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% Addendum 362 (extends Paper 14 geometric hierarchy; consolidates 358a/359a; corrects 361).
% Author: Leon Fernando Vlegels. Text CC BY 4.0.
% Verifier: verify_P362.py (9/9 pass).
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\title{Addendum 362: The $\alpha$-Power Scale Ladder\\
\large The cosmological scale-bridge is $1/H_0=(\pi/2)\,\alpha^{-3}\tau_1$, derived to\\
a single $2\pi$; and the withdrawal of the 361 unifying-floor observation\\[0.3em]
\normalsize (Addendum to Paper 14; consolidates 358a/359a; corrects 361)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-18}

\begin{document}
\maketitle

\begin{abstract}
Addendum~361 conjectured --- flagged as observation, not theorem --- that the Goldbach-weight
null and the Hubble-$2\pi$ floor are ``the same gap,'' the corpus lacking a bridge from its
intrinsic clock (the breath) to external scales. Grounding that conjecture in the corpus's
actual scale machinery yields two findings. \textbf{Positive: the cosmological scale-bridge is
an $\alpha$-power ladder, derived down to a single $2\pi$.} Paper~14's geometric hierarchy
already derives the inner step: multiplication by $\pi$ lifts $\alpha^{-1}$ to the breath,
$\Tb=\pi\alpha^{-1}\tau_1$ (the surviving $G_1=432/\Tb-1$ identity). Addendum~358a gives the
outer step via the de~Sitter horizon $a_0=cH_0/2\pi$: $1/H_0=(\alpha^{-2}/2)\Tb$. Composing,
\[
  \frac1{H_0}=\Bigl(\frac{\alpha^{-2}}{2}\Bigr)\bigl(\pi\alpha^{-1}\bigr)\tau_1
   =\frac{\pi}{2}\,\alpha^{-3}\,\tau_1,
\]
so the Hubble time is $(\pi/2)\,\alpha^{-3}$ fundamental periods (numerically $4.04\times10^6$
against measured $4.2$--$4.6\times10^6$, the $\sim\!10\%$ $2\pi$ slack). The bridge is thus
\emph{derived} except for the single de~Sitter $2\pi$ --- more derived than 357a's ``named
floor, no candidate.'' \textbf{Correction: the 361 unification is withdrawn.} The two steps of
the ladder differ in kind: the inner ($\alpha^{-1}$, fundamental$\to$breath) is Paper~14,
derived; the outer ($\alpha^{-2}$, breath$\to$Hubble) is the de~Sitter horizon, floored at the
$2\pi$. And the Goldbach null is not a scale-bridge floor at all --- it is geometry/arithmetic
\emph{orthogonality} (360/361: the singular series is arithmetic, no geometric signal at any
level). The breath bridges to physical external scales (cosmology) through horizon physics; the
integer line is not a scale it ``fails'' to reach. So the conjectured single floor dissolves
into one real bridge (cosmological) plus one orthogonality (geometry vs arithmetic). All claims
pinned by \texttt{verify\_P362.py} (9/9).
\end{abstract}

\section{The ladder}
\label{sec:ladder}

\begin{proposition}[The inner step is Paper 14's $\pi$-lift, derived]
\label{prop:inner}
Paper~14 \S2 (``The Geometric Hierarchy of Scales'') derives the breath as the next level of a
$\pi$-multiplication hierarchy above $\alpha^{-1}$: $\Tb=\pi\alpha^{-1}$ in fundamental
($\tau_1$) units, the surviving content being the exact identity $G_1=432/\Tb-1$ with
$\Tb=\pi\alpha^{-1}$ (Addendum~266; Paper~37). Numerically $\Tb/\tau_1=430.59=\pi\alpha^{-1}$.
The fundamental$\to$breath step is therefore $\alpha^{-1}$ and corpus-derived.
\end{proposition}

