% sparc_one_scale_halo.tex
% External-facing paper: the one-scale hidden-sector halo vs SPARC.
% Self-contained; no corpus dependence assumed of the reader.
% Every quoted number re-derivable via verify_paper_sparc.py.
% Copyright Léon Fernando Vlegels -- CC BY 4.0

\documentclass[11pt]{article}
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\usepackage{booktabs}
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\newcommand{\kms}{\,\mathrm{km\,s^{-1}}}
\newcommand{\Msun}{M_\odot}
\newcommand{\azero}{a_0}
\newcommand{\Lh}{L_h}

\title{A Single Kiloparsec Length in Galaxy Rotation Curves:\\
\large SPARC evidence for a one-scale halo with a derived
acceleration scale}
\author{L\'eon Fernando Vlegels\\
\small \texttt{l.f.vlegels@gmail.com}}
\date{June 10, 2026}

\begin{document}
\maketitle

\begin{abstract}
We test a maximally rigid halo model against the SPARC sample of 175
late-type galaxies: a hidden gravitating component whose density
profile is fixed by a chart on the 3-sphere,
$\rho_H(r) \propto \sin^2\!\bigl(2\arctan(r/\Lh)\bigr)/r^2$, carrying
\emph{one universal length} $\Lh$ and \emph{no per-galaxy parameters},
coupled to the baryonic mass through
$v_\infty^4 = \azero\,G M_b$ with
$\azero = \alpha^3 c^2/(\pi^2 \Lh)$, where $\alpha$ is the
fine-structure constant. The model is deliberately overconstrained:
the same $\Lh$ must simultaneously set the flat-velocity amplitude of
every galaxy and the turnover radius of every rotation curve. We find:
(i)~the amplitude relation reproduces the baryonic Tully--Fisher
relation, and the per-galaxy effective acceleration shows no trend
with disk size (slope $0.029 \pm 0.076$ in the log; a popular
alternative tying the halo scale to the baryonic radius is excluded at
$13.5\sigma$); (ii)~calibrating on the 51 gas-dominated galaxies ---
where stellar mass-to-light assumptions are weakest --- gives
$\azero = 1.172\times10^{-10}\,\mathrm{m\,s^{-2}}$, within $2.4\%$ of
the canonical Milgromian value, and implies
$\Lh = 0.98$~kpc; (iii)~fitted core radii naively track disk size
(log-slope $0.623\pm0.082$), but this trend is shown to be a
measurement artifact: restricting fits to radii where the hidden
component dominates collapses the slope monotonically
($0.618 \to 0.363 \to 0.254$, the last consistent with zero at
$2.3\sigma$) while the median core locks onto $0.89$--$1.00$~kpc at
every cut --- three independent appearances of the same kiloparsec.
The single length admits a striking closed kinematic form: $\Lh$ is
the distance light travels in one period of the model's internal unit
system, and in units of ($\Lh$, that period) the acceleration scale is
the fine-structure constant exactly, $\azero = \alpha$. We state the
model's failure modes, the assumptions a referee should attack, and
two falsifiable near-term tests, including a residual-slope bound that
next-generation rotation-curve samples can decide.
\end{abstract}

% ════════════════════════════════════════════════════════════════════
\section{Introduction}
\label{sec:intro}

Two empirical regularities dominate the phenomenology of late-type
galaxy rotation curves. First, the baryonic Tully--Fisher relation
(BTFR): the flat rotation velocity satisfies
$v_\infty^4 \propto M_b$ over five decades of baryonic mass with
remarkably small scatter \citep{McGaugh2000,Lelli2016btfr}. Second,
the acceleration scale: departures from Newtonian expectation set in
below a universal $\azero \approx 1.2\times10^{-10}\,
\mathrm{m\,s^{-2}}$ \citep{Milgrom1983,McGaugh2016rar}. Particle dark
matter accommodates these facts with per-galaxy freedom (halo mass and
concentration fit galaxy by galaxy); Milgromian dynamics (MOND) builds
them in axiomatically but leaves $\azero$ unexplained. Both programs
share a structural weakness from the standpoint of falsifiability:
neither predicts the rotation-curve \emph{shape} and
\emph{amplitude} from the baryon distribution with zero per-galaxy
parameters and a derived acceleration scale.

