% 259_Addendum_FoldPerturbationSourcing.tex
% TASK_030 / OI-257-1: what the S3 -> S2 fold completion does to sub-horizon
% perturbations. Hopf-sector mode bookkeeping, transfer brackets, curvature
% pinning, and the alpha^6 near-miss.
% Verifier: Lumen/corpus/addenda/verify/verify_P259.py.
% Copyright Léon Fernando Vlegels -- CC BY 4.0

\documentclass[11pt]{article}
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\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
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\newcommand{\Msun}{M_\odot}
\newcommand{\lH}{\ell_H}
\newcommand{\sigreq}{\sigma_{\rm req}}
\newcommand{\signull}{\sigma_0}

\title{Addendum 259: Fold-Event Perturbation Sourcing\\
\large Hopf-sector bookkeeping of the $S^3 \to S^2$ fold, the transfer
bracket,\\ and curvature pinning with the $\alpha^6$ near-miss}
\author{TOE Corpus}
\date{2026-06-09}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case B$+$ (mechanism bracketed, not closed; one selection
rule missing; new testable consequence extracted) \quad|\quad
\textbf{Task:} TASK\_030 \quad|\quad
\textbf{Verifier:} \texttt{verify\_P259.py} --- all PASS

\begin{abstract}
A257 left OI-257-1 open: can the $S^3 \to S^2$ fold completion source the
$\sigma \approx 0.05$ one-scale perturbation spike that the fold-native
C-$\ln a$ channel requires? This addendum answers with an exact mode
bookkeeping and a two-sided exclusion. \emph{First}, under the Hopf map the
level-$\ell$ eigenspace of $\Delta_{S^3}$ (dimension $(\ell+1)^2$) splits
into a fiber-charge-zero sector of dimension $\ell+1$, which descends
exactly to the $S^2$ harmonics, and a charged sector of dimension
$\ell(\ell+1)$, which the fold annihilates: at large $\ell$ essentially the
whole spectrum is lost, and its energy must go somewhere. \emph{Second}, the
two natural fates bracket the requirement from both sides: level-local
equipartition transfer gives $\sigma \sim \signull\sqrt{\lH} \sim 10^{5}$
--- catastrophic overproduction, excluded by the existence of the universe
--- while charge-selection transfer ($\Delta q = \pm1$ only) gives
$\sigma = \sqrt{3}\,\signull \approx 8\times10^{-5}$, sterile. The
requirement $\sigma \approx 0.06$ sits strictly inside the bracket, so the
mechanism is \emph{possible but not forced}: it needs a transfer fraction
$s$ with $s\,\lH = (\sigreq/\signull)^2 = 1.75\times10^{6}$.
\emph{Third}, because $\lH = k_{J_1} R_{\rm curv}$, fixing $s$ pins the
spatial curvature: the requirement converts a coupling constant into a
curvature prediction. At the Planck flatness bound,
$s_{\rm req} = 0.7$--$1.1\times10^{-13}$, and $\alpha^6 =
1.51\times10^{-13}$ falls inside factor 2 of that range; taking
$s = \alpha^6$ \emph{predicts} $R_{\rm curv} = 7.5\times10^{4}$ Mpc, i.e.
$\Omega_k \approx -3.5\times10^{-3}$ (closed, as $(B^4, S^3)$ requires) ---
in $\sim$2$\sigma$ tension with Planck+BAO and decisively testable by
CMB-S4-class experiments. The identification is flagged as a near-miss
candidate (corpus PSLQ-honesty convention), not a theorem. Verdict:
OI-257-1 \textbf{bracketed}; the missing piece is the transfer selection
rule, filed as OI-259-1.
\end{abstract}

% ────────────────────────────────────────────────────────────────────────────
\section{Hopf-Sector Decomposition of the Fold}

\begin{proposition}[Mode bookkeeping]
\label{prop:bookkeeping}
Write $L^2(S^3) = \bigoplus_\ell E_\ell$ with $\dim E_\ell = (\ell+1)^2$
(P18 spectral decomposition, eigenvalues $\ell(\ell+2)$). Under the Hopf
fibration $S^1 \to S^3 \to S^2$, each $E_\ell$ splits by fiber charge $q$:
the $q = 0$ subspace has dimension $\ell+1$ and descends isometrically to
the $S^2$ harmonic space of the corresponding level, while the $q \neq 0$
complement has dimension
\[
  (\ell+1)^2 - (\ell+1) \;=\; \ell(\ell+1).
\]
The fold completion $S^3 \to S^2$ (P32 \S5) preserves the charge-zero
sector and annihilates the charged sector. The annihilated fraction
$\ell/(\ell+1) \to 1$: at horizon scales ($\ell \sim 10^{19}$, below)
essentially the entire spectrum is charged.
\end{proposition}

