% 261_Addendum_ChargeSuperselection.tex
% OI-259-1 CLOSED: the transfer fraction s is derived from the kernel and
% equals zero. Hopf-charge superselection of the canonical O-hat; the fold
% defect thermalizes as Wheel heat; demotions and inversions that follow.
% Verifier: Lumen/corpus/addenda/verify/verify_P261.py.
% Copyright Léon Fernando Vlegels -- CC BY 4.0

\documentclass[11pt]{article}
\usepackage{amsmath,amssymb,amsthm}
\usepackage[margin=1in]{geometry}
\usepackage{hyperref}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}

\newcommand{\Ohat}{\hat{O}}
\newcommand{\Qhat}{\hat{Q}}
\newcommand{\signull}{\sigma_0}

\title{Addendum 261: Hopf-Charge Superselection\\
\large OI-259-1 closed: the fold-transfer fraction is zero,\\
and the fold defect thermalizes as Wheel heat}
\author{TOE Corpus}
\date{2026-06-09}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case B (negative closure --- and a consistency success:
the kernel independently delivers the decoupling that observation demands)
\quad|\quad \textbf{Verifier:} \texttt{verify\_P261.py} --- all PASS

\begin{abstract}
A259 reduced the fold-native sourcing question to one unknown: the transfer
fraction $s$ from the annihilated Hopf-charged sector into scalar curvature
perturbations, with $s = \alpha^6$ flagged as a near-miss candidate
carrying an $\Omega_k$ prediction. This addendum derives $s$ from the
kernel. The answer is \textbf{$s = 0$ at all orders}: every sector of the
canonical master operator commutes with the Hopf fiber charge $\Qhat$ ---
the radial sectors ($D^2_{B^4}$, $V_{\rm self}^{\rm can}$, $\alpha\rho$,
$\zeta\mathcal{R}$, $\beta\mathcal{M}$) trivially, $\Delta_{S^3}$ as the
isometry Casimir, $\Delta_{S^1}$ as the fiber generator squared, and
$\gamma T_{\rm cycle}$ because the lens $\mathbb{Z}_3$ is a subgroup of
$U(1)_{\rm Hopf}$ (A255, verified numerically) --- while A189's Hopf
Regularity Axiom forbids any admissible extension from breaking the
symmetry, and A218b's assembly uniqueness removes the loophole of
alternative $\Ohat$ forms. The fold map itself is the charge-zero
projection, exactly equivariant. Fiber charge is therefore superselected:
the charged sector cannot leak into scalar perturbations through any
canonical channel, in either the sudden or adiabatic limit of fold
completion. Its energy deposits in the homogeneous background --- the
corpus identity for which is \emph{Wheel heat conservation}
(\texttt{kernel/wheel/wheel.py}). Consequences: the fold-native C-$\ln a$
PBH channel is closed negatively; the $\alpha^6$/$\Omega_k$ pinning of A259
is demoted to void (the $2.2\sigma$ curvature tension evaporates); the
LISA mHz prediction inverts --- the canon now predicts \emph{absence} of
the spike, so a detection would demonstrate structure beyond the canon;
A259's overproduction exclusion is resolved by the kernel itself, a
nontrivial consistency success; and the C-$\eta$ heavy-seed channel
survives unchanged, since it requires only standard perturbations plus the
fold epoch. OI-259-1 is closed.
\end{abstract}

% ────────────────────────────────────────────────────────────────────────────
\section{The Superselection Theorem}

Let $\Qhat = -i\,\partial_\psi$ generate the Hopf $U(1)$ on $S^3$, with
integer charge $q$ labelling the fiber sectors of A259
Prop.~1.1.

\begin{theorem}[Hopf-charge superselection of the canonical $\Ohat$]
\label{thm:super}
$[\Ohat, \Qhat] = 0$, sector by sector:
\begin{itemize}
  \item $D^2_{B^4}$, $V_{\rm self}^{\rm can}(r)$, $\alpha\rho(r)$,
        $\zeta\mathcal{R} = \zeta\, r\partial_r$, $\beta\mathcal{M}$:
        purely radial --- they act on $r$ and spectral weight, not on
        $\psi$; commutation is immediate.
  \item $\Delta_{S^3}$: the Casimir of the isometry group, which contains
        $U(1)_{\rm Hopf}$; a Casimir commutes with every generator.
  \item $\Delta_{S^1}$: equals $-\Qhat^2$ up to the fiber normalization;
        commutes with $\Qhat$ identically.
  \item $\gamma T_{\rm cycle}$: the family cycle is the lens
        $\mathbb{Z}_3$ deck action, and by A255 the lens generator acts as
        $(z_1, z_2) \mapsto (\epsilon z_1, \epsilon z_2)$ with
        $\epsilon = e^{2\pi i/3}$ --- exactly the Hopf $U(1)$ element at
        $\psi = 2\pi/3$. Since $U(1)$ is abelian,
        $[T_{\rm cycle}, \Qhat] = 0$. (Verified numerically; the A255
        distinction is respected: this is the lens $\mathbb{Z}_3$, not
        triality.)
\end{itemize}
Moreover the conclusion is robust beyond the canonical assembly: A189's
Hopf Regularity Axiom requires every admissible sector vector field to be
nowhere-vanishing on $S^3$, and A189 Thm.~4.1 shows the unique such
direction in the centraliser is $H_{U(1)} = \partial_\psi$ itself; and
A218b closes assembly uniqueness, so no alternative admissible $\Ohat$
exists to evade the theorem. Fiber charge is a superselection charge of
the kernel.
\end{theorem}

\begin{corollary}[$s = 0$]
\label{cor:szero}
No canonical operator, at any order of perturbation theory, connects the
charged sector to the $q = 0$ scalar survivors. The transfer fraction of
A259 Cor.~2.3 is $s = 0$ within the canon.
\end{corollary}

