\documentclass[11pt]{article}
% Addendum 393 (the H0 tension is anchor-limited). H0=2alpha^2/T_breath is calibrated through ONE
% dimensionful anchor a0 (P35/366), and the corpus ladder makes H0 ~ sqrt(a0). a0 carries +/-22%
% (388's own 1.2+/-0.26), so the H0 PREDICTION inherits +/-11% (75.8+/-8.2). 388 used that +/-22% to
% pass a0 but treated H0 as exact (~3 sigma) -- an inconsistent budget. Applying it consistently:
% H0 is 0.32 sigma vs SH0ES (0.88 DESI, 1.02 Planck) -- CONSISTENT within the calibration uncertainty,
% central leaning high. A -7% re-anchor of a0 (within SPARC) hits SH0ES. Revises 388's H0 verdict from
% TENSION to consistent-within-anchor; does NOT confirm H0 (central leans high) and does NOT touch the
% separate DE-evolution tension. Author: Leon Fernando Vlegels. CC BY 4.0. Verifier: verify_P393.py (10/10).
\usepackage[margin=1in]{geometry}
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\usepackage{hyperref}
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
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\newcommand{\Tb}{T_{\mathrm{breath}}}

\title{Addendum 393: The $H_0$ Tension Is Anchor-Limited\\
\large $H_0\propto\sqrt{a_0}$ through the chart's single dimensionful anchor, so the\\
$H_0$ prediction inherits the anchor's $\pm22\%$ --- and the $\sim\!3\sigma$ tension was\\
an inconsistent error budget, not a structural over-prediction\\[0.3em]
\normalsize (Digs the chart anchoring left open by 392; corrects the $H_0$ verdict of 388; Papers 35, 39;
Addenda 358a, 362, 366)}
\author{L\'eon Fernando Vlegels}
\date{2026-06-19}

\begin{document}
\maketitle

\begin{abstract}
With the horizon shape (389), the family orbifold (390), and the breath dynamics (392) all closed by
derivation, Addendum~392 re-pointed the $H_0$ over-prediction ($H_0=2\alpha^2/\Tb=75.8$~km/s/Mpc) to the
chart anchoring. This addendum digs there, and finds the tension was overstated by an inconsistent error
budget. \textbf{The chart has one dimensionful anchor.} Paper~35 (via 366): $a_0=\alpha$ is ``the one
dimensionful input \ldots calibrated conditional on the measured $a_0$''; every dimensionful corpus number,
$H_0$ included, inherits its absolute scale from this single anchor. \textbf{The ladder makes
$H_0\propto\sqrt{a_0}$.} Composing the corpus's own scale ladder --- the inner step $a_0\Tb^2/L_h=\alpha$
and the outer step $1/H_0=(\alpha^{-2}/2)\Tb$ (358a/362) --- and eliminating $\Tb$ gives
$1/H_0=\tfrac12\alpha^{-3/2}\sqrt{L_h/a_0}$, so $H_0\propto\sqrt{a_0}$. \textbf{The anchor's uncertainty
propagates.} The measured anchor $a_0=1.2\pm0.26\times10^{-10}\,\mathrm{m/s^2}$ (SPARC --- the value
Addendum~388 itself cites) is $\pm22\%$; since $H_0\propto\sqrt{a_0}$, the $H_0$ \emph{prediction} carries
$\pm11\%$: $H_0=75.8\pm8.2$. \textbf{388's budget was inconsistent.} 388 scored $a_0$ a PASS \emph{using}
that $\pm22\%$, but scored $H_0$ at $\sim\!3\sigma$ using only the measurement error --- implicitly
treating the prediction as exact. But $H_0$ and $a_0$ are the \emph{same} anchor; the error bar that passes
$a_0$ must widen the $H_0$ prediction. Applying it consistently: $H_0$ is $0.32\sigma$ from SH0ES
($0.88\sigma$ DESI, $1.02\sigma$ Planck) --- \emph{consistent} within the calibration uncertainty.
Equivalently, re-anchoring $a_0$ by just $-7\%$ (well inside SPARC's $\pm22\%$) lands $H_0$ exactly on
SH0ES. \textbf{What this does and does not do.} It revises 388's $H_0$ verdict from ``$\sim\!3\sigma$
TENSION'' to ``consistent within the $a_0$ anchor, central value leaning high.'' It does \emph{not} confirm
$H_0$ (the central prediction is still above every datum; sharpening requires pinning $a_0$ below
$\pm22\%$), and it does \emph{not} touch the separate DE-evolution tension (388), which is a shape
disagreement with DESI and stands. All claims pinned by \texttt{verify\_P393.py} (10/10).
\end{abstract}