\begin{remark}[Formal language]
A254's collapse delta is the right invariant: the fold's component map has
kernel rank $\ell(\ell+1)$ per level, the multi-level analogue of the
$n{-}1$ observer-collapse rank. Per A255, the fiber charge here is the lens
$\mathbb{Z}_3$-compatible $U(1)$ charge of the Hopf bundle --- it is
\emph{not} the triality $C_3$ and the bookkeeping never invokes triality.
\end{remark}

% ────────────────────────────────────────────────────────────────────────────
\section{The Transfer Bracket}

Let $\signull = \sqrt{A_s} = 4.6\times10^{-5}$ be the inflationary
amplitude carried by every populated mode before the fold, and let
$\lH = k_{J_1} R_{\rm curv}$ be the harmonic level at the $J_1$ horizon
scale ($k_{J_1} = 1.5\times10^{14}\,{\rm Mpc}^{-1}$ for the C-$\ln a$
survivor at $T = 3$ PeV).

\begin{theorem}[Overproduction exclusion]
\label{thm:over}
Level-local equipartition --- the charged sector's perturbation energy
deposited democratically into the $\ell+1$ survivors --- multiplies the
per-mode variance by $1 + \ell$. At the horizon level this gives
$\sigma = \signull\sqrt{1+\lH} \approx 1.8\times10^{5} \gg 1$ for any
$R_{\rm curv}$ admitted by the flatness bound. The universe would collapse
into black holes at the fold scale. Unsuppressed fold-to-scalar transfer is
therefore \emph{excluded by observation}: the fold must very nearly
decouple from the scalar sector. This is a genuine observational constraint
on fold dynamics --- the first one --- and it holds independently of
whether the fold-native PBH channel is real.
\end{theorem}

\begin{theorem}[Sterility of charge-selection transfer]
\label{thm:sterile}
If conservation of fiber charge at the fold restricts transfer to adjacent
charges ($\Delta q = \pm1$), the donor count per level is $O(2\ell)$ against
$\ell+1$ survivors carrying weight only through the two adjacent sectors:
the variance boost is $O(3)$ and
$\sigma = \sqrt{3}\,\signull \approx 8\times10^{-5}$. Then
$\beta = \mathrm{erfc}(\delta_c/\sqrt{2}\sigma)$ is zero to all relevant
precision ($\delta_c/\sigma \approx 4\times10^{3}$): no population forms.
\end{theorem}

\begin{corollary}[Bracket]
\label{cor:bracket}
The A257 requirement $\sigreq \approx 0.06$ lies strictly between the two
derivable couplings. The fold-native mechanism is possible but not forced:
it requires a transfer fraction
\[
  s \;=\; \frac{(\sigreq/\signull)^2}{\lH}
  \;=\; \frac{1.75\times10^{6}}{\lH},
\]
i.e.\ a selection rule delivering a specific tiny leakage from the charged
sector into the scalar survivors. No such rule is currently derivable from
the corpus; this is the precise missing piece (OI-259-1).
\end{corollary}

% ────────────────────────────────────────────────────────────────────────────
\section{Curvature Pinning and the $\alpha^6$ Near-Miss}

\begin{proposition}[Curvature pinning]
\label{prop:pinning}
Since $\lH = k_{J_1} R_{\rm curv}$, Corollary \ref{cor:bracket} converts any
fixed transfer fraction $s$ into a prediction for the spatial curvature
radius:
\[
  R_{\rm curv} \;=\; \frac{(\sigreq/\signull)^2}{s\,k_{J_1}}.
\]
The mechanism only operates in a closed universe of a definite size ---
consistent with, and now quantitatively sharpening, the corpus's
$(B^4, S^3)$ closed topology. In a flat universe ($R_{\rm curv} \to \infty$)
any fixed $s$ overproduces; requiring $\sigma = \sigreq$ at $J_1$ therefore
\emph{demands} $\Omega_k \neq 0$.
\end{proposition}

