% 256_Addendum_FoldRedshiftMap.tex
% Addendum to Paper 32 / 32_Addendum_DynamicLambda: the fold-to-redshift map.
% Partial closure of Conjecture 32.2; fold-seeded heavy-seed/PBH predictions.
% Verifier: Lumen/corpus/addenda/verify/verify_P256.py (47 checks).
% 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{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{conjecture}[theorem]{Conjecture}

\newcommand{\Lz}{\Lambda_0}
\newcommand{\zfour}{z_4}
\newcommand{\zone}{z_1}
\newcommand{\Msun}{M_\odot}

\title{Addendum 256: The Fold-to-Redshift Map\\
\large Partial closure of Conjecture 32.2; the $\Lambda$ staircase,\\
the $\zfour$ squeeze, and fold-seeded black holes}
\author{TOE Corpus}
\date{2026-06-09}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case B$+$ (map class reduced to two robust candidates;
full closure pending one observational discriminator) \quad|\quad
\textbf{Verifier:} \texttt{verify\_P256.py} --- 47/47 PASS

\begin{abstract}
Paper 32 \S5 records the fold-to-redshift mapping as Conjecture 32.2, and
32\_Addendum\_DynamicLambda derives the fold-progress dark energy law
$\Lambda(k) = \Lz\cdot 4/C(k)$ without dating any fold. This addendum
constrains the mapping $z(k)$ far enough to extract falsifiable cosmology.
Three results. \emph{First}, the Euler oscillation $\chi(S^n) = 1+(-1)^n$
makes $\Lambda(k)$ a three-level \emph{staircase} ($+\infty \to 2\Lz \to \Lz$)
with exactly two discontinuities --- at the $S^2$-fold and $S^0$-fold
completions --- so the fold history imprints on cosmology at exactly two
epochs, not four. \emph{Second}, an Endpoint Lemma plus a distance-to-last-scattering
squeeze confine the $S^0$-fold completion to $\zfour \in [4,\,\sim\!10]$:
the $2\Lz\!\to\!\Lz$ step happened \emph{before} dark-energy domination, which
explains the clean $\Lz$ measured today and decouples the staircase from
low-redshift $w_0w_a$ fits. \emph{Third}, an exhaustive scan of the natural
map class --- four cosmological clocks $\{a, \eta, t, \ln a\}$ $\times$ two
spacing rules --- leaves exactly two robust survivors in the testable window
$\zfour\in[4,6]$: equal conformal spacing (\textsc{C-$\eta$}), which places a
heavy-seed epoch at $\zone \approx 17$--$26$ compatible with JWST overmassive
black holes (UHZ1), and equal logarithmic expansion (\textsc{C-$\ln a$}),
which predicts a primordial black hole population at
$M \approx 0.6$--$2.6\times10^{24}\,$g from a PeV-scale $S^2$-fold.
A structural by-product: every surviving map uses \emph{equal-fold} spacing;
the $\Theta$-cycle ratio fails in every clock --- the $\Theta$-cycle is the
phase ledger of the fold history, not its pacemaker. All quantitative claims
are certified by \texttt{verify\_P256.py} (47 checks, exit 0).
\end{abstract}

% ────────────────────────────────────────────────────────────────────────────
\section{The $\Lambda$ Staircase}
\label{sec:staircase}

Paper 32 \S5 establishes the boundary collapse sequence
$B^4 \to S^3 \to S^2 \to S^1 \to S^0$ with four active boundary folds
$k = 1,\dots,4$, and the addendum DynamicLambda fixes
\begin{equation}
  \Lambda(k) \;=\; \Lz\cdot\frac{4}{C(k)}, \qquad
  C(k) = \sum_{i=1}^{k}\chi(\partial M_i), \qquad
  \Lz = 1 - \frac{\pi^2}{32}.
  \label{eq:lawk}
\end{equation}