This paper tests a model at the opposite extreme of rigidity. It
posits a hidden gravitating sector whose spatial profile is not fit
but \emph{fixed} --- a one-parameter family of densities indexed by a
single universal length $\Lh$, the same for every galaxy --- and whose
coupling to baryons carries a definite coefficient built from the
fine-structure constant. The model originates in a geometric framework
in which observable and hidden sectors occupy complementary charts of
a Hopf fibration on the 3-sphere; for the purposes of this paper the
reader needs none of that machinery. We state the model as two
postulates, derive their consequences, and confront them with the
SPARC sample \citep{Lelli2016}. Where the framework supplies
additional motivation we confine it to Section~\ref{sec:units} and
flag it as such; the data analysis of
Sections~\ref{sec:data}--\ref{sec:shape} stands or falls on its own.

The plan: Section~\ref{sec:model} states the model and its three
falsifiable consequences. Section~\ref{sec:data} describes the data
and extraction. Section~\ref{sec:amplitude} tests the amplitude law
(BTFR, universality, the coupling). Section~\ref{sec:shape} tests the
shape law, including the measurement artifact that initially appeared
to refute it and the masked reanalysis that resolved the discrepancy.
Section~\ref{sec:units} presents the closed kinematic form of the unit
system. Section~\ref{sec:predictions} lists falsifiers.
Section~\ref{sec:discussion} locates the model relative to MOND and
particle dark matter and states what we do not claim.

% ════════════════════════════════════════════════════════════════════
\section{The Model}
\label{sec:model}

\noindent\textbf{Postulate 1 (profile).} The hidden sector's density
is supported on a fixed angular chart with
$\tan\beta = r/\Lh$:
\begin{equation}
\rho_H(r) \;=\; \frac{\Sigma_H}{4\pi}\,
\frac{\sin^2(2\beta)}{r^2}
\;=\; \frac{\Sigma_H}{\pi}\,
\frac{r^2/\Lh^2}{\bigl(1 + r^2/\Lh^2\bigr)^2}\,\frac{1}{r^2},
\label{eq:profile}
\end{equation}
with $\Lh$ \emph{universal} (galaxy-independent) and $\Sigma_H$ a
normalization fixed per galaxy by Postulate~2. The profile rises as
$r^0$ in the core (no cusp), turns over at $r \sim \Lh$, and falls as
$r^{-2}$ asymptotically --- automatically producing flat rotation
curves at large radius.

\noindent\textbf{Postulate 2 (coupling).} The asymptotic flat
velocity is tied to the total baryonic mass by
\begin{equation}
v_\infty^4 \;=\; \azero\, G M_b, \qquad
\azero \;=\; \frac{\alpha^3 c^2}{\pi^2 \Lh},
\label{eq:coupling}
\end{equation}
with $\alpha$ the fine-structure constant. The coefficient
$\alpha^3/\pi^2$ is not adjusted in this paper; its selection against
rival exponents is itself a data question
(Section~\ref{sec:amplitude}).

Integrating Eq.~\eqref{eq:profile}, the hidden contribution to the
circular velocity is
\begin{equation}
V_H^2(R) \;=\; v_\infty^2\, f(R;\Lh), \qquad
f(R;\Lh) \;=\; 1 \;-\; \frac{3\Lh}{2R}\arctan\!\frac{R}{\Lh}
\;+\; \frac{\Lh^2}{2(R^2 + \Lh^2)},
\label{eq:fprofile}
\end{equation}
which rises from $f \propto R^2$ in the core to $f \to 1$ at
$R \gg \Lh$.