\begin{remark}[The $\alpha^6$ near-miss --- candidate, not theorem]
\label{rem:alpha6}
At the Planck flatness bound ($|\Omega_k| \le 2\times10^{-3}$ to
$7\times10^{-4}$), the required fraction is
$s_{\rm req} = 1.1\times10^{-13}$ to $6.7\times10^{-14}$. The TOE constant
$\alpha^6 = 1.51\times10^{-13}$ falls within a factor $1.3$--$2.2$ of this
range ($\alpha^5$ and $\alpha^7$ miss by $10^{2}$). Taking $s = \alpha^6$
exactly:
\[
  R_{\rm curv} = 7.5\times10^{4}\ {\rm Mpc}, \qquad
  \Omega_k \approx -3.5\times10^{-3} \ \text{(closed)},
\]
in $\sim$2.2$\sigma$ tension with Planck+BAO
($\Omega_k = 0.0007 \pm 0.0019$) and decisively testable at
CMB-S4 / next-generation BAO precision ($\sigma(\Omega_k) \sim 10^{-3}$).
Per the corpus PSLQ-honesty convention (A245, A180), this is recorded as a
\emph{near-miss candidate}: the exponent 6 has plausible readings
($2\cdot\dim S^3$; one factor of $\alpha$ per fold operation in the
octonionic count $k_8 = 2k_4$ at $k_4 = 3$ remaining folds --- both
post-hoc) but no derivation. If OI-259-1 lands on $s = \alpha^6$, the
$\Omega_k$ prediction becomes the framework's \textbf{fifth kill-shot}, and
its sign (closed) is already forced by the arena.

\emph{Superseded (2026-06-09, same day; A261).} The transfer fraction has
since been derived: $s = 0$ by Hopf-charge superselection of the canonical
$\Ohat$ (A261 Thm.~1.1). This remark's candidate and its $\Omega_k$
pinning are void within the canon; the bracket theorems of
\S\ref{sec:bracket}\,--- and Thm.~\ref{thm:over}'s decoupling demand, which
A261 shows the kernel satisfies exactly --- stand.
\end{remark}

% ────────────────────────────────────────────────────────────────────────────
\section{Scale Selectivity}

The SIGW companion (A257 Prop.~3) requires a narrow spike. Narrowness
survives the bookkeeping: the deposit lands on modes crossing the horizon
within the fold-completion window. An event-like completion (less than a
Hubble time in the fold clock --- the natural reading of ``fold
completion'' in P32 \S5, where $k$ is a discrete counter, not a continuous
parameter) restricts the receiving band to
$\Delta\ell/\lH \lesssim 1$, preserving the narrow-feature character in
both surviving clocks. A continuous-process reading would smear the spike
and weaken both the PBH formation efficiency and the SIGW visibility; the
discrete reading is canonical in the corpus.

% ────────────────────────────────────────────────────────────────────────────
\section{Verdict and Open Items}

\paragraph{OI-257-1 verdict: bracketed (Case B$+$).} The fold-native
sourcing mechanism is neither derived nor excluded. What is now proven:
(i) the fold annihilates the charged Hopf sector, $\ell(\ell+1)$ modes per
level (Prop.~\ref{prop:bookkeeping}); (ii) unsuppressed transfer is
observationally excluded (Thm.~\ref{thm:over}) --- the fold nearly
decouples from the scalar sector; (iii) charge-selection transfer is
sterile (Thm.~\ref{thm:sterile}); (iv) the viable window requires a
specific transfer fraction that pins the curvature radius
(Prop.~\ref{prop:pinning}), with $s = \alpha^6$ as the flagged candidate
(Rem.~\ref{rem:alpha6}).

\paragraph{Open items.}
\begin{enumerate}
  \item \textbf{OI-259-1 (transfer selection rule).} Derive $s$ from the
        kernel --- what fraction of charged-sector energy couples to the
        scalar survivors at fold completion. The Wheel's heat-conservation
        identity (Fork/Weld cycles; Weld $\leftrightarrow \Delta_{S^3}$) is
        the natural corpus anchor for the fold's energy bookkeeping and has
        not yet been applied to it.
  \item \textbf{OI-259-2 (= OI-257-2).} Matter-era analogue for C-$\eta$:
        the annihilated sector's imprint on the halo mass function at
        $z \approx 17$--$26$.
\end{enumerate}

\paragraph{Verifier.} \texttt{Lumen/corpus/addenda/verify/verify\_P259.py}
--- all checks pass, exit 0. Task: \texttt{Codex/Write/TASK\_030} (executed
by Cowork directly, 2026-06-09).

\end{document}
