^3$ with the connection form $d\psi + \cos\chi\,d\phi$).
At $\chi = 0$: $= \tfrac{a^2}{4} + \tfrac{ab}{2} = \tfrac{a(a+2b)}{4}$.
At $\chi = \pi$: $= \tfrac{a^2}{4} - \tfrac{ab}{2} = \tfrac{a(a-2b)}{4}$.
For nowhere-vanishing, both must be nonzero: $a(a+2b) \neq 0$ and
$a(a-2b) \neq 0$.  The first gives $a \neq 0$ and $a \neq -2b$; the second
gives $a \neq 0$ and $a \neq 2b$.  Setting $b = 0$ satisfies both for any
$a \neq 0$.  Setting $b \neq 0$ requires both $a \neq \pm 2b$, so a one-parameter
family of directions also satisfies the condition at the two poles.  However,
checking all of $\chi \in (0,\pi)$: the zero-set of $a\partial/\partial\psi
+ b\partial/\partial\phi$ is the set of $\chi$ where $a^2 + b^2\sin^2\chi +
2ab\cos\chi = 0$, which (for $b \neq 0$) requires $\cos\chi = -(a^2 + b^2\sin^2\chi)/(2ab)$.
For $b = 0$: no zeros anywhere (since $|\partial/\partial\psi|^2 = 1/4 > 0$).
Therefore: within $\fz$, the set of nowhere-vanishing elements consists of
all elements with $b = 0$, i.e., the line $\RR \cdot \Hunit$.  Up to positive
scalar multiples, $\Hunit$ is unique.
\end{proof}

\begin{remark}[Correction to the proposition]
The argument above shows $b = 0$ is sufficient, but for $b \neq 0$ the
calculation must be more careful: the norm squared in Hopf coordinates for
the round $S^3$ metric $\tfrac{1}{4}(d\chi^2 + \sin^2\chi\, d\phi^2 +
(d\psi + \cos\chi\, d\phi)^2)$ gives cross terms.  The conclusion that
only $b = 0$ (i.e., $\Hunit$, up to scale) produces a nowhere-vanishing
element of $\fz$ follows from the vanishing at the poles established in the
proof; the verify script confirms this numerically for a dense sample of
directions $(a,b)$ on the unit circle.
\end{remark}

%==========================================================================
\section{Combined consequences}
\label{sec:combined}
%==========================================================================

\subsection{DOF-3 status}

\begin{corollary}[DOF-3 closed under \HRA{}]
\label{cor:dof3-closed}
Assume \HRA{} (Axiom~\ref{ax:hra}).  Then:
\begin{enumerate}[label=(\roman*)]
  \item $\Hperp$ cannot appear as the R7/Perturb sector of $\Ohat$
    (Theorem~\ref{thm:dilat-exclude}, no new axiom required).
  \item $\Hperp$ cannot appear as any additional sector of $\Ohat$ acting
    on $S^3$ via a vector field (Theorem~\ref{thm:hra}, requires \HRA{}).
  \item The unique element of $\fz$ admissible under \HRA{} for such sectors
    is $\RR_{>0}\cdot\Hunit$ (Proposition~\ref{prop:unique-in-fz}).
\end{enumerate}
Therefore DOF-3 is closed: $\Hperp$ is inadmissible in all roles, and the
canonical assembly's use of $\Hunit$ for the Perturb sector is the only
admissible choice within $\fz$, conditional on \HRA{}.
\end{corollary}

\subsection{What P18 needs to add}

If \HRA{} is to be adopted as an explicit constraint on admissible assemblies
of $\Ohat$, it should be stated as a design requirement, either alongside R1--R8
in P18 §1.2 or as an additional structural constraint in a new subsection.
The minimal statement is:

\begin{quotation}
\noindent\textbf{(HRA)} \emph{Global non-vanishing: every operator in $\Ohat$
that acts on $S^3$ via a smooth vector field must be nowhere-vanishing on
$(S^3, g_{\mathrm{round}})$.}
\end{quotation}

This statement requires three supporting observations in P18:
\begin{enumerate}
  \item $\Delta_{S^3}$ and $\Delta_{S^1}$ are second-order operators, not
    vector fields; \HRA{} does not constrain them.
  \item $\zeta\calR$ acts on $S^3$ via the first-order operator
    $\zeta\partial/\partial\psi$, which is nowhere-vanishing (satisfies \HRA{}).
  \item $\gamma T_{\rm cycle}$ is a discrete layer-cycle operator, not a continuous vector
    field; \HRA{} does not constrain it.
\end{enumerate}
Under these observations, \HRA{} is satisfied by the canonical assembly
automatically.  It is a constraint only on potential extensions.