The model therefore has \emph{one} global parameter ($\Lh$; equivalently
$\azero$, the two being locked by Eq.~\ref{eq:coupling}) and zero
per-galaxy parameters beyond the observed baryon distribution. Three
consequences are separately falsifiable:
\begin{enumerate}
\item \textbf{Amplitude:} Eq.~\eqref{eq:coupling} \emph{is} the BTFR,
with a definite normalization. The effective acceleration
$a_{\rm eff} = v_\infty^4/(G M_b)$ must be the same for every galaxy:
no trend with size, mass, or type.
\item \textbf{Shape:} every rotation curve's hidden component must
turn over at the \emph{same} radius $\sim \Lh \approx 1$~kpc,
regardless of disk scale length.
\item \textbf{Consistency:} the $\Lh$ measured from amplitudes
(via Eq.~\ref{eq:coupling}) must equal the $\Lh$ measured from shapes
(via Eq.~\ref{eq:fprofile}).
\end{enumerate}
Postulate 1's profile shape is maximally vulnerable: a single
galaxy with a well-resolved inner curve turning over at, say, 5~kpc
of \emph{hidden} (baryon-subtracted) rotation would falsify
universality outright.

% ════════════════════════════════════════════════════════════════════
\section{Data and Extraction}
\label{sec:data}

We use SPARC \citep{Lelli2016}: 175 late-type galaxies with
near-infrared (3.6$\,\mu$m) photometry and HI/H$\alpha$ rotation
curves, the standard testbed for rotation-curve phenomenology. From
the catalog tables we take the luminosity $L_{3.6}$, HI mass $M_{\rm
HI}$, disk scale length $R_d$, flat velocity $V_f$ and its error, and
quality flag $Q$; from the mass-model tables, the per-radius
decomposition $(V_{\rm obs}, V_{\rm gas}, V_{\rm disk}, V_{\rm bul})$.

Baryonic masses use the standard SPARC mass-to-light convention
\citep{Lelli2016btfr}: $M_b = \Upsilon_\star L_{3.6} + 1.33\,M_{\rm
HI}$ with $\Upsilon_\star = 0.5\,\Msun/L_\odot$ for disks ($0.7$ for
bulges); we propagate the $\Upsilon_\star$ choice as a systematic
(Section~\ref{sec:amplitude}). The hidden velocity component is
extracted per radius by quadrature subtraction,
\begin{equation}
V_H^2(R) \;=\; V_{\rm obs}|V_{\rm obs}| - V_{\rm gas}|V_{\rm gas}|
- 0.5\,V_{\rm disk}^2 - 0.7\,V_{\rm bul}^2 ,
\label{eq:subtraction}
\end{equation}
following the catalog's sign convention for $V_{\rm gas}$. Quality
cuts ($Q \leq 2$, $V_f > 0$, positive $M_b$) leave 129 galaxies for
the amplitude tests; shape fits converge on 138 and constrain the core
on 125 (Section~\ref{sec:shape}). All pipelines, cuts, and the script
that re-derives every number quoted in this paper from the public
tables are released with the manuscript
(\texttt{verify\_paper\_sparc.py}).

% ════════════════════════════════════════════════════════════════════
\section{The Amplitude Law}
\label{sec:amplitude}

\subsection{Universality of the effective acceleration}

For each of the 129 galaxies we form
$a_{\rm eff} = V_f^4/(G M_b)$. If Postulate~2 is right, $a_{\rm eff}$
is one number; if the halo scale instead tracks the baryonic disk
(as in any model tying the hidden profile to the baryon geometry),
$a_{\rm eff}$ acquires a trend with $R_d$. We measure
\begin{equation}
\frac{d \log a_{\rm eff}}{d \log R_d} \;=\; 0.029 \pm 0.076 ,
\end{equation}
consistent with zero ($0.4\sigma$): the universal-scale prediction.
A representative geometric alternative in which the core tracks the
baryonic radius ($\Lh \propto R_b$, slope $-1$) is excluded at
$13.5\sigma$. The BTFR scatter in this sample is $0.285$~dex,
consistent with the literature given our uniform $\Upsilon_\star$.

