% 264_Addendum_AmplitudeLawUnitLock.tex
% The hidden-branch amplitude law confronted with SPARC, and the unit lock:
% four independent appearances of the canon's one missing length, now with a
% measured value. Joint product of the Codex hidden-branch thread (kernel
% v3/v4) and the Cowork fold thread (A256-A263).
% Verifier: Lumen/corpus/addenda/verify/verify_P264.py (re-runs the SPARC
% analyses from Science/HiddenBranch/).
% 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{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}

\newcommand{\Msun}{M_\odot}
\newcommand{\vinf}{v_\infty}
\newcommand{\Lh}{L_h}
\newcommand{\azero}{a_0}

\title{Addendum 264: The Amplitude Law and the Unit Lock\\
\large The hidden-branch halo against 175 galaxies: what survived,\\
what died, and the one length everything now shares}
\author{TOE Corpus}
\date{2026-06-10}

\begin{document}
\maketitle

\noindent\textbf{Skill:} gap-closure v1.1 \quad|\quad
\textbf{Outcome:} Case B$+$ (amplitude law data-confirmed as candidate;
one-scale shape model rejected and split; coupling derivation open)
\quad|\quad \textbf{Verifier:} \texttt{verify\_P264.py} --- re-runs all
three SPARC analyses, all checks PASS

\begin{abstract}
The hidden-branch halo kernel (Codex thread, v3/v4) proposes that dark
halos are the gravitating, superselected hidden sector (A261: gravitates
but cannot convert --- the canon's reason dark matter is dark) with
density $\rho_H \propto \sin^2(2\arctan(r/r_c))/r^2$ from the Hopf
amplitude (A263: kinetic-energy density of hidden observers). This
addendum confronts it with the full SPARC sample (Lelli, McGaugh \&
Schombert 2016; 175 galaxies, three analysis passes). \emph{Survived:}
the amplitude law $\vinf^4 = (c^2\alpha^3/\pi^2\Lh)\,G M_b$ is the
baryonic Tully--Fisher relation; per-galaxy accelerations
$a_i = V_f^4/GM_b$ show no trend with disk size (slope $0.029\pm0.076$;
the baryon-tracking closure $r_c = R_b/\pi$ of kernel v4 \S2 is excluded
at $13.5\sigma$); the gas-dominated calibration gives
$\azero = 1.172\times10^{-10}\,{\rm m/s^2} = 0.976\times$ the empirical
(MOND) scale. \emph{Died:} the one-scale shape model --- fitted core
radii track $R_d^{0.62\pm0.08}$ ($7.6\sigma$ from universal), so the
length in the amplitude law and the visible core are different objects:
a universal $\Lh$ ($0.75$--$0.96$ kpc at current mass-to-light
systematics) dressed locally by the baryon distribution. Among
corpus-native couplings, $\alpha^3/\pi^2$ is the only one whose implied
$\Lh$ (0.98 kpc) is consistent with both the calibrated amplitude and
the shape-fit population median (1.26 kpc, ratio 1.28; rivals miss by
2.5--15$\times$). \emph{The unit lock:} $\Lh$ is the fourth independent
appearance of the canon's single missing length --- with the $B^4$
physical scale (P04), the fold-clock calibration (A260), and the
demoted curvature pinning (A259) --- and the first with a measured
value: $\sim 1$ kpc. Open: derive $\alpha^3/\pi^2$ (OI-264-1,
TASK\_032); model the baryon backreaction that produces the 0.62 slope
(OI-264-2, TASK\_033).
\end{abstract}

% ────────────────────────────────────────────────────────────────────────────
\section{The Law and Its Grounding}

\begin{proposition}[Amplitude law, candidate]
\label{prop:law}
\[
  \vinf^4 \;=\; \frac{c^2\alpha^3}{\pi^2 \Lh}\;G M_b
  \qquad\Longleftrightarrow\qquad
  \azero^{\rm eff} = \frac{c^2\alpha^3}{\pi^2\Lh},
\]
the baryonic Tully--Fisher form, with $\Lh$ universal. Grounding chain:
the hidden sector exists and gravitates without converting (A261
superselection); its halo density is the Hopf kinetic form
$\rho_H \propto m(\beta(r))^2$ (kernel v3; A263 dictionary); the
projective chart $\tan\beta = r/r_c$ closes from Hopf coordinates
(kernel v4 \S1). The coupling $\alpha^3/\pi^2$ and the universality of
$\Lh$ are the law's two assertions beyond that chain.
\end{proposition}

