Download at 10.5281/zenodo.20373369
The Dark Matter Resolution
How Galaxies really work
The K(X) Regime of the Gravitational Fabric
A companion to Temporal Congestion Mechanics
Matthew Ward-Broadfield
Independent researcher, England
Abstract
The phenomena attributed to dark matter — flat galactic rotation curves, the Baryonic Tully-Fisher Relation, the radial acceleration relation, the universal asymptotic galactic rotation velocity at 149.67 km/s — are recovered structurally by the K(X) regime of the gravitational fabric in Temporal Congestion Mechanics (Ward-Broadfield 2026a). The K(X) regime is forced by the framework's action; from it follow the BTFR slope-4 exactly, the Ward Constant v_∞ = c²/√(2πGλ) = 149.67 km/s, and the mass-dependent gradient direction confirmed in the SPARC sample of 175 disk galaxies (Ward-Broadfield 2026b; independent confirmation at p ≈ 3.9 × 10⁻²¹ in §3). The present paper develops the dark matter resolution: no halo, no particle, no missing-satellite problem, no Keplerian decline to be hidden. The five-decade null result of direct-detection experiments is structural confirmation.
Contents
1. The Dark Matter Problem
2. The Framework's Gravitational Apparatus
3. From the Black Hole to 149.67 km/s
3.1 The black hole interior
3.2 The Newtonian region
3.3 The knee
3.4 The Wardonian region — stellar band
3.5 The Wardonian region — gas-only band
3.6 Convergence at 149.67 km/s
3.7 Light, heavy, crossover
4. The Dark Matter Implications
5. Parameter Count Against ΛCDM
6. The Direct-Detection Null Results
7. Falsification Conditions
8. Conclusion
References
Data Availability
1 The Dark Matter Problem
Rubin and Ford (1970, 1980) established that the rotation curves of spiral galaxies do not fall off Keplerian at large radius. Orbital velocities stay approximately constant past the visible stellar disc. Newtonian gravity acting on the observed baryons alone does not predict this. Across the 175-galaxy SPARC sample (Lelli, McGaugh & Schombert 2016) the pattern is universal.
The conventional resolution since the early 1980s introduces dark matter — an unobserved mass component, approximately five times the baryonic density, distributed in a halo around every galaxy. Each galaxy is fitted with its own halo using two or three free parameters. The resolution has not produced a particle in five decades of dedicated searching, and does not derive the BTFR slope, the radial acceleration relation deviation scale, or the universal asymptotic rotation velocity from first principles.
Temporal Congestion Mechanics (Ward-Broadfield 2026a) treats space as a physical medium with congestion index n(x,t). The framework derives the K(X) regime of the fabric's constitutive law from a single action; the regime governs every observed galactic rotation curve and produces parameter-free what dark matter physics has been fitting case-by-case. The companion rotation-curve paper (Ward-Broadfield 2026b) tests this against SPARC and confirms 168 of 175 galaxies at p < 4 × 10⁻⁴¹. The present paper develops what these results structurally imply about dark matter as a hypothesis.
2 The Framework's Gravitational Apparatus
Space is a physical medium with a single real scalar field n(x,t) — the congestion index. Resting fabric has n = 1; baryonic mass compresses the fabric so n > 1 nearby. Earth's surface sits at n ≈ 1.0000000007. The maximum congestion the fabric can sustain, set by the framework's Saturation Law (Ward-Broadfield 2026a, §10), is the Broadfield Constant n_H = √e ≈ 1.6487. The interior of every black hole sits at n_H — the universal ceiling of fabric compression. Between these two extremes the congestion index varies by a factor of nearly a billion.
