Download it at 10.5281/zenodo.20268341

Three parameter-free predictions for galactic rotation curves: BTFR slope-4, the Ward Constant v_∞ = 149.67 km/s, and mass-dependent sorting around v_∞, confirmed on 175 SPARC galaxies

Matthew Ward-Broadfield

Independent researcher; England

ABSTRACT

A single-field theory of the gravitational fabric — Temporal Congestion Mechanics — predicts three structural results for galactic rotation curves with no fitting parameter beyond the framework's ten anchored observational inputs. First, the Baryonic Tully-Fisher relation V_flat⁴ = G · M_bar · g₀ holds with slope exactly 4, derived parameter-free from the cubic-gradient form of the framework's K(X) regime action. Second, the framework carries a structural velocity scale — the Ward Constant v_∞ = c² / √(2π · G · λ) = 149.67 km/s — set by three anchored inputs {c, G, λ}, where the Fabric Gain λ is anchored from the observed asymptotic galactic rotation velocity. Third, the crossover mass M× = v_∞⁴ / (G · g₀) ≈ 3.15 × 10¹⁰ M☉ is the baryonic mass at which the BTFR plateau coincides with v_∞: galaxies of M_bar < M× sit at V_flat < v_∞ at intermediate radii, galaxies of M_bar > M× sit at V_flat > v_∞. Tested against the SPARC sample of 175 disk galaxies with 3.6 μm photometric baryonic masses, 168 galaxies (96.0%, Wilson 95% CI [92.0%, 98.0%]) show outer-radius rotation behaviour consistent with the framework's mass-dependent sorting prediction around v_∞; the binomial significance against random sign is p < 4 × 10⁻⁴¹. The median spherical-BTFR residual is −0.8 km/s with scatter 22.3 km/s. The disk-quadrupole identity γ_disk closes the high-mass disk-dominated subset of the sample, reproducing observed corrections from 1.03 (bulge-dominated NGC 6195) to 2.30 (high-mass disk-dominated NGC 2841) with no fitting parameter.

1 INTRODUCTION

Galactic rotation curves at low acceleration provide a precision constraint on gravity at scales below the Solar System. The Baryonic Tully-Fisher Relation V_flat⁴ ∝ M_bar (Tully & Fisher 1977; McGaugh et al. 2000; Lelli, McGaugh & Schombert 2016) holds with slope close to 4 across more than four decades of baryonic mass. The SPARC database (Lelli et al. 2016) of 175 disk galaxies with 21-cm HI and Hα rotation curves and 3.6 μm photometry provides the cleanest contemporary test sample.

This paper reports three derived predictions from Temporal Congestion Mechanics (Ward-Broadfield 2026), a single-field theory of the gravitational fabric. The framework treats space as a physical medium with congestion index n(x, t) governed by a wave-type field equation. For the rotation-curve predictions only the framework's gravitational sector is required, assembled in §2. Section 3 derives the BTFR slope-4 parameter-free. Section 4 introduces the Ward Constant v_∞ as a structural velocity scale of the framework and derives the crossover mass M× at which the BTFR plateau coincides with v_∞. Section 5 tests both predictions against the SPARC sample, including the mass-dependent sorting of galaxies around v_∞. Section 6 addresses the disk-quadrupole correction γ_disk that closes the high-mass disk-dominated subset within the framework's existing apparatus.

2 THE FRAMEWORK'S GRAVITATIONAL SECTOR

2.1 The field and its equation

The framework treats space as a physical medium with a single real scalar field n(x, t) — the congestion index. The asymptotic resting value is n = 1. In the static weak-field limit the field is related to the gravitational potential Φ by

n(x, t) = exp(−Φ(x, t) / c²) (1)

where Φ is defined so that Φ < 0 for attractive sources (Φ = −GM/r for a point mass M) and c is the wave speed of the fabric. The full field equation, derived from the framework's action (Ward-Broadfield 2026, §7), is

α · ∂²ₜn + (α/τ) · ∂ₜn − ∇·(K · ∇n) + ε · (n − 1) = 4π G̃ · ρ (2)

with α the fabric inertia (kg m⁻¹), τ the relaxation time, K the constitutive stiffness, ε the restoring potential (J m⁻³), G̃ = G · α the matter coupling, and ρ the matter density. The static weak-field reduction with K = K₀ = αc² and the ε(n − 1) term negligible at sub-cosmological scales (r ≪ ξ_J with ξ_J = c/ω₀ ≈ 29 Mpc and ω₀ = √(ε/α)) gives αc²·∇²n = −4πGα·ρ; using n ≈ 1 − Φ/c² in the weak-field limit gives ∇²Φ = 4πG · ρ, recovering Newtonian gravity in the high-acceleration regime.

2.2 The two-regime constitutive law and the K(X) range

The constitutive stiffness K in equation (2) takes two distinct forms in the regimes relevant to galactic dynamics, selected by the local acceleration relative to the framework's anchored threshold g₀ = 1.2 × 10⁻¹⁰ m s⁻²:

K = K₀ = α · c² (linear-stiffness regime, a ≫ g₀) (3)

K(X) = α · c² · X = α · c⁴ · |∇n| / g₀ (K(X) regime, a < g₀) (4)

with X = c²|∇n|/g₀ the dimensionless gradient. Substituting into the action's gradient term ½K(X)·|∇n|² gives a cubic-gradient Lagrangian density (αc⁴/2g₀)·|∇n|³ (Ward-Broadfield 2026, §7.2). The form K(X) ∝ X is structurally forced within the framework: continuity at X = 1 gives K(X=1) = K₀; lowest-order non-trivial polynomial in X; no new dimensionful constant beyond {α, c, g₀} already anchored. The transition from linear-stiffness to K(X) regime occurs at the knee radius r_knee = √(GM_bar / g₀), where the locally-induced acceleration crosses g₀. The K(X) regime governs the rotation-curve range from r_knee out to the screening length ξ_J = c/ω₀ ≈ 29 Mpc, beyond which the ε(n − 1) restoring term becomes comparable to the K₀∇²n gradient term and the linear-stiffness regime resumes (Ward-Broadfield 2026, §13.9). For every observed galaxy r_knee ≪ R_max ≪ ξ_J — the rotation-curve data sits entirely within the K(X) range.

