% 257_Addendum_FoldNativeMechanism.tex
% Addendum to A256: the formation mechanism must be fold-native, and what it
% must deliver. Lambda-negligibility at J1; abundance/amplitude requirements;
% the mandatory SIGW companion in the LISA band.
% Verifier: Lumen/corpus/addenda/verify/verify_P257.py.
% Copyright Léon Fernando Vlegels -- CC BY 4.0

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

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

\newcommand{\Lz}{\Lambda_0}
\newcommand{\Msun}{M_\odot}
\newcommand{\fpbh}{f_{\rm PBH}}

\title{Addendum 257: The Fold-Native Formation Mechanism\\
\large $\Lambda$-negligibility at $J_1$, the amplitude requirement,\\
and the mandatory LISA-band gravitational-wave companion}
\author{TOE Corpus}
\date{2026-06-09}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case A$'$ (A256 assumption (c) corrected by computation;
mechanism requirements quantified; new falsifiable companion prediction)
\quad|\quad \textbf{Verifier:} \texttt{verify\_P257.py} --- all PASS

\begin{abstract}
Addendum 256 carried the collapse-burst hypothesis as bridge assumption (c):
that the $\Lambda$-staircase discontinuities trigger gravitational-collapse
bursts. This addendum tests and corrects it. \emph{First}, the
$\Lambda$-mediated reading is dead: at the $J_1$ jump the vacuum-energy
fraction is $\rho_{\rm DE}/\rho_{\rm rad} \approx 2\times10^{-74}$ for the
C-$\ln a$ survivor (PeV era) and $\rho_{\rm DE}/\rho_m \approx
2$--$8\times10^{-4}$ for C-$\eta$ ($z \approx 17$--$26$); a factor-2 jump in
a $10^{-74}$ component moves no collapse threshold. The staircase marks fold
completions; it cannot cause formation. If fold-seeding is real, the
mechanism is the fold event itself --- \emph{fold-native}. \emph{Second}, the
mechanism's job is sharply quantified: producing the C-$\ln a$ population at
$M \approx 10^{24}\,$g requires formation fraction
$\beta \approx 1.5\times10^{-13}\fpbh$ and perturbation amplitude
$\sigma \approx 0.05$--$0.06$ at the single $J_1$ horizon scale --- a
$\sim\!10^3$ boost over the inflationary baseline, nearly independent of
$\fpbh$ (Gaussian-tail statistics make the amplitude almost binary).
\emph{Third}, such a one-scale spike necessarily induces a narrow
scalar-induced gravitational-wave background peaking at
$f \approx 0.5$--$0.9\,$mHz --- mid-LISA band --- with
$\Omega_{\rm GW} \sim \sigma^4$ well above LISA sensitivity for a narrow
feature. Because $\sigma$ barely depends on $\fpbh$, LISA non-detection at
that frequency kills the fold-native C-$\ln a$ channel outright,
independently of microlensing. All numbers certified by
\texttt{verify\_P257.py}.
\end{abstract}

% ────────────────────────────────────────────────────────────────────────────
\section{$\Lambda$-Negligibility at $J_1$}

\begin{proposition}[The staircase is marker, not mechanism]
\label{prop:marker}
At the $J_1$ epoch of either A256 survivor, the dark-energy fraction of the
total density is
\[
  \frac{\rho_{\rm DE}}{\rho_{\rm rad}}\Big|_{T = 3\,{\rm PeV}}
    = 1.8\times10^{-74}, \qquad
  \frac{\rho_{\rm DE}}{\rho_{\rm rad}}\Big|_{T = 6\,{\rm PeV}}
    = 1.1\times10^{-75}
\]
for C-$\ln a$ (with $\rho_{\rm DE} = 2\Lz\rho_{\rm crit,0}$ and
$\rho_{\rm rad} = \tfrac{\pi^2}{30}g_*T^4$, $g_* = 106.75$), and
\[
  \frac{\rho_{\rm DE}}{\rho_m}\Big|_{z=17} = 7.5\times10^{-4}, \qquad
  \frac{\rho_{\rm DE}}{\rho_m}\Big|_{z=26} = 2.2\times10^{-4}
\]
for C-$\eta$. Collapse-threshold shifts scale with the dark-energy fraction;
the $J_1$ discontinuity in $\Lambda$ therefore has no dynamical effect on
formation at either epoch. The $\Lambda$ staircase is the \emph{marker} of
fold completions, not the formation \emph{mechanism}.
\end{proposition}

\begin{remark}[Scope of the correction]
A256's map-class results (Endpoint Lemma, $z_4$ squeeze, grid theorem,
survivor signatures) are unaffected: none invoked the mechanism. What changes
is the causal story: formation bursts at $J_1$, if real, must be sourced by
the fold event itself --- the topological collapse $S^3 \to S^2$ acting on
sub-horizon perturbations --- a kernel-level question filed below as
OI-257-1 and adjacent to OI-256-1 (the fold clock).
\end{remark}