In the static weak-field limit the field is related to the gravitational potential through n(x,t) = exp(−Φ(x,t)/c²). The dynamical field equation is the Master PDE:
α·∂²ₜn + (α/τ)·∂ₜn − ∇·(K·∇n) + ε·(n − 1) = 4πG̃·ρ (1)
with α the fabric inertia, ε the restoring potential, G̃ = G·α the matter-fabric coupling, ρ the matter density. The constitutive stiffness K has two regimes selected by the local gravitational acceleration relative to a threshold g₀:
K = K₀ = α·c² (linear regime, a ≫ g₀) (2)
K(X) = α·c²·X = α·c⁴·|∇n|/g₀ (K(X) regime, a < g₀) (3)
with X = c²|∇n|/g₀ the dimensionless congestion change. The K(X) form is forced by writing the action correctly (Ward-Broadfield 2026a, Appendix C): continuity at X = 1 fixes K(X=1) = K₀; it is the lowest-order polynomial in X compatible with this; it introduces no new dimensionful constant. No other functional form satisfies all three. The threshold g₀ = 1.2 × 10⁻¹⁰ m·s⁻² is anchored from observation; it is the same scale McGaugh, Lelli & Schombert (2016) identified empirically as the radial acceleration relation deviation scale.
3 From the Black Hole to 149.67 km/s
A galaxy sits inside a medium. The visible mass of the galaxy — stars, gas, dust, the central black hole — compresses the fabric around it, pulling n above its resting value of 1. The fabric responds in two ways. Its stiffness pushes back against the compression. And at every point it also tries to return to n = 1, through the restoring term ε·(n − 1) in the Master PDE. The galaxy you see is what stars do when they sit inside that competition between compression by visible matter and the fabric’s tendency to return to rest.
The picture is traced most cleanly from the centre outward. The Milky Way provides a concrete journey; the same structure repeats around every gravitating mass, with the numerical scales set by the mass.
3.1 The black hole interior
At the centre of the Milky Way sits Sgr A*, a black hole of approximately 4.15 × 10⁶ M☉. Inside the black hole’s saturation surface, the fabric is at its maximum permitted compression — n = n_H = √e ≈ 1.6487. This is the universal ceiling of the framework. Every black hole, regardless of mass, has the same interior congestion. Outside the saturation surface, n falls toward 1 with distance. The fabric’s compression by the central mass — together with the bulge and inner disc that build up around it — sets the gravitational structure of the inner galaxy. The total enclosed mass at each radius is what the fabric responds to.
For comparison, Earth's surface sits at n − 1 ≈ 7 × 10⁻¹⁰. The black hole interior is approximately 9 × 10⁸ times more congested than Earth's surface. This is the dynamical range of the fabric.
3.2 The Newtonian region
From the black hole outward, through the Milky Way's central bulge and the inner disc, the fabric is well above the K(X) threshold. The locally-induced gravitational acceleration is much greater than g₀, and the fabric is in the linear regime — K = K₀ = α·c². The static reduction of equation (1) gives Poisson's equation, ∇²Φ = 4πGρ. Newtonian gravity is recovered exactly.
The restoring term ε·(n − 1) of the Master PDE is acting here too, pulling the fabric back toward n = 1. In this regime the stiffness response K₀·∇²n dominates the field equation, and the restoring term sits quietly in the background. It becomes structurally important only further out, where the stiffness response weakens under the K(X) law.
Stars in this region orbit at speeds set by the enclosed mass at their radius. Deep in the bulge, orbital velocities can reach 1000 km/s and above. At the Sun’s distance — about 26,000 light-years from the Galactic Centre — the orbital velocity is approximately 229 km/s. The Newtonian region extends out to about 28,000 light-years (≈ 8.4 kpc), where the locally-induced acceleration drops to g₀.
Every classical test of gravity — Mercury's perihelion advance, the bending of starlight by the Sun, the Cassini timing residuals, the Hulse-Taylor binary decay, gravitational-wave propagation — has been measured at parts-per-thousand to parts-per-million precision in the Newtonian region. The framework reproduces every test through fabric mediation, with no curved spacetime invoked (Ward-Broadfield 2026a, Appendix L).
3.3 The knee
At a radius of approximately 28,000 light-years from the Milky Way's centre, the locally-induced acceleration drops to g₀ = 1.2 × 10⁻¹⁰ m·s⁻². At this radius the constitutive law of the fabric switches form — from K = K₀ to K(X) = α·c²·X. The two forms join continuously: K(X = 1) = K₀ exactly. The knee is a mode-switch in how the fabric responds to congestion change, not a sudden boundary in any other observable.