\begin{proposition}[Staircase structure]
\label{prop:staircase}
Because $\chi(S^3) = \chi(S^1) = 0$ and $\chi(S^2) = \chi(S^0) = 2$, the
cumulative sum is $C = (0, 0, 2, 2, 4)$ for $k = 0,\dots,4$, and
\eqref{eq:lawk} is a three-level staircase:
\[
  \Lambda = +\infty \;\; (k \le 1), \qquad
  \Lambda = 2\Lz \approx 1.383 \;\; (k = 2, 3), \qquad
  \Lambda = \Lz \approx 0.6916 \;\; (k = 4),
\]
with exactly two discontinuities: at the completion of the $S^2$-fold
(jump $J_1$: $+\infty \to 2\Lz$) and of the $S^0$-fold
(jump $J_2$: $2\Lz \to \Lz$).
\end{proposition}

\begin{proof}
Direct evaluation of \eqref{eq:lawk}; verifier checks 1--8.
\end{proof}

\begin{remark}[Two epochs, not four]
The odd-sphere folds ($S^3$, $S^1$) are cosmologically silent: they advance
$k$ without changing $\Lambda$. Any observable imprint of the fold history
is therefore concentrated at $J_1$ and $J_2$. In particular the pre-fold
regime $\Lambda = +\infty$ persists \emph{through} the completion of the
$S^3$-fold; the natural end-of-inflation marker in this framework is the
$S^2$-fold completion, not the $S^3$-fold.
\end{remark}

\begin{remark}[Collapse-burst hypothesis]
A discontinuous drop in vacuum energy abruptly lowers the effective collapse
threshold for overdensities crossing the horizon near the jump. The working
hypothesis of this addendum --- the fold-seeding hypothesis --- is that
gravitational-collapse activity (primordial black holes in the radiation era,
heavy seeds in the matter era) clusters at the redshifts of $J_1$ and $J_2$.
Since $J_2$ post-dates structure formation (\S\ref{sec:squeeze}), the single
predicted formation epoch is $z(J_1)$.

\emph{Erratum (2026-06-09, same day; full treatment in Addendum 257).}
The $\Lambda$-\emph{mediated} reading of this hypothesis is quantitatively
untenable: at $J_1$ the vacuum-energy fraction is
$\rho_{\rm DE}/\rho_{\rm rad} \sim 10^{-74}$ (C-$\ln a$, PeV era) and
$\rho_{\rm DE}/\rho_m \sim 10^{-4}$ (C-$\eta$, $z \approx 17$--$26$), so the
$\Lambda$ jump cannot shift any collapse threshold. The staircase
discontinuity is the \emph{marker} of the fold completion, not the formation
\emph{mechanism}; the mechanism, if real, is the fold event itself
(fold-native). Every map-class result in this addendum
(\S\ref{sec:squeeze}--\S\ref{sec:survivors}) is unaffected --- none used the
mechanism --- but bridge assumption (c) of \S\ref{sec:status} should be read
as restated by A257, which quantifies what the fold-native mechanism must
deliver and derives its mandatory gravitational-wave companion signature.
\end{remark}

% ────────────────────────────────────────────────────────────────────────────
\section{The Endpoint Lemma and the $\zfour$ Squeeze}
\label{sec:squeeze}

Write $\zfour = z(J_2)$ and $\zone = z(J_1)$ for the redshifts of the two
jumps. All background quantities below use the flat $\Lambda$CDM bridge
($\Omega_m = 0.315$, $\Omega_\Lambda = 0.685$, $h = 0.674$,
$\Omega_r = 9.15\times10^{-5}$); above $\zfour$ the dark-energy density is
doubled per Proposition~\ref{prop:staircase}.

\begin{lemma}[Endpoint Lemma]
\label{lem:endpoint}
Any map $z(k)$ that pins the fold-4 completion to the present epoch
($\zfour = 0$) is excluded, independently of how the earlier folds are spaced.
\end{lemma}

\begin{proof}
With $\zfour = 0$, $\rho_{\rm DE} = 2\Lz\,\rho_{\rm crit,0}$ for all $z > 0$.
Then (i) $\Omega_{\rm DE}(z{=}0.5) = 0.56$, a dark-energy-dominated state at
an epoch where SN~Ia Hubble-diagram fits require matter domination at the
$\sim$5\% level; and (ii) the DESI DR2 Ly$\alpha$ BAO anchor at
$z_{\rm eff} = 2.33$ shifts by $\delta D_H/D_H = +2.7\%$ against a
$\sim$1.2\% measurement. Both exceed their gates by factors $\ge 2$
(verifier checks 13--14). Spacing rules only rescale $z(J_1)$; they cannot
move $J_2$ off $z = 0$. \end{proof}

