A Single Kiloparsec Length in Galaxy Rotation Curves · SPARC evidence for a one-scale halo with a derived acceleration scale
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 one universal length $\Lh$ and 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.
1 Introduction
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 . Second, the acceleration scale: departures from Newtonian expectation set in below a universal \(a_0\approx 1.2\times10^{-10}\, \mathrm{m\,s^{-2}}\) . 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 \(a_0\) unexplained. Both programs share a structural weakness from the standpoint of falsifiability: neither predicts the rotation-curve shape and 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 fixed — a one-parameter family of densities indexed by a single universal length \(L_h\), 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 . Where the framework supplies additional motivation we confine it to Section Section 6 and flag it as such; the data analysis of Sections Section 3–Section 5 stands or falls on its own.
The plan: Section Section 2 states the model and its three falsifiable consequences. Section Section 3 describes the data and extraction. Section Section 4 tests the amplitude law (BTFR, universality, the coupling). Section Section 5 tests the shape law, including the measurement artifact that initially appeared to refute it and the masked reanalysis that resolved the discrepancy. Section Section 6 presents the closed kinematic form of the unit system. Section Section 7 lists falsifiers. Section Section 8 locates the model relative to MOND and particle dark matter and states what we do not claim.
2 The Model
Postulate 1 (profile). The hidden sector’s density is supported on a fixed angular chart with \(\tan\beta = r/L_h\): \[\rho_H(r) \;=\; \frac{\Sigma_H}{4\pi}\, \frac{\sin^2(2\beta)}{r^2} \;=\; \frac{\Sigma_H}{\pi}\, \frac{r^2/L_h^2}{\bigl(1 + r^2/L_h^2\bigr)^2}\,\frac{1}{r^2}, \label{eq:profile}\] with \(L_h\) 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 \(r \sim L_h\), and falls as \(r^{-2}\) asymptotically — automatically producing flat rotation curves at large radius.
Postulate 2 (coupling). The asymptotic flat velocity is tied to the total baryonic mass by \[v_\infty^4 \;=\; a_0\, G M_b, \qquad a_0\;=\; \frac{\alpha^3 c^2}{\pi^2 L_h}, \label{eq:coupling}\] 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 Section 4).
Integrating Eq. \(\eqref{eq:profile}\), the hidden contribution to the circular velocity is \[V_H^2(R) \;=\; v_\infty^2\, f(R;L_h), \qquad f(R;L_h) \;=\; 1 \;-\; \frac{3L_h}{2R}\arctan\!\frac{R}{L_h} \;+\; \frac{L_h^2}{2(R^2 + L_h^2)}, \label{eq:fprofile}\] which rises from \(f \propto R^2\) in the core to \(f \to 1\) at \(R \gg L_h\).
The model therefore has one global parameter (\(L_h\); equivalently \(a_0\), the two being locked by Eq. \(\ref{eq:coupling}\)) and zero per-galaxy parameters beyond the observed baryon distribution. Three consequences are separately falsifiable:
Amplitude: Eq. \(\eqref{eq:coupling}\) 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.
Shape: every rotation curve’s hidden component must turn over at the same radius \(\sim L_h\approx 1\) kpc, regardless of disk scale length.
Consistency: the \(L_h\) measured from amplitudes (via Eq. \(\ref{eq:coupling}\)) must equal the \(L_h\) measured from shapes (via Eq. \(\ref{eq:fprofile}\)).
Postulate 1’s profile shape is maximally vulnerable: a single galaxy with a well-resolved inner curve turning over at, say, 5 kpc of hidden (baryon-subtracted) rotation would falsify universality outright.
3 Data and Extraction
We use SPARC : 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 : \(M_b = \Upsilon_\star L_{3.6} + 1.33\,M_{\rm
HI}\) with \(\Upsilon_\star = 0.5\,M_\odot/L_\odot\) for disks (\(0.7\) for bulges); we propagate the \(\Upsilon_\star\) choice as a systematic (Section Section 4). The hidden velocity component is extracted per radius by quadrature subtraction, \[V_H^2(R) \;=\; V_{\rm obs}|V_{\rm obs}| - V_{\rm gas}|V_{\rm gas}|
- 0.5\,V_{\rm disk}^2 - 0.7\,V_{\rm bul}^2 ,
\label{eq:subtraction}\] 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 Section 5). All pipelines, cuts, and the script that re-derives every number quoted in this paper from the public tables are released with the manuscript (verify_paper_sparc.py).
