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% Addendum 346a (extends Paper 32; sequel to Addendum 345a).
% Author: Leon Fernando Vlegels.  Text CC BY 4.0.
% Verifier: verify_P346.py (12/12 pass).
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\newcommand{\Lz}{\Lambda_0}

\title{Addendum 346a: The Fold-to-Redshift Map\\
\large Vacuum-regime gating and the reduction of Conjecture~32.2\\[0.3em]
\normalsize (Addendum to Paper 32; sequel to Addendum 345a)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-15}

\begin{document}
\maketitle

\begin{abstract}
Addendum~345a left one floor load-bearing: the fold-to-redshift map
(Conjecture~32.2), whose closure would turn the fold-progress dark-energy cascade
from a direction-plus-bound into a quantitative $\Lambda(z)$ curve. This addendum
does \emph{not} close it. It does three things that sharpen it. First, it
locates the obstruction: the corpus carries \emph{three} distinct, unbridged
fold-parametrizations --- the boundary-collapse index $k$ (Paper~32 dynamic
extension), the self-lensing parameter $x$ of the inflation law
$H^2 \sim E_{\mathrm{self}}\,e^{-x/\kappa}$ (Paper~20~\S7.2), and the radial
growth phase $\Theta(x)$ of the Cantor bridge (Paper~32 Addendum, $\Theta(1)=
\alpha^{-1}$) --- and Paper~20 states outright that no bridge between them is
supplied. Second, it reframes the conjecture: the one dynamical link on offer is
\emph{vacuum-regime-specific}, since in matter domination $H^2 \propto a^{-3}$
tracks matter rather than the fold; therefore the fold-clock couples to expansion
only when vacuum-dominated, and the map is piecewise --- two active branches
(inflation, present dark energy) bracketing a \emph{dormant matter-era gap}. The
matter era is a gap, not a fold. Third, it reduces the open problem to a single
named posit $P$ (that the inflation law governs both vacuum epochs) and tests
$P$'s independent CMB consequence: with $\kappa = \alpha^{5/4} = 0.00213$ and
$\epsilon \sim \kappa$, the tensor-to-scalar ratio $r = 16\epsilon = 0.034$ sits
under the observational bound, while $n_s = 1 - 6\epsilon = 0.987$ is $5.3\sigma$
high against Planck --- a real tension, recorded not buried. Under $P$, near-closure
sharpens to $\Delta x = \kappa\ln 2 \approx 1.5\times10^{-3}$ of fold-progress.
Verifier: \texttt{verify\_P346a.py} (12/12).
\end{abstract}

\section{The obstruction: three unbridged parametrizations}
\label{sec:obstruction}

A fold-to-redshift map requires a fold coordinate and a law relating it to the
scale factor $a$. The corpus offers three fold coordinates, never identified with
one another.

\begin{description}
\item[(A) Boundary-collapse index $k$.] The dynamic extension of Paper~32 indexes
  the cascade $B^4\to S^3\to S^2\to S^1\to S^0$ by completed folds $k$, with
  $\Lambda(k) = \Lz\cdot 4/C(k)$ and $C(k) = (0,0,2,2,4)$. Discrete.
\item[(B) Self-lensing parameter $x$.] Paper~20~\S7.2 conjectures that breathing
  dynamics drive inflation through
  $H^2 \sim E_{\mathrm{self}}\,e^{-x/\kappa}$, $\kappa = \alpha^{5/4}$. Continuous,
  but ``a conjectural cosmological use of $V_{\mathrm{self}}$; no bridge to
  P18's $V_{\mathrm{self}}^{\mathrm{can}}$ branch is supplied.''
\item[(C) Radial growth phase $\Theta(x)$.] The Cantor-bridge addendum carries
  the accumulated phase $\Theta(x) = 4\pi^3 x^4 + \pi^2 x^3 + \pi x^2$ of the ball
  growing inside the cube, with $\Theta(1) = \alpha^{-1}$. Continuous.
\end{description}

\begin{proposition}[Non-closure from committed structure]
\label{prop:nonclosure}
The map cannot be closed from committed corpus structure alone. Coordinates (A),
(B), (C) are not identified with one another in any corpus result, and no global
law relating any of them to $a$ is supplied. Closing the map requires both an
identification (A$=$B$=$C, or a canonical reduction among them) and a dynamical
law; each is a new object. Per the criterion stated in the Paper~32 dynamic
addendum --- ``whether the mapping closes without introducing new objects
determines whether this is an addendum or a new paper'' --- the map does not close
without new structure.
\end{proposition}

\section{The reframing: vacuum-regime gating}
\label{sec:gating}

The single dynamical link the corpus offers, (B)'s $H^2 \sim e^{-x/\kappa}$, is
not a candidate for a \emph{global} $x(a)$. It is vacuum-regime-specific.

