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% Addendum 347a (resolves Addendum 346a F1; uses Paper 18 canonical V_self).
% Author: Leon Fernando Vlegels.  Text CC BY 4.0.
% Verifier: verify_P347.py (11/11 pass).
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\title{Addendum 347a: The Inflation Sector from the Canonical\\
Self-Lensing Potential\\
\large $n_s = 1 - 2/N$ and the resolution of the Addendum~346a tension\\[0.3em]
\normalsize (Addendum to Papers 18, 20, 32)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-15}

\begin{document}
\maketitle

\begin{abstract}
Addendum~346a reduced the fold-to-redshift map to a single dynamical posit and
tested its inflation-sector consequence using the heuristic decaying exponential
$H^2 \sim e^{-x/\kappa}$ of Paper~20~\S7.2, finding a $5.3\sigma$ tension in the
scalar spectral index ($n_s = 0.987$ against Planck's $0.9649$). That tension was
an artefact of the heuristic. The \emph{canonical} self-lensing potential is given
in Paper~18~\S3 as
\[
  V_{\mathrm{self}}^{\mathrm{can}}(r) = \frac{E_{\mathrm{self}}}{m_0^2}
  \bigl(1 - e^{-r/r_0}\bigr), \qquad r_0 = \kappa = \alpha^{5/4},
\]
a \emph{saturating} potential whose shape is concave (a plateau), not the convex
decay used in Addendum~346a. Plateau potentials are the observationally favoured
class. We compute its slow-roll observables and find the universal plateau-attractor
result $n_s = 1 - 2/N$, giving $n_s = 0.9649$ at $N = 57$ e-folds --- an exact
match to Planck across the standard window $N \in [50,60]$ ($n_s \in
[0.960,0.967]$) --- with a tensor-to-scalar ratio $r = 8r_0^2/N^2 \approx
10^{-8}$, far below any foreseeable detection. The Addendum~346a inflation tension
is removed by using the correct potential; the model's standing in the inflation
sector upgrades from strained to a clean, falsifiable prediction.
Verifier: \texttt{verify\_P347a.py} (11/11).
\end{abstract}

\section{The correct potential}
\label{sec:potential}

Paper~18~\S3 fixes the self-lensing potential of the master operator as
\begin{equation}
  V_{\mathrm{self}}^{\mathrm{can}}(r) = \frac{E_{\mathrm{self}}}{m_0^2}
  \bigl(1 - e^{-r/r_0}\bigr),
  \qquad r_0 = \kappa = \alpha^{5/4} = 0.00213,\quad E_{\mathrm{self}} = 13.177.
  \label{eq:vself}
\end{equation}
This rises from $0$ at $r=0$ to the plateau $E_{\mathrm{self}}/m_0^2$ as
$r\to\infty$: it is \emph{concave} ($V''<0$). The heuristic $H^2\sim e^{-x/\kappa}$
used in Addendum~346a is the opposite shape --- a \emph{convex}, monotonically
decaying exponential --- and convex/decaying potentials generically overproduce
$n_s$ (or $r$). The substitution of \eqref{eq:vself} for the heuristic is the whole
content of this addendum.

\section{Slow-roll on the plateau}
\label{sec:slowroll}

Take the inflaton to be the canonically normalised radial field $\phi \equiv r$
($M_{\mathrm{Pl}}=1$) and write $u = e^{-\phi/r_0}$. From \eqref{eq:vself},
\[
  \epsilon = \tfrac12\!\left(\tfrac{V'}{V}\right)^2 = \frac{u^2}{2r_0^2(1-u)^2},
  \qquad
  \eta = \frac{V''}{V} = -\frac{u}{r_0^2(1-u)}.
\]
On the plateau ($u\ll1$): $\epsilon \approx u^2/2r_0^2$ and $\eta \approx -u/r_0^2$,
so $|\eta|\gg\epsilon$. The e-fold count is
\[
  N = \int \frac{V}{V'}\,d\phi \approx \frac{r_0^2}{u_*},
  \qquad\text{hence}\qquad u_* = \frac{r_0^2}{N},
  \quad \eta_* = -\frac1N .
\]

\begin{theorem}[Plateau-attractor spectral index]
\label{thm:ns}
For the canonical self-lensing potential \eqref{eq:vself},
\[
  \boxed{\,n_s = 1 + 2\eta_* - 6\epsilon_* = 1 - \frac{2}{N}\,}
\]
to leading order on the plateau, independent of $r_0$. The tensor-to-scalar ratio
is $r = 16\epsilon_* = 8r_0^2/N^2$, which \emph{does} depend on $r_0$.
\end{theorem}

The $n_s$ result is the same universal value carried by Starobinsky inflation,
Higgs inflation, and the $\alpha$-attractors: the plateau shape, not the detailed
$r_0$, fixes it.

\section{Planck confrontation}
\label{sec:planck}

\begin{proposition}[Match to Planck]
\label{prop:match}
$n_s = 1 - 2/N$ equals the Planck~2018 central value $0.9649 \pm 0.0042$ at
$N = 57$ e-folds, within the standard reheating window. Across $N\in[50,60]$,
$n_s \in [0.960, 0.967]$, all within $\sim\!1.2\sigma$. The tensor-to-scalar ratio
is
\[
  r = \frac{8r_0^2}{N^2} \approx 1.1\times10^{-8}\quad(N=57,\ r_0=\kappa),
\]
far below the bound $r<0.06$ and below any foreseeable sensitivity: the model
predicts effectively no primordial gravitational waves.
\end{proposition}

\begin{corollary}[Resolution of Addendum 346a, F1]
\label{cor:resolve}
The $5.3\sigma$ tension reported in Addendum~346a used the convex heuristic
$e^{-x/\kappa}$, which gives $n_s \approx 0.99$. Replacing it with the canonical
concave potential \eqref{eq:vself} gives $n_s = 1-2/N = 0.9649$. The tension was a
property of the heuristic, not of the corpus; Addendum~346a's falsifier F1 is
discharged.
\end{corollary}

\section*{Status, falsifiers, and caveats}

\textbf{Status.} The inflation sector of the fold cascade now carries a clean,
parameter-free spectral prediction $n_s = 1 - 2/N$ and a definite (tiny) tensor
amplitude. No quantities are fit: $r_0 = \kappa = \alpha^{5/4}$,
$E_{\mathrm{self}} = 13.177$, and the form \eqref{eq:vself} are fixed upstream in
Paper~18; Planck data enter only as the target.

\textbf{Falsifiers.} (F1$'$) A future measurement pinning $n_s$ outside
$[0.960,0.967]$ (the $N\in[50,60]$ band) falsifies the plateau prediction.
(F2$'$) A detection of primordial tensors at $r \gtrsim 10^{-3}$ falsifies the
$r = 8r_0^2/N^2$ prediction (with $r_0 = \kappa$): this model lives at $r\sim
10^{-8}$.

\textbf{Caveats.} (i) The computation takes the inflaton to be the canonically
normalised $B^4$ radial field; the $n_s = 1-2/N$ result is robust to the value of
$r_0$, but a non-flat field-space metric would rescale the map between $\phi$ and
$r$ and could shift $r$ (not $n_s$). (ii) $n_s = 1-2/N$ is the prediction; pinning
the central $0.9649$ requires $N\approx57$, which depends on the reheating history
and is not independently derived here. (iii) The estimate is leading-order
slow-roll; higher-order corrections are below the Planck error bar at these
$\epsilon,\eta$.

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

\end{document}