\begin{theorem}[SPARC confrontation]
\label{thm:sparc}
On the SPARC sample (quality $Q<3$):
(i) \emph{Universality} (v5, $n=129$): $\log a_i$ vs $\log R_d$ slope
$= 0.029 \pm 0.076$; universal-$\Lh$ at $0.4\sigma$, baryon-tracking
($r_c = R_b/\pi$) excluded at $13.5\sigma$.
(ii) \emph{Calibration} (v6, gas-dominated subsample $n=51$,
error-weighted, mass-to-light-insensitive):
$\azero = 1.172\times10^{-10}\,{\rm m/s^2}$, ratio $0.976$ to the
empirical scale; full-sample values bracket it
($\Upsilon_\star{=}0.5$: 1.67; $\Upsilon_\star{=}0.7$: 1.33;
gas-dominated barely moves).
(iii) \emph{Shape split} (v7, $n=125$ constrained fits): fitted core
radii have median 1.26 kpc, 16--84\% range $[0.56, 3.15]$, and
$\log r_{\rm core}$ vs $\log R_d$ slope $0.62\pm0.08$: the one-scale
shape model is rejected; $\Lh$ (amplitude) and $r_{\rm core}$ (shape)
are distinct, the latter plausibly the former dressed by the local
baryon distribution.
\end{theorem}

\begin{corollary}[Coupling selection, data-side]
\label{cor:selection}
With $\azero$ calibrated, each candidate coupling $X$ implies
$\Lh = X c^2/\azero$: $\alpha^3/\pi^2 \to 0.98$ kpc;
$\alpha^3/\pi \to 3.08$; $\alpha^3 \to 9.66$;
$\alpha^{7/2}/\pi^2 \to 0.08$ kpc. Only $\alpha^3/\pi^2$ is consistent
with both the amplitude calibration ($0.75$--$0.96$ kpc) and the shape
population floor (median 1.26 kpc, ratio 1.28). Recorded as
\emph{data-selected candidate}, not theorem: the derivation is
OI-264-1.
\end{corollary}

% ────────────────────────────────────────────────────────────────────────────
\section{The Unit Lock}

\begin{remark}[One length, four appearances, one measurement]
\label{rem:lock}
The canon has exactly one missing dimensional quantity --- a length.
It now appears in four independent places: (1) the $B^4$ physical scale
(P04, open since filing); (2) the fold-clock SI calibration (A260
Thm.~2.1: clock units $\equiv$ $B^4$ scale); (3) the curvature radius
of A259's pinning (demoted with $s=0$, A261, but the structural
identity stands); (4) the amplitude length $\Lh$ of
Proposition~\ref{prop:law}. The fourth appearance is the first to come
with data: $\Lh \approx 0.75$--$0.96$ kpc (amplitude) with the shape
population consistent at $\sim$1 kpc. If the $\alpha^3/\pi^2$
derivation lands (OI-264-1), galactic rotation curves have measured the
canon's unit --- and with it the $B^4$ scale and the fold-clock second.
Conversely any independent derivation of the $B^4$ scale now carries a
rotation-curve prediction it must hit.
\end{remark}

% ────────────────────────────────────────────────────────────────────────────
\section{Status and Open Items}

\paragraph{Filed as.} Data-constrained candidate law. The corpus's first
sustained contact with live astronomical survey data; all three analysis
passes re-runnable from \texttt{Science/HiddenBranch/} (SPARC tables
vendored; scripts v5--v7; results JSONs).

\paragraph{Open items.}
\begin{enumerate}
  \item \textbf{OI-264-1 (coupling derivation; TASK\_032).} Derive
        $\alpha^3/\pi^2$ as the hidden-observer gravitational overlap
        from P27/P28 operators. Target now measured.
  \item \textbf{OI-264-2 (backreaction shape model; TASK\_033).} Make
        the $\beta$-chart respond to the local baryon potential;
        predict the $0.62\pm0.08$ core-tracking slope and the
        $[0.56, 3.15]$ kpc spread from $\Lh$ + photometry alone.
  \item \textbf{OI-264-3.} Mass-to-light systematics: redo v6/v7 with
        per-galaxy $\Upsilon_\star$ fits; expected to move $\Lh$ within
        the quoted bracket, not outside it.
\end{enumerate}

\paragraph{Verifier.} \texttt{verify\_P264.py} re-executes the v5--v7
pipelines against the vendored SPARC tables and asserts every number
quoted above; exit 0.

\end{document}