\section{One anchor, and $H_0\propto\sqrt{a_0}$}
\label{sec:anchor}

\begin{proposition}[The chart's single dimensionful anchor]
\label{prop:anchor}
Paper~35, as quoted in Addendum~366: ``$a_0=\alpha$ is the corpus's dimensionful anchor \ldots the unit
system is calibrated conditional on the measured $a_0$ \ldots it is the one dimensionful input and is not
derivable.'' The dimensionless geometry fixes every \emph{ratio}; converting any ratio to a physical
(dimensionful) value requires exactly this one anchor. $H_0$, a dimensionful quantity, is no exception: its
absolute scale in km/s/Mpc is set entirely by $a_0$.
\end{proposition}

\begin{theorem}[$H_0\propto\sqrt{a_0}$]
\label{thm:scaling}
The corpus's scale ladder (358a/362) has two steps: the galactic step $a_0\Tb^2/L_h=\alpha$ (exact) and
the cosmological step $1/H_0=(\alpha^{-2}/2)\,\Tb$. Eliminating $\Tb=\sqrt{\alpha L_h/a_0}$ from the first
and substituting into the second,
\begin{equation}
\frac{1}{H_0}=\frac{\alpha^{-2}}{2}\sqrt{\frac{\alpha L_h}{a_0}}
=\frac12\,\alpha^{-3/2}\sqrt{\frac{L_h}{a_0}},
\qquad\text{so}\qquad
H_0=2\,\alpha^{3/2}\sqrt{\frac{a_0}{L_h}}.
\end{equation}
$H_0$ is therefore proportional to $\sqrt{a_0}$ (and to $1/\sqrt{L_h}$). The dimensionless factor
$2\alpha^{3/2}$ is pinned; the entire physical scale of $H_0$ rides on the anchor $a_0$ (and the galactic
length $L_h$).
\end{theorem}

\section{The propagated uncertainty, and 388's inconsistent budget}
\label{sec:prop}

\begin{proposition}[The $H_0$ prediction inherits $\pm11\%$]
\label{prop:pm11}
The anchor is the measured MOND/SPARC acceleration $a_0=1.2\pm0.26\times10^{-10}\,\mathrm{m/s^2}$, the value
Addendum~388 cites in its own scorecard --- a $\pm22\%$ uncertainty. By Theorem~\ref{thm:scaling},
$H_0\propto\sqrt{a_0}$, so the fractional uncertainty halves: the $H_0$ \emph{prediction} carries
$\pm11\%$, i.e.\ $H_0=75.8\pm8.2$~km/s/Mpc. This is a property of the prediction, independent of any
measurement of $H_0$.
\end{proposition}

\begin{theorem}[Applying the anchor error consistently dissolves the tension]
\label{thm:consistent}
Addendum~388 scored $a_0$ a PASS by \emph{using} its $\pm22\%$ error bar, but scored $H_0$ at
$\sim\!3\sigma$ by using only the measurement error ($\pm0.86$ for SH0ES), which treats the prediction
$75.8$ as exact. Because $H_0$ and $a_0$ are the \emph{same} anchor (Theorem~\ref{thm:scaling}), the error
bar that makes $a_0$ a PASS must equally widen the $H_0$ prediction (Proposition~\ref{prop:pm11}). Combining
the prediction uncertainty ($\pm8.2$) with each measurement error in quadrature:
\begin{equation}
\text{SH0ES }73.17:\ 0.32\sigma,\qquad
\text{DESI }68.53:\ 0.88\sigma,\qquad
\text{Planck }67.4:\ 1.02\sigma.
\end{equation}
(388 reported $3.1$, $9.1$, $16.8\sigma$ respectively, data-error only.) The $H_0$ prediction is consistent
with all three within the calibration uncertainty. Equivalently, since $H_0\propto\sqrt{a_0}$, re-anchoring
$a_0$ by $-7\%$ --- well within SPARC's $\pm22\%$ --- lands $H_0$ exactly on SH0ES, so the bridge fits the
galactic and the local-$H_0$ data simultaneously.
\end{theorem}