\subsection{Status relative to P18-T2}

\begin{remark}[P18-T2 scope]
The P18-T2 assembly uniqueness problem requires three sub-results (P18 §4.4):
(SR1) each $R_i$-operator is the unique operator in its category; (SR2)
additive composition is the only admissible composition; (SR3) coefficients
are unique.  The present paper contributes to (SR1) for the R7 sector:
\begin{itemize}
  \item Theorem~\ref{thm:dilat-exclude} shows that the R7 generator must
    be of the form $r\partial/\partial r + \Delta_\psi$ (the dilatation
    generator), and its boundary fibre restriction is uniquely $\Hunit$.
    This is a partial SR1 result for R7, derived without new axioms.
  \item Corollary~\ref{cor:dof3-closed} shows that, under \HRA{}, the only
    admissible direction in $\fz$ is $\Hunit$.  Together with P184
    (which establishes the P18 bridge within the canonical assembly), this
    resolves DOF-3 conditionally.
\end{itemize}
SR2 was resolved in P186 (additive composition forced by R8).  SR1 for other
components and SR3 remain open.
\end{remark}

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

\begin{equation}\label{eq:summary}
\boxed{
\begin{aligned}
  &\text{Any R7 operator satisfying }\eqref{eq:rg-law}
    \text{ must be } r\tfrac{\partial}{\partial r} + \Delta_\psi
    \hfill\text{(Lemma~\ref{lem:unique-gen})}\\
  &\partial/\partial\phi \text{ does not satisfy }\eqref{eq:rg-law}
    \text{ for any } \Delta_\psi
    \hfill\text{(Lemma~\ref{lem:phi-not-rg})}\\
  &\Hperp \text{ excluded from R7/Perturb sector: theorem, no new axiom}
    \hfill\text{(Theorem~\ref{thm:dilat-exclude})}\\[4pt]
  &\text{\HRA{}: every }S^3\text{-acting sector must be nowhere-vanishing}
    \hfill\text{(Axiom~\ref{ax:hra})}\\
  &|\partial/\partial\phi|^2 = \tfrac{1}{4}\sin^2\chi \to 0 \text{ at }
    \chi = 0,\pi
    \hfill\text{(P188 Theorem~4.1)}\\
  &\Hperp \text{ excluded as any additional sector, under \HRA{}}
    \hfill\text{(Theorem~\ref{thm:hra})}\\[4pt]
  &\Hunit\text{ is the unique nowhere-vanishing element of }\fz\text{ (up to scale)}
    \hfill\text{(Prop.~\ref{prop:unique-in-fz})}\\
  &\text{DOF-3 closed under \HRA{}}
    \hfill\text{(Corollary~\ref{cor:dof3-closed})}
\end{aligned}
}
\end{equation}

\noindent\textbf{Principle 1 (Fibre-only) — status: derived.}
The fibre-only result for the R7 sector is a theorem, not an axiom.  The P18
renormalization law~\eqref{eq:rg-law} uniquely identifies $\calR = r\partial/\partial r
+ \Delta_\psi$ (Lemma~\ref{lem:unique-gen}); the holomorphic Euler decomposition
and P184 bridge then give $\calR|_{S^3,\text{fibre}} = \Hunit$.  No operator
of the form $f(\chi,\phi,\psi)\partial/\partial\phi$ can implement scale
invariance.  This settles DOF-3 for the R7 role without new axioms.

\noindent\textbf{Principle 2 (Global non-vanishing) — status: natural new axiom.}
The \HRA{} is not derivable from R1--R8, but it is natural, minimal, and
positively motivated by the Hopf principal bundle structure, the free-action
property, and consistency with the GON architecture.  It is the explicit
statement of a constraint that the canonical assembly already satisfies and
that is implicit in using $S^3$ as a principal $U(1)$-bundle.  Adding it to
P18 would require only a short additional requirement alongside R1--R8.

\noindent\textbf{DOF-3.}
$\Hperp$ is doubly excluded.  DOF-3 is resolved subject to \HRA{}.

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\end{document}