% ────────────────────────────────────────────────────────────────────────────
\section{What the Fold-Native Mechanism Must Deliver}

\begin{proposition}[Amplitude requirement]
\label{prop:amplitude}
For the C-$\ln a$ mass $M = 1.1\times10^{24}\,$g, the standard radiation-era
abundance relation $\beta \approx 6.4\times10^{-9}\fpbh\sqrt{M/\Msun}$ and
Gaussian-peaks statistics $\beta = \mathrm{erfc}\!\bigl(\delta_c/\sqrt{2}\sigma\bigr)$
with $\delta_c = 0.45$ give
\[
\begin{array}{lll}
  \fpbh = 1: & \beta = 1.5\times10^{-13}, & \sigma = 0.061,\\
  \fpbh = 10^{-2}: & \beta = 1.5\times10^{-15}, & \sigma = 0.056,\\
  \fpbh = 10^{-4}: & \beta = 1.5\times10^{-17}, & \sigma = 0.053,
\end{array}
\]
a boost of $\sim\!1.1$--$1.2\times10^3$ over the inflationary baseline
$\sigma \approx 5\times10^{-5}$, concentrated at the single $J_1$ horizon
scale.
\end{proposition}

\begin{corollary}[Near-binarity]
\label{cor:binary}
Because $\sigma \propto 1/\sqrt{\ln(1/\beta)}$, the required amplitude moves
only $\sim$15\% while $\fpbh$ moves four orders of magnitude. The fold either
delivers $\sigma \approx 0.05$ and produces a population, or it does not and
produces essentially nothing. There is no tuning freedom in which the
mechanism is real but the population is unobservably small.
\end{corollary}

% ────────────────────────────────────────────────────────────────────────────
\section{The Mandatory Gravitational-Wave Companion}

\begin{proposition}[SIGW companion]
\label{prop:sigw}
A narrow scalar-power spike at the horizon scale of $M$ induces a stochastic
gravitational-wave background peaking at
$f_{\rm peak} \approx 3\,{\rm nHz}\,(M/30\Msun)^{-1/2}$:
\[
  M = 6\times10^{23}\,{\rm g}: f = 0.95\,{\rm mHz}; \qquad
  M = 1.1\times10^{24}\,{\rm g}: f = 0.70\,{\rm mHz}; \qquad
  M = 2.6\times10^{24}\,{\rm g}: f = 0.45\,{\rm mHz}
\]
--- the entire C-$\ln a$ band sits inside the LISA window
($10^{-4}$--$10^{-1}$\,Hz). The amplitude scales as
$\Omega_{\rm GW} \sim A^2$ with spectral amplitude
$A = \sigma^2 \approx 2.5$--$3.7\times10^{-3}$, far above LISA's narrow-band
sensitivity.
\end{proposition}

\begin{corollary}[Fourth kill-shot]
\label{cor:killshot}
Combining Corollary \ref{cor:binary} with Proposition \ref{prop:sigw}: if the
fold-native C-$\ln a$ channel produces \emph{any} PBH population at all, a
narrow SIGW peak must be present at $0.45$--$0.95$\,mHz. LISA non-detection
at that frequency and amplitude falsifies the channel independently of M31
microlensing. Conversely, a LISA detection of a narrow mHz feature with no
inflationary explanation would be the framework's first positive signature.

\emph{Inverted (2026-06-09, same day; A261).} A261 derives $s = 0$
(Hopf-charge superselection): the canon produces no fold-native population
and hence predicts the \emph{absence} of this spike. The frequency
computation above stands; its logical role inverts --- detection at
0.45--0.95 mHz would demonstrate structure beyond the canon.
\end{corollary}

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

\paragraph{What this addendum establishes.} Assumption (c) of A256 is
corrected (Proposition \ref{prop:marker}); the fold-native mechanism's
requirements are quantified (Proposition \ref{prop:amplitude}); and the
framework gains its sharpest falsifier yet (Corollary \ref{cor:killshot}).
The dark-matter half of the original fold-seeding picture is now precise:
fold-seeding supplies dark matter only if the fold event sources a
$\sigma \approx 0.05$ one-scale spike, and that claim is LISA-decidable.

\paragraph{Open items.}
\begin{enumerate}
  \item \textbf{OI-257-1 (mechanism derivation).} Derive from the kernel
        what the $S^3 \to S^2$ fold completion does to sub-horizon
        perturbations --- specifically whether it can source
        $\sigma \approx 0.05$ at the horizon scale. Adjacent to OI-256-1;
        both reduce to the dynamics of the fold event, which the corpus
        currently treats kinematically.
  \item \textbf{OI-257-2 (C-$\eta$ analogue).} The C-$\eta$ survivor's $J_1$
        is matter-era; its fold-native signature would be a feature in the
        halo mass function at $z \approx 17$--$26$ rather than SIGW.
        Quantify once 21\,cm forecasts firm up.
\end{enumerate}

\paragraph{Verifier.} \texttt{Lumen/corpus/addenda/verify/verify\_P257.py}
--- all checks pass, exit 0. Derivation script:
\texttt{Ged/step\_outputs/fold-dm-mechanism/s1\_s3\_quantify.py}.
Plan: \texttt{Ged/plans/fold-dm-mechanism.json}.

\end{document}