\begin{theorem}[The $\alpha$-power scale ladder]
\label{thm:ladder}
Combining Proposition~\ref{prop:inner} with the de~Sitter outer step of Addendum~358a
($1/H_0=(\alpha^{-2}/2)\Tb$),
\[
  \frac1{H_0}=\frac{\pi}{2}\,\alpha^{-3}\,\tau_1,
\]
the Hubble time is $(\pi/2)\,\alpha^{-3}$ fundamental periods. This evaluates to
$4.04\times10^6\,\tau_1$ against the measured $4.20\times10^6$ (SH0ES) and $4.55\times10^6$
(Planck) --- consistent within the $\sim\!10\%$ slack that is the un-derived de~Sitter $2\pi$
(Addendum~359a). The cosmological scale is thus pinned to the fundamental period by a single
$\alpha^{-3}$ and an $O(1)$ coefficient $\pi/2$, derived except for that $2\pi$.
\end{theorem}

\section{The correction}
\label{sec:correction}

\begin{remark}[The ladder is not a single principle]
\label{rmk:notuniform}
The two steps differ in power \emph{and} in status: the inner step is $\alpha^{-1}$ and derived
(Paper~14); the outer step is $\alpha^{-2}$ and floored at the de~Sitter $2\pi$ (358a/359a). The
$\alpha^{-3}$ of Theorem~\ref{thm:ladder} factors as $\alpha^{-1}\cdot\alpha^{-2}$, two distinct
mechanisms, not one uniform lift. There is no single scale-bridge principle generating the whole
ladder.
\end{remark}

\begin{remark}[Withdrawal of the 361 unifying-floor observation]
\label{rmk:withdraw}
Addendum~361 observed (explicitly as conjecture, not theorem) that the Goldbach-weight null and
the Hubble-$2\pi$ floor are the same gap. Grounding withdraws this. The cosmological floor is the
de~Sitter $2\pi$ --- a \emph{physical} horizon factor, with the rest of the bridge derived
(Paper~14 $\pi$-lift plus 358a). The Goldbach null (360/361) is geometry/arithmetic
\emph{orthogonality}: the representation counts are the arithmetic singular series, and no
geometric signal appears at any level; the integer line is not a physical scale the breath
bridges to at all. The two are different in kind. What 361 got right is narrow --- both involve
the breath and an external structure --- but the cosmological case is a (mostly derived) physical
bridge, while the prime case is an orthogonality, not a missing bridge. The unification was
over-reach and is withdrawn here.
\end{remark}

\section*{Status}
\textbf{Positive.} The cosmological scale-bridge is the $\alpha$-power ladder
$1/H_0=(\pi/2)\,\alpha^{-3}\tau_1$ (Theorem~\ref{thm:ladder}): the inner step
($\times\pi\alpha^{-1}$, fundamental$\to$breath) is derived by Paper~14's geometric hierarchy
(Proposition~\ref{prop:inner}), and the outer step ($\times\alpha^{-2}/2$, breath$\to$Hubble) is
the 358a de~Sitter bridge, floored only at the single $2\pi$ (359a). The cosmological scale is
therefore pinned to the corpus fundamental period \emph{derived down to one horizon factor} ---
sharper than 357a's original ``named floor, no candidate.'' \textbf{Correction.} The ladder is
not a single principle (Remark~\ref{rmk:notuniform}), and the 361 conjecture that the Goldbach
null and the Hubble floor are one gap is withdrawn (Remark~\ref{rmk:withdraw}): the cosmological
case is a mostly-derived physical bridge with a $2\pi$ residual, the prime case is
geometry/arithmetic orthogonality. The honest picture is one real bridge plus one orthogonality,
not a unification. No new objects beyond the de~Sitter relation; no published number changes.
Verifier: \texttt{verify\_P362.py}, 9/9 pass.

\paragraph{What remains.}
The lone residual of the cosmological bridge is unchanged and now isolated: the de~Sitter $2\pi$
(the horizon-thermodynamics program of 359a). Everything else in $1/H_0=(\pi/2)\alpha^{-3}\tau_1$
is corpus-derived. Closing the $2\pi$ would make the Hubble time a clean $\alpha^{-3}$ prediction
from the fundamental period.

\end{document}