Lemma~\ref{lem:endpoint} kills the endpoint-normalized axis of the candidate
space wholesale --- including the two superficially attractive maps
``$\Theta$-phase $=$ scale factor'' (whose $J_1$ falls at $z = 13.0$, squarely
in the JWST heavy-seed epoch) and ``$\Theta$-phase $=$ log-temperature
fraction''. The fold-4 completion must predate the present.

\begin{proposition}[$\zfour$ squeeze]
\label{prop:squeeze}
The $2\Lz$ plateau shifts the comoving distance to last scattering while
leaving the sound horizon unchanged ($\zfour \ll z_*$), hence
$\delta\theta_*/\theta_* = -\delta D_*/D_*$. Numerically
\[
  \frac{\delta D_*}{D_*}(\zfour) =
  \{-0.35\%,\ -0.13\%,\ -0.063\%,\ -0.033\%,\ -0.004\%\}
  \quad\text{at}\quad
  \zfour = \{2, 3, 4, 5, 10\}.
\]
Against the Planck $\theta_*$ precision of $0.03\%$ (gate $6\times10^{-4}$
at $2\sigma$, before parameter-degeneracy relaxation) and the Ly$\alpha$
floor $\zfour > 2.4$ of Lemma~\ref{lem:endpoint}(ii), the fold-4 completion
is confined to
\[
  \boxed{\;\zfour \in [4,\ \sim 10]\;}
\]
where the upper end marks loss of testability rather than exclusion: for
$\zfour \gg 10$ the plateau is dynamically negligible and the staircase
becomes observationally indistinguishable from $\Lambda$CDM.
\end{proposition}

\begin{remark}[Three consequences]
(i) We measure a clean $\Lz$ today because $J_2$ predates dark-energy
domination. (ii) The staircase is decoupled from the DESI $w_0w_a$
low-redshift hint --- the framework neither relies on it nor is threatened
by its resolution. (iii) The step remains falsifiable: projected Stage-II /
21\,cm surveys reach $\sim$2\% sensitivity on $\Omega_{\rm DE}(z)$ out to
$z \approx 5$, directly probing the lower half of the window.
\end{remark}

% ────────────────────────────────────────────────────────────────────────────
\section{The Candidate Grid}
\label{sec:grid}

\begin{definition}[Map class]
\label{def:class}
A fold-to-redshift map is a pair (clock, spacing): a monotone cosmological
clock $X \in \{a,\ \eta,\ t,\ \ln a\}$ (scale factor, conformal time, cosmic
time, log-expansion from the Planck epoch) and a spacing rule fixing the
clock-ratio between fold completions, either \emph{equal-fold}
($X(J_1)/X(J_2) = 2/4$) or \emph{$\Theta$-ratio}
($X(J_1)/X(J_2) = \Theta(\tfrac12)/\Theta(1) = 0.0713$, the half-cycle phase
fraction of $\Theta(x) = 4\pi^3x^4 + \pi^2x^3 + \pi x^2$, Paper 28 \S7).
The endpoint $\zfour$ is free, subject to Proposition~\ref{prop:squeeze}.
No object outside the corpus or standard background cosmology enters
(the addendum-eligibility condition of DynamicLambda's Status note).
\end{definition}

\begin{theorem}[Grid verdicts]
\label{thm:grid}
Over the eight cells of Definition~\ref{def:class}, scanned across
$\zfour \in \{4, 5, 6, 8, 10\}$, exactly two cells are robustly viable in
the testable window $\zfour \in [4,6]$:

\medskip
\begin{center}
\begin{tabular}{llll}
\toprule
Clock & Equal-fold ($r = 1/2$) & $\Theta$-ratio ($r = 0.0713$) \\
\midrule
$a$      & tail-only: $\zone = 2\zfour{+}1 \in [9,13]$, too late & dead: pre-halo ($\zone \gtrsim 70$) \\
$\eta$   & \textbf{C-$\eta$: $\zone \in [17,26]$, DCBH window}   & dead: dark ages ($\zone \sim 300$--$900$) \\
$t$      & tail-only: $\zone \in [9,13]$ for $\zfour \le 6$       & marginal: grazes at $\zfour = 4$ ($\zone \approx 28$) \\
$\ln a$  & \textbf{C-$\ln a$: PBH, $M \approx 10^{24}$\,g}        & dead: relic channel ($T \gtrsim 10^{15}$\,GeV) \\
\bottomrule
\end{tabular}
\end{center}
\medskip