The knee position is set by the total enclosed baryonic mass through r_knee = √(G·M_bar/g₀). It is not set by Sgr A* alone (which would give a knee at only 70 parsecs) or by the bulge alone (4.2 kpc) — it is set by the full baryonic mass of the galaxy enclosed at that radius. The threshold g₀ is universal; what varies between galaxies is the physical radius at which the locally-induced acceleration reaches it.
At the knee, the pulls from either side have matched. Inside, the stiffness response of the fabric dominates and the visible mass governs through Poisson’s equation. Outside, the medium’s tendency to return to rest is starting to win. Past the knee the gravitational pull falls off as 1/r rather than 1/r², and stars are gradually declining in rotation speed the further out they sit. This is the structural mechanism behind what observers see as flat rotation curves. Conventional physics has no concept of this transition — Newton’s law simply continues to fall as 1/r² with no threshold, which is why dark matter had to be invented to hold the outer stars in place.
3.4 The Wardonian region — Zone 1 - stellar band
Past the knee, the medium is starting to win. The fabric is in the K(X) regime — the cubic-gradient form of the constitutive law — and the static-reduction profile in the matter-free exterior gives |∇n| ∝ 1/r. Circular orbits satisfy V²(r) = a(r)·r, and stars decline gently in rotation speed across the Wardonian region.
Matching to the linear regime at the knee gives the Baryonic Tully-Fisher Relation:
V_flat⁴ = G · M_bar · g₀ (BTFR slope-4 exactly) (4)
Slope 4 exactly. No fitting. Set by G and g₀ alone, both anchored independently. The framework derives from a single action what Tully & Fisher (1977), McGaugh et al. (2000), and Lelli, McGaugh & Schombert (2016) have established observationally across four decades of galactic mass.
In the Milky Way, this stellar band of the Wardonian region runs from about 28,000 light-years (the knee) out to roughly 50,000 light-years, where the visible stellar disc ends — a band approximately 22,000 light-years wide. Stars at the knee orbit at around 225 km/s. By the outer stellar edge, they have eased down to around 200 km/s. The fall is so shallow that Vera Rubin’s early rotation-curve measurements in the 1970s looked essentially flat.
3.5 The Wardonian region — Zone 2 - gas-only band
Beyond about 50,000 light-years from the Milky Way's centre, the gas density in the disc has fallen below the threshold required for stars to form. Stars cannot exist past this radius — but the fabric continues to be in K(X) handover mode. The Wardonian region is not bounded by where stars happen to form; it is bounded by where the fabric finishes returning toward its asymptotic structure.
In this gas-only band, neutral hydrogen continues to orbit in the K(X) regime. Radio observations of extended HI discs see exactly this — HI routinely extends past the visible stellar disc by a factor of two or more, with rotation velocities continuing to trace the K(X) flat-rotation pattern as they descend toward the asymptotic limit. The 21 cm line gives direct Doppler velocities for this gas, mapping the descent of the fabric's handover through the band where no stars exist.
The Milky Way’s gas-only band extends to approximately 80,000–100,000 light-years from the centre. The hydrogen orbits at speeds that continue the gentle descent the stars began — measurements past the visible stellar edge show the rotation curve dropping into the 170–180 km/s range, on its way toward the asymptotic universal value of 149.67 km/s. Past the last gas, the fabric is no longer being held in K(X) stretch and returns the rest of the way to its resting value n = 1 — the Relaxonian zone.
For an isolated galaxy in empty space, this return to rest would extend a long way outward. But galaxies are not alone. Each galaxy’s relaxing fabric eventually meets a neighbour’s — the same continuous n(x,t) field connects every gravitating mass in the universe. The Milky Way and Andromeda are about 2.5 million light-years apart, and their fabric configurations meet somewhere in the gap between them. The cosmic web of galaxies seen in surveys is the patchwork of these meeting boundaries.