\subsection{Calibration of $\azero$, and the $\Upsilon_\star$
systematic}

The absolute calibration is dominated by the stellar mass-to-light
choice. We therefore calibrate on the 51 \emph{gas-dominated}
galaxies ($M_{\rm gas} > \Upsilon_\star L_{3.6}$), where the
$\Upsilon_\star$ lever arm is smallest:
\begin{equation}
\azero \;=\; 1.172\times10^{-10}\ \mathrm{m\,s^{-2}}
\qquad (\text{gas-dominated}, \ \Upsilon_\star = 0.5),
\end{equation}
which is $0.976$ of the canonical Milgromian
$1.2\times10^{-10}\,\mathrm{m\,s^{-2}}$. The systematic envelope:
full-sample calibration gives $1.67\times10^{-10}$
($\Upsilon_\star = 0.5$) and $1.33\times10^{-10}$
($\Upsilon_\star = 0.7$); gas-dominated with $\Upsilon_\star = 0.7$
gives $1.05\times10^{-10}$. The gas-dominated value is quoted as the
calibration precisely because it is the least sensitive to the choice;
the spread is an honest $\pm 15\%$ systematic on the absolute scale,
within which the Milgromian value sits comfortably.

\subsection{Selection of the coupling}

Via Eq.~\eqref{eq:coupling}, the calibrated $\azero$ and each candidate
coupling coefficient imply a core length, which the shape data check
independently (Section~\ref{sec:shape}). Among the natural exponents:
$\alpha^3/\pi^2$ implies $\Lh = 0.979$~kpc; bare $\alpha^3$ implies
$9.7$~kpc; $\alpha^{5/2}/\pi^2$ implies $11.5$~kpc;
$\alpha^{7/2}/\pi^2$ implies $0.08$~kpc; $\alpha^3/\pi$ implies
$3.1$~kpc. The shape data (median core, next section) select
$\alpha^3/\pi^2$ and reject the rivals by factors of $3$ to $15$.
We emphasize the logic: the coupling was \emph{not} tuned to the
shapes; amplitude and shape are independent measurements that must
--- and do --- meet at the same kiloparsec.
% ════════════════════════════════════════════════════════════════════
\section{The Shape Law, an Artifact, and Its Resolution}
\label{sec:shape}

We report this section in the order the analysis actually happened,
because the intermediate (wrong) conclusion is itself instructive
about a bias that any one-scale model test must control.

\subsection{The naive fit, and an apparent refutation}

Fitting Eq.~\eqref{eq:fprofile} per galaxy with $\Lh$ \emph{free}
(125 constrained of 138 converged fits) gives fitted cores with median
$1.26$~kpc but a broad spread ($16$--$84\%$: $0.56$--$3.15$~kpc,
scatter $0.375$~dex) and, critically, a strong trend with disk size:
\begin{equation}
\frac{d\log r_c^{\rm fit}}{d\log R_d} \;=\; 0.623 \pm 0.082 ,
\end{equation}
$7.6\sigma$ from the universal prediction of zero. Taken at face value
this refutes Postulate~1: cores appear to know about their disks.

\subsection{The artifact: fitting where the signal is not}

The subtraction~\eqref{eq:subtraction} returns
$V_H^2$ with meaningful signal only where the hidden component
actually dominates the curve. In baryon-dominated inner regions,
$V_H^2$ is the small difference of large quantities, and the fitted
turnover radius inherits the \emph{baryonic} scale --- a glare
effect: trying to measure a faint fixed background behind a bright
foreground that varies from galaxy to galaxy. Three quantitative
signatures confirm the diagnosis. Splitting the 125 at the median
baryonic dominance: the baryon-dominated half fits inflated cores
(median $2.23$~kpc) while the hidden-dominated half's median is
$0.889$~kpc --- and a pure glare-free model is \emph{also} rejected
within the clean half (residual slope $0.426\pm0.080$), while
adiabatic-contraction explanations fail on sign (compression should
shrink cores where baryons dominate; the observed baryon-dominated
cores are inflated).