\section{What this revises, and what it does not}
\label{sec:scope}

\begin{remark}[Not a confirmation]
\label{rmk:notconfirm}
This dissolves an overstated tension; it does not confirm $H_0$. The central prediction, $75.8$, still lies
above every current datum --- the bridge leans high. What changed is the error bar attached to it: the
prediction is $\pm11\%$, not exact, because it is anchored on a quantity known to $\pm22\%$. The honest
statement is ``consistent within a wide calibration uncertainty, central value high,'' not ``correct.''
Sharpening $H_0$ into a decisive test requires pinning $a_0$ below $\pm22\%$, or introducing an independent
dimensionful anchor that does not route through $a_0$.
\end{remark}

\begin{remark}[The DE-evolution tension is untouched]
\label{rmk:de}
388's other tension --- the dark-energy \emph{evolution} shape against DESI DR2 --- does not involve the
$a_0$ anchor (it is a comparison of $w(z)$ shapes, dimensionless). This addendum says nothing about it; it
stands as recorded. The dig localizes and corrects only the $H_0$ residual.
\end{remark}

\begin{remark}[The H_0 program, re-pointed]
\label{rmk:program}
389/390/392 closed the three near-surface axes by derivation; this addendum shows the remaining residual is
not structural but anchor-limited. The program is therefore re-pointed once more: the open question is no
longer ``what lowers $75.8$?'' but ``how well is the anchor $a_0$ known, and is there a tighter dimensionful
anchor?'' A future MOND/SPARC determination of $a_0$ to $\pm5\%$ would tighten the $H_0$ prediction to
$\pm2.5\%$ and turn it back into a sharp test --- which the corpus, with $H_0\propto\sqrt{a_0}$ fixed, would
then either pass or fail cleanly.
\end{remark}

\section*{Status}
Digging into the chart anchoring (the axis 392 left open), the $H_0$ over-prediction is found to be
anchor-limited, not structural. $H_0=2\alpha^2/\Tb$ is calibrated through the chart's single dimensionful
anchor $a_0$ (Paper~35/366; Proposition~\ref{prop:anchor}), and composing the corpus's own ladder gives
$H_0\propto\sqrt{a_0}$ (Theorem~\ref{thm:scaling}). The measured $a_0=1.2\pm0.26$ ($\pm22\%$, 388's own
cited value) therefore gives the $H_0$ \emph{prediction} a $\pm11\%$ uncertainty, $75.8\pm8.2$
(Proposition~\ref{prop:pm11}). 388 used the $\pm22\%$ to pass $a_0$ but treated the $H_0$ prediction as
exact when scoring $\sim\!3\sigma$ --- an inconsistent budget for what is one shared anchor. Applying the
anchor error consistently revises $H_0$ to $0.32\sigma$ from SH0ES ($0.88\sigma$ DESI, $1.02\sigma$
Planck): consistent within the calibration uncertainty, central value leaning high
(Theorem~\ref{thm:consistent}); equivalently a $-7\%$ re-anchor of $a_0$, within SPARC, hits SH0ES exactly.
This corrects 388's $H_0$ verdict from TENSION to consistent-within-anchor; it does \emph{not} confirm
$H_0$ (the central still leans high; Remark~\ref{rmk:notconfirm}) and does \emph{not} touch the separate
DE-evolution tension (Remark~\ref{rmk:de}). The $H_0$ program is re-pointed from ``what lowers $75.8$'' to
``pin the anchor'' (Remark~\ref{rmk:program}). No published number changes; 388's $H_0$ significance is
corrected. Verifier: \texttt{verify\_P393.py}, 10/10 pass.

\end{document}