3.6 Convergence at 149.67 km/s
At the asymptotic outer limit of the Wardonian region — far enough out that the fabric’s easing has run its course — two terms in the Master PDE come into balance. The K(X) gradient term sustains the outward stretch of the fabric established by the visible matter at the centre. The restoring term ε·(n − 1) pulls the fabric back toward its resting value n = 1. These two competing terms reach equilibrium at a specific universal scale, derived from the anchored inputs {c, G, λ} and independent of any individual galaxy’s mass:
v_∞ = c² / √(2π·G·λ) = 149.67 km/s (Ward Constant) (5)
Every galaxy’s Wardonian region ends at 149.67 km/s. The fabric far from any galaxy is structured the same way, regardless of what mass produced the inner compression. The Ward Constant is a property of the medium itself. It is what the Master PDE delivers when stretching by matter and return to rest reach equilibrium at large radius — the balance condition depends only on properties of the fabric, not on which galaxy produced the inner compression.
3.7 Light, heavy, crossover
Equation (4) sets the velocity inside the knee at V_flat = (G·M_bar·g₀)^¼. Equation (5) sets the velocity at the asymptotic outer limit at v_∞ = 149.67 km/s. These two are generally unequal — equal only at the specific baryonic mass at which they coincide:
M× = v_∞⁴ / (G·g₀) ≈ 3.15 × 10¹⁰ M☉ (crossover mass) (6)
The K(X)-regime Master PDE in the matter-free exterior carries these two as boundary conditions: V_flat at the inner edge, v_∞ at the outer. The solution must interpolate between them. The direction of the interpolation is forced by the sign of (M_bar − M×). Light galaxies (M_bar < M×) sit below v∞ at the inner edge and the rotation curve rises through the Wardonian region toward 149.67 km/s. Heavy galaxies (M_bar > M×, the Milky Way at M_bar/M× ≈ 1.9 is one; Andromeda at M_bar/M× ≈ 3.5–4.8 across published baryonic-mass estimates is another) sit above v∞ and the rotation curve falls toward 149.67 km/s. At M_bar = M× exactly, the plateau coincides with v∞ and the rotation curve is flat throughout.
Tested across the SPARC sample using Lelli's published photometric baryonic masses (datafile1.txt, ϒ★ = 0.5 at 3.6 μm), the 147 cross-matched galaxies partition into three categories with the outer-half trend in Table 1.
Mass category
Predicted trend
Rising
Falling
Light (M_bar < 0.5 M×, n = 89)
Rise to v_∞
75 (84%)
4 (4%)
Crossover (0.5 < M_bar/M× < 2, n = 26)
Flat near v_∞
5 (19%)
13 (50%)
Heavy (M_bar > 2 M×, n = 32)
Fall to v_∞
3 (9%)
20 (62%)
Table 1. SPARC outer-half trend by photometric baryonic mass relative to M× = 3.15 × 10¹⁰ M☉. Among the 102 light + heavy galaxies with decisive (non-flat) outer trends, 95 trend in the framework-predicted direction and 7 against. Binomial significance against random sign assignment: p ≈ 3.9 × 10⁻²¹. The simple methodology used here is reproducible directly from the public SPARC tables; Ward-Broadfield 2026b uses significance-tested outer slopes and reports 168 of 175 at p < 4 × 10⁻⁴¹.
NGC 3198, the canonical example, has photometric M_bar = 3.39 × 10¹⁰ M☉ — within 8% of M×. The framework predicts V_flat ≈ v_∞ = 149.67 km/s in the outer K(X) regime. Lelli’s published asymptotic V_flat = 150.1 km/s, measured from the outer HI in the gas-only band. Match to 0.29%.
No other framework predicts this number. Dark matter models fit each galaxy with its own halo and produce no shared scale across the sample. MOND uses the same threshold acceleration g₀ as TCM but predicts no universal velocity. The Ward Constant is uniquely a TCM result, and the framework places it at the value the SPARC galaxies converge toward.
4 The Dark Matter Implications
The structural picture of §3 carries direct consequences for the dark matter hypothesis. Each of the dark matter framework's standing problems has a structural counterpart in TCM that does not require any unobserved substance.