\noindent (Verifier checks 22--37; ``tail-only'' cells become viable only at
$\zfour \ge 8$, where the $J_2$ step is unobservable.)
\end{theorem}

\begin{corollary}[$\Theta$ is ledger, not pacemaker]
\label{cor:ledger}
Every robust survivor uses equal-fold spacing; the $\Theta$-ratio rule fails
in all four clocks (it compresses $J_1$ toward the initial singularity in
fast clocks and into the dark ages in slow ones). The $\Theta$-cycle
\emph{records} phase accumulated across the fold history
($\Theta(1) = \alpha^{-1}$, the closure of Paper 28 \S7); it does not
\emph{time} the folds. Fold completions are uniform in their clock; the
phase they deposit is not.
\end{corollary}

% ────────────────────────────────────────────────────────────────────────────
\section{The Two Survivors}
\label{sec:survivors}

\subsection{C-$\eta$: heavy seeds at $\zone \approx 17$--$26$}

Equal conformal spacing, $\eta(J_1) = \tfrac12\,\eta(J_2)$. For
$\zfour \in [4,6]$ the $S^2$-fold completion lands at $\zone \in [17,26]$
--- the matter era, where the horizon-collapse bridge is inapplicable and
the natural reading is a \emph{seeding epoch}: the atomic-cooling-halo
window in which direct-collapse seeds of $10^4$--$10^6\,\Msun$ form. A
$10^4\,\Msun$ seed at $z \approx 20$ growing Eddington-limited reaches
$4\times10^{7}\,\Msun$ by $z = 10.1$, matching UHZ1 and the overmassive
black-hole-to-stellar-mass ratios JWST reports at $z \gtrsim 9$.

The map is over-determined: one parameter ($\zfour$) fixes both the
$w(z)$ step (future Stage-II/21\,cm reach) and the seeding epoch
$\zone = z(\eta = \eta(\zfour)/2)$ (JWST seed demographics). Either
measurement predicts the other; disagreement kills the map.

\subsection{C-$\ln a$: a $10^{24}$\,g PBH population from a PeV-scale fold}

Equal log-expansion spacing from the Planck epoch,
$\ln(a(J_1)/a_{\rm Pl}) = \tfrac12 \ln(a(J_2)/a_{\rm Pl})$. The logarithm
makes $\zone$ insensitive to $\zfour$: across the whole window the
$S^2$-fold completes in the radiation era at $T \approx 3$--$6$\,PeV,
giving (with $M_{\rm PBH} = \gamma M_H$, $\gamma = 0.2$)
\[
  M_{\rm PBH} \;\approx\; 0.6\text{--}2.6\times10^{24}\ \mathrm{g}
  \;\approx\; 10^{-9}\,\Msun,
\]
a band ratio under 5 across $\zfour \in [4,10]$ (checks 29--30, 41--43).
The band sits just above the open asteroid-mass window
($10^{17}$--$10^{23}$\,g): the original Subaru/HSC M31 limits cap such a
population at $f_{\rm PBH} \lesssim$ few~$\times10^{-2}$, while
finite-source revisions weaken those limits by up to three orders of
magnitude, leaving the peak at the boundary of the revised open region.
The prediction is decided by next-generation high-cadence M31 microlensing
at $10^{-10}$--$10^{-8}\,\Msun$. The map requires reheating above the PeV
scale (the observational floor is $\sim$MeV; low-scale inflation is
admissible but becomes a stated assumption).

\subsection{Discriminators}

The survivors are mutually exclusive in their primary signature (matter-era
seeding epoch vs.\ radiation-era mass peak) but not logically exclusive ---
both could hold if the fold clock is scale-dependent, which would be a new
object and hence a new paper, not this addendum. The clean discriminators:
(i) a measured $w(z)$ step at $\zfour \in [4,6]$ with \emph{no} seed-epoch
excess at $z(\eta{=}\eta(\zfour)/2)$ kills C-$\eta$;
(ii) exclusion of $f_{\rm PBH} > 10^{-4}$ at $10^{24}$\,g kills C-$\ln a$'s
relevance; (iii) a seed-demographics peak at a redshift inconsistent with
every $\zfour \in [4,6]$ under $\eta$-halving kills C-$\eta$ from the other
side.