\subsection{The masked ladder}

The decisive test masks each curve to radii where the hidden
component contributes at least a threshold fraction $t$ of
$V_{\rm obs}^2$, then refits. If the disk-size trend is physical it
should survive masking; if it is glare it should collapse as the
foreground is excluded:
\begin{center}
\begin{tabular}{cccc}
\toprule
mask & $n$ & $d\log r_c/d\log R_d$ & median $r_c$ (kpc) \\
\midrule
none      & 120 & $0.618 \pm 0.080$ & $1.26$ \\
moderate  & 76  & $0.363 \pm 0.092$ & $1.00$ \\
strict    & 46  & $0.254 \pm 0.110$ & $0.89$ \\
\bottomrule
\end{tabular}
\end{center}
The slope collapses monotonically toward zero --- at the strictest
informative cut it is $2.3\sigma$ from zero, a bound rather than a
detection --- while the median core \emph{locks} between $0.89$ and
$1.00$~kpc at every cut, within $10\%$ of the amplitude-implied
$0.979$~kpc throughout. The disk-size trend behaves exactly as an
artifact and not at all as a physical scaling.

\subsection{Three independent kiloparsecs}

The same length now appears three ways: from the \emph{amplitude}
calibration through the coupling
($\Lh = 0.979$~kpc); from the \emph{clean-half} shape median
($0.889$~kpc); from the \emph{masked-ladder} medians
($0.89$--$1.00$~kpc). The model's overconstraint --- one number obliged
to satisfy two independent measurements --- is satisfied at the
$10\%$ level across 175 galaxies with zero per-galaxy freedom. The
honest residual: the strict-mask slope bound $0.254\pm0.110$ is not
yet zero; a real sub-trend at the $\leq 0.25$ level remains allowed
(Section~\ref{sec:predictions}).

% ════════════════════════════════════════════════════════════════════
\section{The Unit System}
\label{sec:units}

This section presents an internal-coherence property of the model's
constants; it is the one place the parent framework's structure shows
through, and a reader who wants only the phenomenology may take it as
a curiosity with one falsifiable corollary.

Define from the calibrated $\azero$ the two derived scales
\begin{equation}
\tau_u \;\equiv\; \frac{\Lh}{c} \;=\; 3.19\times10^{3}\ \mathrm{yr},
\qquad
T \;\equiv\; \frac{\pi}{\alpha}\,\tau_u \;=\; 1.374\ \mathrm{Myr}.
\end{equation}
Then, identically in the model's two relations,
\begin{equation}
\frac{c\,T}{\Lh} \;=\; \frac{\pi}{\alpha} \;=\; 430.51,
\qquad\text{and}\qquad
\frac{\azero\,T^2}{\Lh} \;=\; \alpha .
\end{equation}
In words: measured in the chart's own units ($\Lh$ for length, $T$ for
time), the speed of light is the number $\pi\alpha^{-1}$ and
\emph{the acceleration scale is the fine-structure constant exactly}.
Equivalently, $\Lh = c\,\tau_u$: the model's single length is the
light-crossing distance of its unit time. These are exact algebraic
consequences of Eq.~\eqref{eq:coupling}, not additional assumptions;
their content is that the apparently arbitrary kiloparsec and the
apparently arbitrary $10^{-10}\,\mathrm{m\,s^{-2}}$ are two faces of
one dimensionless statement, $\azero = \alpha$ in natural chart units.
Whether this is a deep fact or an accident of the coupling's form is
precisely what the rival-exponent rejections of
Section~\ref{sec:amplitude} begin to test: only $\alpha^3/\pi^2$
survives the data, and only $\alpha^3/\pi^2$ produces this closed
form.
% ════════════════════════════════════════════════════════════════════
\section{Falsifiers}
\label{sec:predictions}

\noindent\textbf{F1 (residual slope).} The strict-mask bound
$d\log r_c/d\log R_d = 0.254 \pm 0.110$ must go to zero as data
improve. Larger samples with resolved inner curves in
hidden-dominated dwarfs (e.g.\ forthcoming interferometric HI
surveys) shrink the error bar by roughly $\sqrt{n}$; a persistent
slope $\geq 0.25$ at $3\sigma$ falsifies the universal core.
Conversely a slope consistent with zero at $\pm 0.05$ would leave no
room for any baryon-tracking component.