% ────────────────────────────────────────────────────────────────────────────
\section{The Fold Map Cannot Rescue It}

\begin{proposition}[Equivariance of the fold]
\label{prop:equiv}
The fold completion $S^3 \to S^2$ acts on modes as the Hopf pushforward ---
the charge-zero projection $P_0$. The Hopf base point
$(2 z_1 \bar z_2,\, |z_1|^2 - |z_2|^2)$ is invariant under the $U(1)$
action (verified numerically to $10^{-12}$ over random states), so $P_0$
is exactly equivariant: the fold itself conserves the charge it
annihilates --- it removes the charged sector's \emph{support}, not its
\emph{charge}.

In the \emph{sudden} limit, the post-fold state is $P_0|{\rm pre}\rangle$;
the charged components have no image in $L^2(S^2)$ and their energy enters
the homogeneous trace. In the \emph{adiabatic} limit, charged energies
$\sim q^2/r_{\rm fiber}^2$ blueshift as the fiber collapses, and the
adiabatic invariant keeps them in charged eigenstates until they exit the
spectrum --- again into the background. In both limits the deposit is
homogeneous: \textbf{zero scalar transfer}.
\end{proposition}

\begin{remark}[The corpus identity for the deposit: Wheel heat]
The kernel already owns the bookkeeping object for ``ambiguity/energy that
is conserved but not attached to any face'': Wheel heat
(\texttt{kernel/wheel/wheel.py} line 30: ``Heat conservation: total
ambiguity is conserved across Fork/Weld pairs''; tested in
\texttt{kernel/wheel/tests/test\_wheel.py}, TestHeatConservation). The
fold is a Weld-sector event ($\text{Weld} \leftrightarrow \Delta_{S^3}$,
Pin 5); the annihilated sector's energy is Wheel heat. The fold defect
\emph{thermalizes}; it does not gravitate as structure.
\end{remark}

\begin{remark}[NEW THEORY boundary]
A nonzero $s$ would require a non-equivariant correction to the fold map
--- a preferred angular direction during completion. The canon not only
lacks such structure; HRA exists precisely to exclude fixed-point-bearing
angular directions from the assembly. Any future mechanism delivering
$s \neq 0$ is therefore NEW THEORY by construction, and must repeal or
weaken HRA explicitly.
\end{remark}

% ────────────────────────────────────────────────────────────────────────────
\section{Consequences}

\begin{corollary}[Channel closure and demotions]
\label{cor:consequences}
(i) $\sigma_{\rm fold} = \signull$ exactly: no enhancement. With
$\delta_c/(\sqrt2\,\signull) \approx 6.9\times10^{3}$,
$\log_{10}\beta \sim -10^{7}$: the fold-native C-$\ln a$ PBH channel is
\textbf{closed negatively}. (ii) The $\alpha^6$ candidate and its
$\Omega_k \approx -3.5\times10^{-3}$ pinning (A259 Rem.~3.2) are
\textbf{demoted to void} --- the $2.2\sigma$ curvature tension never
materializes; A259's bracket theorems stand. (iii) The 0.45--0.95 mHz SIGW
spike (A257 Prop.~3) \textbf{inverts}: the canon predicts its
\emph{absence}; a LISA detection of a narrow feature there would
demonstrate structure beyond the canon (a falsifier turned discovery
channel). (iv) A259's overproduction exclusion (Thm.~2.1) demanded that
the fold nearly decouple from the scalar sector; Theorem~\ref{thm:super}
shows the kernel \emph{guarantees exact} decoupling --- HRA, adopted in
A189 on spectral grounds, independently protects the universe from
shredding at fold completions. A consistency success of the assembly.
(v) The C-$\eta$ heavy-seed channel survives untouched: it requires only
standard inflationary perturbations plus the fold \emph{epoch}
(A256/A260); its tests (JWST demographics at $z_1 \approx 17$--$26$
coupled to the $w(z)$ step at $z_4 \approx 4$--$6$) are now the thread's
primary observational programme.
\end{corollary}

% ────────────────────────────────────────────────────────────────────────────
\section{Status}

\paragraph{OI-259-1: CLOSED} (negative for sourcing; positive for
consistency). The ``one key, four locks'' structure resolves as: the key
exists and reads $s = 0$ --- it locks the PBH door rather than opening it,
releases the curvature lock ($R_{\rm curv}$ unconstrained by this
mechanism), and leaves the clock-unit lock to its A260 reduction
($\equiv B^4$ scale, open as before).

\paragraph{Remaining open in this thread.} OI-257-2/259-2 (C-$\eta$
matter-era demographics), OI-260-1 (fold pacing from the Wheel), OI-260-2
(per-fold arc audit, now lower priority with the anchor demoted), and the
B$^4$-scale problem (P04), unchanged.

\paragraph{Verifier.} \texttt{Lumen/corpus/addenda/verify/verify\_P261.py}
--- all checks pass, exit 0.

\end{document}
