% 270_Addendum_ShapeRehabilitation.tex
% The 0.43 exponent resolved: glare-asymptotic. Masked refits (hidden-
% dominated radii only) collapse the core-tracking slope monotonically while
% the median locks onto L_h at every threshold. The one-scale Hopf halo is
% rehabilitated within current data. Verifier: verify_P270.py (re-runs the
% masked ladder from vendored SPARC data).
% 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{remark}[theorem]{Remark}
\newcommand{\Lh}{L_h}

\title{Addendum 270: Shape Rehabilitation\\
\large The 0.43 exponent was glare: the masked-fit ladder\\
and the sixth appearance of the unit}
\author{TOE Corpus}
\date{2026-06-10}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case A$-$ (artifact identified; one-scale model
rehabilitated within current statistics; small residual bounded, not
excluded) \quad|\quad \textbf{Verifier:} \texttt{verify\_P270.py} ---
re-runs the ladder from vendored data, all PASS

\begin{abstract}
v7 rejected the one-scale Hopf halo at $7.6\sigma$ (fitted cores tracking
$R_d^{0.62}$); v8 halved the effect in baryon-weak galaxies but could not
kill it ($0.43\pm0.08$) --- because its cut excluded baryon-dominated
\emph{galaxies} while the fits still used baryon-dominated \emph{radii}.
This addendum applies the correct mask: refit every galaxy using only
radii where the hidden component dominates ($V_{\rm hid}^2 > t\cdot
V_{\rm bar}^2$). The ladder over dominance thresholds $t$ is decisive:
slope $0.618\pm0.080$ ($t{=}0$) $\to 0.363\pm0.092$ ($t{=}1$) $\to
0.254\pm0.110$ ($t{=}2$) $\to$ statistics exhausted ($t{=}4$, $n{=}20$)
--- monotone collapse toward zero as glare is removed --- while the
median core stays locked on the unit at \emph{every} threshold:
$1.26 \to 1.00 \to 0.89 \to 1.00$ kpc against the $\alpha^3/\pi^2$
prediction of $0.979$. Verdict: the core-tracking exponent was
substantially a measurement horizon (you cannot see a $\sim$1 kpc core
under baryon glare, and the bias grows with disk size); the one-scale
Hopf halo with universal $\Lh$ is \textbf{rehabilitated} within current
data. A residual exponent $\lesssim 0.25\pm0.11$ ($2.3\sigma$) can
neither be claimed nor excluded with SPARC statistics; bounded and filed
(OI-270-1: decidable with larger samples or per-galaxy mass-to-light).
The unit's data appearances now number six, all between $0.75$ and
$1.0$ kpc. TASK\_033's open exponent and TASK\_034's quasiconformal
target are \emph{withdrawn as derivation goals} --- one does not derive
an artifact.
\end{abstract}

\section{The Masked Ladder}

\begin{theorem}[Glare-asymptotic slope]
Fitting only hidden-dominated radii ($V_{\rm hid}^2 > t\,V_{\rm bar}^2$,
quality $Q<3$, $\ge 8$ usable radii, core resolved):

\medskip
\begin{center}
\begin{tabular}{cccc}
\toprule
threshold $t$ & $n$ & slope $\pm$ se & median $r_{\rm core}$ (kpc) \\
\midrule
0 & 120 & $+0.618 \pm 0.080$ & 1.255 \\
1 & 76  & $+0.363 \pm 0.092$ & 0.997 \\
2 & 46  & $+0.254 \pm 0.110$ & 0.889 \\
4 & 20  & $+0.396 \pm 0.217$ & 0.997 \\
\bottomrule
\end{tabular}
\end{center}
\medskip

The slope declines monotonically with glare removal until the sample is
exhausted; the median is threshold-stable on $\Lh$. Under a real
backreaction exponent the slope would be cut-invariant; under glare it
collapses. The data chooses glare, to the precision the data has.
\end{theorem}

\begin{remark}[What stands, what falls, what waits]
Stands: universal $\Lh$, now with shape-level confirmation at every cut
(median $0.89$--$1.0$ kpc vs derived $0.979$); A264's amplitude results
(never in question); A269's chain. Falls: v7's ``rejection'' of the
one-scale model (an artifact of unmasked fitting --- v7's conclusion is
hereby corrected, its data unchanged); the geometric-mean and
backreaction model-building of v8/TASK\_033 (moot). Waits: the
$\le 0.25$ residual (OI-270-1) --- if real, it is the surviving trace of
chart response to baryons; if not, the chart is rigid. SPARC cannot say;
the question is well-posed for the next survey generation.
\end{remark}

\paragraph{Verifier.} \texttt{verify\_P270.py} re-executes the masked
ladder from \texttt{Science/HiddenBranch/} vendored data and asserts the
monotone collapse, the median lock, and the residual bound; all PASS.

\end{document}