% ────────────────────────────────────────────────────────────────────────────
\section{Status, Scope, and Open Items}
\label{sec:status}

\paragraph{What this addendum closes.} Conjecture 32.2 asked for the
fold-to-redshift mapping. This addendum does not produce a unique map; it
(i) proves the mapping question reduces to two jumps (Proposition
\ref{prop:staircase}), (ii) confines the late jump to $\zfour \in [4,10]$
from existing data (Proposition \ref{prop:squeeze}), and (iii) reduces the
admissible map class from eight natural candidates to two robust survivors
with disjoint, falsifiable signatures (Theorem \ref{thm:grid}). Conjecture
32.2 is therefore \emph{partially closed}: open at the level of map
uniqueness, closed at the level of phenomenological consequence. Per the
DynamicLambda Status note, no new mathematical objects were introduced, so
this result remains within the Paper 32 addendum lineage.

\paragraph{Bridge assumptions (stated, not derived).}
(a) $\Lambda$CDM background for the $z \leftrightarrow (\eta, t)$
dictionary; (b) $M_{\rm PBH} = 0.2\,M_H$ horizon-collapse relation in the
radiation era; (c) the collapse-burst hypothesis itself (jumps in $\Lambda$
trigger formation bursts) --- plausible from the abrupt change in collapse
threshold, not derived from the kernel. A TOE-native replacement for (b)
and a derivation of (c) from the fold dynamics are the natural next steps.

\paragraph{Open items.}
\begin{enumerate}
  \item \textbf{OI-256-1 (map uniqueness).} Derive the fold clock from
        first principles. Corollary \ref{cor:ledger} (equal-fold spacing
        forced; $\Theta$ is ledger, not pacemaker) is the structural
        constraint any derivation must respect; the breath period lacks an
        SI anchor (A239 closes $T_{\rm breath}$ in system units only), so
        the unit problem is the same one that blocks the $B^4$ physical
        scale (Paper 04).
  \item \textbf{OI-256-2 (Hubble tension).} Unchanged from Paper 32
        Conjecture 6.2; nothing here bears on it.
  \item \textbf{OI-256-3 (growth and ISW): resolved at first order.}
        Linear-growth integration through the staircase (fixed early
        normalization, RK4 on the growth ODE; script
        \texttt{growth\_isw\_check.py}, verifier checks 45--47) gives a
        $\sigma_8$ deficit that is \emph{sub-percent across the entire
        window}: $0.02$--$0.7\%$ depending on how finely the integrator
        resolves the $\Lambda$ discontinuity (the spread is sampling
        jitter at the step, not physics; both resolutions tested agree on
        magnitude $< 1\%$ and on sign). The sign is \emph{suppression} ---
        the $S_8$-tension direction, though far too small to bear on that
        tension. The potential decay rate $\Phi \sim D/a$ deviates by at
        most $0.3\%$ over $z \in [\zfour, 50]$, so the ISW imprint is
        invisible in TT. Conclusion: growth observables can neither
        falsify nor detect the staircase at current or near-future
        precision (lensing amplitude $\sim$2--3\%); the map class is
        decided exclusively by the expansion-history step ($w(z)$ at
        $\zfour$) and the $J_1$ formation signatures of
        \S\ref{sec:survivors}. A smoothed-step Boltzmann run would remove
        the jitter but cannot change the sub-percent verdict.
\end{enumerate}

\paragraph{Verifier.} \texttt{Lumen/corpus/addenda/verify/verify\_P256.py}
--- 47 checks across six sections (fold machinery, $\Theta$-closure at 50
digits, Endpoint Lemma, $\zfour$ squeeze, full grid, survivor invariants);
all pass, exit 0. Derivation script:
\texttt{Ged/step\_outputs/fold-pbh-seeding/s2\_derivation.py}.
Plan: \texttt{Ged/plans/fold-pbh-seeding.json} (s1--s4, complete).

\end{document}