4 The Amplitude Law
4.1 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 \[\frac{d \log a_{\rm eff}}{d \log R_d} \;=\; 0.029 \pm 0.076 ,\] consistent with zero (\(0.4\sigma\)): the universal-scale prediction. A representative geometric alternative in which the core tracks the baryonic radius (\(L_h\propto R_b\), 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 \(\Upsilon_\star\).
4.2 Calibration of \(a_0\), and the \(\Upsilon_\star\) systematic
The absolute calibration is dominated by the stellar mass-to-light choice. We therefore calibrate on the 51 gas-dominated galaxies (\(M_{\rm gas} > \Upsilon_\star L_{3.6}\)), where the \(\Upsilon_\star\) lever arm is smallest: \[a_0\;=\; 1.172\times10^{-10}\ \mathrm{m\,s^{-2}} \qquad (\text{gas-dominated}, \ \Upsilon_\star = 0.5),\] 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.
4.3 Selection of the coupling
Via Eq. \(\eqref{eq:coupling}\), the calibrated \(a_0\) and each candidate coupling coefficient imply a core length, which the shape data check independently (Section Section 5). Among the natural exponents: \(\alpha^3/\pi^2\) implies \(L_h= 0.979\) 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 not tuned to the shapes; amplitude and shape are independent measurements that must — and do — meet at the same kiloparsec.
5 The Shape Law, an Artifact, and Its Resolution
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.
5.1 The naive fit, and an apparent refutation
Fitting Eq. \(\eqref{eq:fprofile}\) per galaxy with \(L_h\) 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: \[\frac{d\log r_c^{\rm fit}}{d\log R_d} \;=\; 0.623 \pm 0.082 ,\] \(7.6\sigma\) from the universal prediction of zero. Taken at face value this refutes Postulate 1: cores appear to know about their disks.
5.2 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 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 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).
5.3 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:
| mask | \(n\) | \(d\log r_c/d\log R_d\) | median \(r_c\) (kpc) |
|---|---|---|---|
| none | 120 | \(0.618 \pm 0.080\) | \(1.26\) |
| moderate | 76 | \(0.363 \pm 0.092\) | \(1.00\) |
| strict | 46 | \(0.254 \pm 0.110\) | \(0.89\) |
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 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.
5.4 Three independent kiloparsecs
The same length now appears three ways: from the amplitude calibration through the coupling (\(L_h= 0.979\) kpc); from the clean-half shape median (\(0.889\) kpc); from the 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 Section 7).
6 The Unit System
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 \(a_0\) the two derived scales \[\tau_u \;\equiv\; \frac{L_h}{c} \;=\; 3.19\times10^{3}\ \mathrm{yr}, \qquad T \;\equiv\; \frac{\pi}{\alpha}\,\tau_u \;=\; 1.374\ \mathrm{Myr}.\] Then, identically in the model’s two relations, \[\frac{c\,T}{L_h} \;=\; \frac{\pi}{\alpha} \;=\; 430.51, \qquad\text{and}\qquad \frac{a_0\,T^2}{L_h} \;=\; \alpha .\] In words: measured in the chart’s own units (\(L_h\) for length, \(T\) for time), the speed of light is the number \(\pi\alpha^{-1}\) and the acceleration scale is the fine-structure constant exactly. Equivalently, \(L_h= c\,\tau_u\): 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, \(a_0= \alpha\) 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 Section 4 begin to test: only \(\alpha^3/\pi^2\) survives the data, and only \(\alpha^3/\pi^2\) produces this closed form.
7 Falsifiers
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.
F2 (the worst single object). Universality dies by counterexample: one clean, gas-dominated, well-resolved galaxy whose hidden component demonstrably turns over at \(r_c > 3\,L_h\) or \(< L_h/3\) ends the model. We could not find one in SPARC; the masked fits’ full distribution at the strict cut spans \(0.5\)–\(2\) kpc.
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 \(a_0= (1.05\)–\(1.35)\times10^{-10}\,\mathrm{m\,s^{-2}}\) would break the consistency between amplitude and shape kiloparsecs.
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.
8 Discussion
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 derives the scale: \(a_0\) is not a constant of nature here but the combination \(\alpha^3 c^2/\pi^2L_h\), with the burden moved to one length whose closed kinematic form (Section Section 6) ties it to the fine-structure constant. Unlike MOND interpolation functions, the inner-curve shape is fixed with zero freedom.
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.
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 \((a_0, L_h)\) locked to one by Eq. \(\eqref{eq:coupling}\) and judge the economy on the fits alone.
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.
Data and code availability
All inputs are the public SPARC tables . The complete analysis pipeline and a single verification script (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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