No halo. Each SPARC galaxy in ΛCDM analysis is fitted with its own NFW (or Einasto, or DC14) halo of 2–3 free parameters. There is no NFW halo in the framework — the K(X) regime is what produces flat outer rotation. The mechanism is universal; the per-galaxy fit is replaced by a single calibration of g₀ used across the sample.
No particle. The K(X) regime is a property of the medium, not a substance. The framework predicts no dark matter particle exists. Five decades of direct-detection searches have produced no confirmed detection — consistent with the framework's structural prediction (§6).
No missing satellites. ΛCDM N-body simulations predict an order of magnitude more dark matter satellite halos than the observed satellite galaxies of the Milky Way and Andromeda. The framework has no halos to count. Observed satellite galaxies are baryonic structures; the predicted overabundance does not exist in the framework.
No core-cusp problem. Dwarf galaxy inner density profiles are observed to be cored rather than cuspy as ΛCDM predicts. The framework predicts neither core nor cusp; the inner profile of any galaxy is set by its baryonic distribution and the linear regime, not by a fitted halo. There is no tension to resolve.
No too-big-to-fail. ΛCDM expects the most massive subhalos around the Milky Way to host the brightest satellite galaxies, contrary to observation. No subhalos exist in the framework; the problem does not arise.
Cluster lensing without missing mass. Clusters sit beyond their knee radius (r_knee ≈ 0.34 Mpc for 10¹⁴ M☉ baryonic mass). Cluster-scale gravitational structure is the K(X) regime acting on the baryonic distribution, producing the lensing signal and the velocity dispersions attributed to dark matter halos. Strong lensing in cluster cores remains a Newtonian-regime calculation in baryon-dominated regions.
Each item above has been a problem for the dark matter framework for between fifteen and forty years. Each item dissolves structurally in TCM. The framework requires no separate dark sector beyond the fabric itself.
5 Parameter Count Against ΛCDM
A theory that explains the same data with fewer free parameters is structurally cleaner. The contrast between TCM and ΛCDM on the SPARC sample is stark.
Framework
Free parameters across 175 galaxies
Per-galaxy free parameters
ΛCDM, NFW 3-parameter
~525
3
ΛCDM, NFW 2-parameter
~350
2
TCM K(X) regime
2 (universal)
0
Ratio
~260:1
—
Table 2. Parameter-count comparison. TCM uses approximately 260 times fewer free parameters than 3-parameter NFW halo fitting, with zero per-galaxy fitting.
The two universal calibrations in TCM are g₀ and λ. Both are anchored once to independent observations — g₀ to the radial acceleration relation deviation scale (McGaugh, Lelli & Schombert 2016), λ to the asymptotic galactic rotation velocity. Once anchored, they are applied across the sample with no adjustment. The ΛCDM halo fitting is independently performed per galaxy, with no universal parameter doing structural work across the sample.
6 The Direct-Detection Null Results
The framework structurally predicts that no dark matter particle exists. The history of direct-detection experiments is consistent with this prediction.
LUX-ZEPLIN (LZ). Ten tonnes of liquid xenon at the Sanford Underground Research Facility. The December 2025 result, based on 417 live days of data taken March 2023 – April 2025, sets world-leading exclusion limits across the WIMP parameter space including new bounds in the 3–9 GeV/c² range. No dark matter detection. In the same dataset, LZ began observing boron-8 solar neutrinos via coherent elastic neutrino-nucleus scattering at 4.5σ — confirming the detector's sensitivity to standard-model interactions. The dark matter interaction is not present at the level the framework requires it to be present.
XENONnT. Eight tonnes of liquid xenon at Gran Sasso. Null results across multiple dark matter searches; solar neutrinos detected at 2.73σ.
PandaX-4T. Five and a half tonnes of liquid xenon at Jinping. Null results from commissioning run and first science run; 1.54 tonne·year exposure reaching the irreducible neutrino background. Solar neutrinos at 2.64σ.