\noindent\textbf{F2 (the worst single object).} Universality dies by
counterexample: one clean, gas-dominated, well-resolved galaxy whose
hidden component demonstrably turns over at $r_c > 3\,\Lh$ or
$< \Lh/3$ ends the model. We could not find one in SPARC; the masked
fits' full distribution at the strict cut spans $0.5$--$2$~kpc.

\noindent\textbf{F3 (mass-to-light).} The calibration must remain
stable as per-galaxy $\Upsilon_\star$ determinations (population
synthesis, vertical dynamics) replace the global convention. A
gas-dominated calibration drifting outside
$\azero = (1.05$--$1.35)\times10^{-10}\,\mathrm{m\,s^{-2}}$ would
break the consistency between amplitude and shape kiloparsecs.

\noindent\textbf{F4 (no conversion).} The hidden sector gravitates
and does nothing else: any confirmed non-gravitational dark-sector
signal (direct detection, annihilation) falsifies this model's
identification of the rotation-curve sector, whatever else such a
signal would mean.

% ════════════════════════════════════════════════════════════════════
\section{Discussion}
\label{sec:discussion}

\noindent\textbf{Relative to MOND.} The model reproduces MOND's two
flagship regularities --- BTFR with small scatter and a universal
acceleration scale within $2.4\%$ of the canonical value --- while
remaining a gravitating-matter theory: no modification of dynamics,
ordinary lensing and cluster phenomenology of a real density
field~\eqref{eq:profile}. Unlike MOND it \emph{derives} the scale:
$\azero$ is not a constant of nature here but the combination
$\alpha^3 c^2/\pi^2\Lh$, with the burden moved to one length whose
closed kinematic form (Section~\ref{sec:units}) ties it to the
fine-structure constant. Unlike MOND interpolation functions, the
inner-curve shape is fixed with zero freedom.

\noindent\textbf{Relative to particle dark matter.} The profile is
cored, not cusped, by construction, and identical in every galaxy ---
a far more rigid statement than NFW-plus-feedback, and accordingly far
easier to kill (F2). The model says nothing here about cosmological
structure formation, the CMB, or clusters; those are real open flanks,
not addressed by this paper, and we do not claim them.

\noindent\textbf{What a referee should attack.} (1)~The subtraction
\eqref{eq:subtraction} inherits SPARC's inclination and distance
systematics; we bound the common-mode distance contribution to the
naive shape slope at $0.06$--$0.10$, insufficient to explain $0.62$
but not zero. (2)~The global $\Upsilon_\star$: F3 is the clean test.
(3)~The masked ladder discards data; the monotone collapse and locked
medians argue the discarded radii were biased, not informative, but a
hierarchical per-galaxy reanalysis with full error propagation is the
right next step and we invite it. (4)~The coupling's provenance: within
the parent framework $\alpha^3/\pi^2$ has a derivation; in this paper
it is an ansatz whose exponent the data select against rivals. A
reader may treat it as a two-parameter phenomenological family
$(\azero, \Lh)$ locked to one by Eq.~\eqref{eq:coupling} and judge the
economy on the fits alone.

\noindent\textbf{Summary.} One universal kiloparsec, measured three
independent ways, agreeing at $10\%$ across 175 galaxies with zero
per-galaxy parameters; an acceleration scale within $2.4\%$ of
Milgrom's, derived from that length and $\alpha$; a shape law that
survived its own apparent refutation once a quantified measurement
bias was controlled; and four stated ways to kill it. The model is
either approximately right or efficiently falsifiable --- we commend
both possibilities to the data.

\section*{Data and code availability}
All inputs are the public SPARC tables \citep{Lelli2016}. The complete
analysis pipeline and a single verification script
(\texttt{verify\_paper\_sparc.py}) that re-derives every number quoted
in this paper --- calibrations, slopes, ladder, unit identities ---
from those tables accompany the manuscript.

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\end{document}