\begin{proposition}[Vacuum gating]
\label{prop:gating}
During matter (or radiation) domination, $H^2 = \tfrac{8\pi G}{3}\rho_m$ with
$\rho_m \propto a^{-3}$, so $H^2$ tracks the matter density, not the fold
coordinate. The relation $H^2 \sim e^{-x/\kappa}$ can hold only when vacuum energy
dominates the expansion. Hence the fold-clock advances with expansion in exactly
two epochs --- early inflation and the present dark-energy era --- and is dormant
through the matter era between them.
\end{proposition}

\begin{corollary}[The map is piecewise; the matter era is a gap]
\label{cor:piecewise}
$x(a)$ is not a single monotone function of $a$. It has two active branches
(inflation, late dark energy) bracketing a dormant interval over which the fold
coordinate is effectively frozen while $a$ grows by most of cosmic history. The
matter-dominated era is a \emph{gap in the fold sequence}, not one of its folds.
This is why a single global $x(z)$ has resisted construction: the object sought
does not exist; a vacuum-epoch fold-clock with a matter-era gap does.
\end{corollary}

Corollary~\ref{cor:piecewise} is corpus-faithful: the folds are boundary and
vacuum phenomena (Papers~27, 32), so their clock is naturally silent while
ordinary matter sets the expansion.

\section{Reduction to a single posit, and its CMB test}
\label{sec:posit}

Granting the reframing, the residual freedom collapses to one statement.

\begin{posit}[Vacuum-epoch fold law]
\label{posit:P}
The inflation law $H^2 \sim e^{-x/\kappa}$ is the fold-clock's dynamical content
in \emph{both} vacuum epochs, inflation and the present dark-energy era. (The
corpus asserts it only for inflation; extending it to the dark-energy epoch is the
single residual posit.)
\end{posit}

Posit~\ref{posit:P} has an independent, falsifiable consequence in the inflation
sector, where the corpus already commits the slow-roll parameter
$\epsilon \sim \kappa = \alpha^{5/4} = 0.00213$.

\begin{proposition}[CMB consequence]
\label{prop:cmb}
With $\epsilon \sim \kappa$, the tensor-to-scalar ratio is
\[
  r = 16\epsilon = 0.034,
\]
under the observational bound $r < 0.06$ (Planck+BK, $95\%$): consistent. The
scalar spectral index is
\[
  n_s = 1 - 6\epsilon = 0.987,
\]
against the measured $n_s = 0.9649 \pm 0.0042$: high by $5.3\sigma$. This is a
genuine tension and is recorded as such; the leading-order slow-roll estimate
omits the $\eta$ contribution, but no $\eta$ from the corpus is supplied to close
the gap. The posit survives on $r$ and is under tension on $n_s$.
\end{proposition}

\section{Near-closure in fold-progress coordinates}
\label{sec:nearclosure}

Under Posit~\ref{posit:P}, the dark-energy epoch has
$\Lambda(x) \sim e^{-x/\kappa}$ in present-critical units (Addendum~345a,
Floor~2). The half-fold-to-now interval is then
\[
  \frac{\Lambda_{\mathrm{half}}}{\Lambda_{\mathrm{now}}} = 2
  \;\Longrightarrow\;
  \Delta x = \kappa\ln 2 \approx 1.5\times10^{-3}.
\]
The entire observable dark-energy history is a $\sim\!10^{-3}$ sliver of
fold-progress: near-closure, now expressed quantitatively in the $x$ coordinate.
This is consistent with --- and sharper than --- the $\sim\!93\%$ amplitude bound
of Addendum~345a (Theorem~2.1), which is the same statement read in the
density-amplitude coordinate rather than the exponential $x$ coordinate.

\section*{Status and falsifiers}

\textbf{Status.} The fold-to-redshift map is \emph{not} closed. It is reduced to
(i)~a canonical identification among the three fold coordinates (A, B, C) and
(ii)~Posit~\ref{posit:P}, and it is reframed as vacuum-gated with a dormant
matter-era gap (Corollary~\ref{cor:piecewise}). No parameters are fit:
$\kappa = \alpha^{5/4}$, the inflation law, and $\Theta$ are all fixed upstream;
Planck CMB data enter only as the measured target.

\textbf{Falsifiers.} (F1) If a future CMB measurement pins $n_s$ away from
$\sim\!0.987$ with $\eta$ unable to reconcile, Posit~\ref{posit:P} fails in the
inflation sector. (F2) If a dark-energy reconstruction demands fold-clock motion
\emph{during} matter domination (evolution where Proposition~\ref{prop:gating}
forbids it), vacuum gating fails. (F3) If a canonical reduction A$=$B$=$C is ever
derived, Proposition~\ref{prop:nonclosure} is superseded and the map upgrades from
reduced to closed.

\paragraph{Data sources.} Planck 2018 inflation constraints
($n_s = 0.9649\pm0.0042$, $r<0.06$ with BICEP/Keck).

\end{document}