Historical detectors. XENON1T, LUX, PandaX-II, CDMS-II, EDELWEISS, ADMX (axion search), KIMS, COUPP, PICO, and DAMA/LIBRA (whose modulation signal has not been confirmed by any other experiment) — no confirmed detection across five decades of dedicated effort.
Within ΛCDM, each null result narrows the allowed parameter space and the framework is preserved by retreating to weaker couplings, lower masses, or alternative candidates. Within TCM, each null result is structural confirmation. The detectors are sensitive — they see solar neutrinos. What they don't see is what the framework predicts isn't there.
7 Falsification Conditions
The framework is not protected from refutation. Each of the following observations, if confirmed, would falsify the dark matter resolution presented here.
1. Any confirmed dark matter detection consistent with the conventional ΛCDM galactic mass budget, replicated across collaborations.
2. Any low-mass galaxy (M_bar ≪ M×) found to sit well above v_∞ in the K(X) regime, in a clean sample with controlled systematics.
3. Any measurement of the universal asymptotic galactic rotation velocity at the crossover mass significantly inconsistent with v_∞ = 149.67 km/s.
4. Any wide binary star pair at separations where the mutual gravitational acceleration falls below g₀ (approximately 7,000–10,000 AU depending on combined mass) showing purely Newtonian rather than K(X)-regime orbital dynamics, in a sample with controlled systematics.
5. A homogeneous galaxy sample showing BTFR slope statistically inconsistent with 4 at high confidence.
6. Extended HI rotation curves past the visible stellar edge that fail to track the Ward Constant convergence appropriate to the galaxy's baryonic mass — light rising, crossover flat, heavy descending from above — in a sample with controlled photometric M_bar.
Each is specific, observable with current or near-current technology, and unconditional on the framework. No escape valve.
8 Conclusion
Dark matter as a hypothesis solved one problem — the flat outer rotation curves of spiral galaxies — by introducing an unobserved mass component and tuning a separate halo to each galaxy in observed samples. Five decades on, the particle has not been found, the empirical regularities (BTFR slope-4, the radial acceleration relation, the universal asymptotic rotation velocity) remain underived from first principles, and a list of structural problems (missing satellites, core-cusp, too-big-to-fail) has accumulated.
The K(X) regime of the gravitational fabric is what the dark matter hypothesis was approximating. The framework derives slope-4 from action principles. The Ward Constant 149.67 km/s emerges from the fabric's own properties. The mass-dependent gradient direction is forced by the boundary-condition structure of the Master PDE. No fitting, no halo, no particle. The 175-galaxy SPARC sample, deeply tested by external researchers over the past decade, confirms the structural picture at p < 4 × 10⁻⁴¹ in the companion paper and p ≈ 3.9 × 10⁻²¹ in the independent test of §3.
The path forward is testing — six specific falsifiers in §7, each accessible with current technology. The wide-binary K(X) signature in current Gaia data, the extended HI rotation pattern of galaxies with well-characterised photometric M_bar, the next generation of direct-detection results (each null result is further structural confirmation). If any falsifier returns positive, the framework is refuted. Until then, what was attributed to dark matter for fifty years is the fabric, doing what the action requires it to do.
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Ward-Broadfield M., 2026a, Temporal Congestion Mechanics — A Theory of Everything. Zenodo DOI: 10.5281/zenodo.20128541.
Ward-Broadfield M., 2026b, Galactic rotation curves from a single-field theory of the gravitational fabric. Zenodo DOI: 10.5281/zenodo.20268341.
Data Availability
The SPARC database is publicly available at astroweb.cwru.edu/SPARC; the Zenodo mirror is at zenodo.org/records/16284118. The framework is documented at www.temporalcongestionmechanics.com. The parent paper is deposited at Zenodo DOI 10.5281/zenodo.20128541; the companion rotation-curve paper at Zenodo DOI 10.5281/zenodo.20268341. The per-galaxy analysis behind Table 1 is reproducible from the public SPARC tables using the photometric baryonic masses of datafile1.txt and the rotation-curve data of datafile2.txt.