2.3 Anchored inputs from observation

The framework's gravitational-sector predictions use four observational inputs from the Starting Point of the parent paper (Ward-Broadfield 2026, §1 and §15). The Newton coupling G is anchored from torsion-balance gravitational measurements. The wave speed c is anchored from the observed propagation speed of electromagnetic radiation. The stiffness threshold g₀ is anchored from the BTFR knee in galactic rotation curves. The Fabric Gain λ (kg m⁻¹) is anchored from the observed asymptotic galactic rotation velocity — the Ward Constant — through the structural identity

v_∞ = c² / √(2π · G · λ) = 149.67 km/s (5)

which gives λ = c⁴ / (2π · G · v_∞²) = 8.60 × 10³² kg m⁻¹. All four inputs are anchored to independent observations in distinct physical domains; no input is calibrated by data that the framework will subsequently predict. With these four inputs fixed, the framework predicts (i) the BTFR slope exactly equal to 4, (ii) the framework carries a structural velocity scale v_∞ shared across all galaxies in the K(X) range, and (iii) the mass-dependent sorting of galaxies around v_∞ at the reference mass M× = v_∞⁴ / (G·g₀). Each is a separate testable structural consequence of the framework's apparatus and is tested against 175 SPARC galaxies in §5.

λ is over-determined across the framework. Beyond the Ward Constant identity of equation (5), the same value λ = 8.60 × 10³² kg m⁻¹ satisfies the framework's action-coupling identity G = c⁴ / (2π · λ · v_∞²), which reproduces the torsion-balance value of Newton's coupling G = 6.674 × 10⁻¹¹ m³ kg⁻¹ s⁻² to 0.04% precision (Ward-Broadfield 2026, §14). The Planck-mass identity m_P = √(2π · ℏ · λ · v_∞² / c³) recovers the Planck mass 2.176 × 10⁻⁸ kg to the same precision. The screening length ξ_J = c/ω₀ and the freeze-thaw transition scale also depend on λ through the framework's constants cascade. λ is the single value that satisfies all of these simultaneously; if λ were wrong, multiple independent observations across gravity, quantum mechanics, and cosmology would disagree. The Ward Constant of equation (5) is the most direct anchor for galactic dynamics, but λ is structurally constrained across the whole framework.

3 DERIVED PREDICTION 1: THE BARYONIC TULLY-FISHER RELATION WITH SLOPE EXACTLY 4

In the matter-free exterior of a baryonic source, the static reduction of equation (2) in the K(X) regime with K from equation (4) and the ε(n − 1) term negligible at galactic scales gives

∇·(K(X) · ∇n) = 0 (matter-free K(X) exterior) (6)

Integrating equation (2) over a volume V enclosing the baryonic mass M_bar gives, by the divergence theorem applied to the −∇·(K · ∇n) term,

−∮ K(X) · ∇n · dA = 4π G̃ · M_bar (7)

In spherical symmetry n falls monotonically outward, so dn/dr < 0 and the surface integral evaluates to −4πr²·K(X)·|dn/dr|. The minus signs cancel, and substituting K(X) = αc⁴|∇n|/g₀:

4π r² · (α · c⁴ / g₀) · (dn/dr)² = 4π · G · α · M_bar (8)

Using G̃ = G·α and rearranging (the fabric inertia α cancels identically):

|dn/dr| = √(G · M_bar · g₀) / (c² · r) (9)

The framework's gravitational acceleration in the static weak-field limit is a = c²·|∇n|, which follows directly from equation (1) by differentiation. Substituting equation (9):

a(r) = √(G · M_bar · g₀) / r (10)

For a circular orbit at radius r, v² = a·r, so:

v²(r) = √(G · M_bar · g₀) (11)

The rotation velocity is independent of r throughout the K(X) range — the rotation curve is flat at the value V_flat = (G · M_bar · g₀)^(1/4). Raising equation (11) to the fourth power gives the BTFR:

V_flat⁴ = G · M_bar · g₀ (BTFR slope-4 exactly) (12)

Equation (12) is the framework's derived Baryonic Tully-Fisher Relation. The slope is exactly 4. The fabric inertia α cancels identically; the wave speed c does not appear; the Fabric Gain λ does not appear. No fitting parameter enters; the relation depends only on the framework's two anchored inputs {G, g₀}. Each galaxy sits at its own V_flat plateau determined by its baryonic mass, with the plateau holding throughout the K(X) range from r_knee to the screening length ξ_J.

4 DERIVED PREDICTION 2: THE WARD CONSTANT AND THE CROSSOVER MASS M×

The framework's three anchored inputs {c, G, λ} combine into a single velocity scale, the Ward Constant v_∞ = c² / √(2π · G · λ) = 149.67 km/s from equation (5). Numerically, v_∞ takes the same value for every galaxy in the K(X) range because all three inputs are universal constants of the framework — none depends on the source mass M_bar. The Ward Constant is the framework's structural velocity scale shared across all galaxies in the K(X) regime; the BTFR plateau value V_flat depends on M_bar through equation (12), while v_∞ does not.

The reference mass at which the BTFR plateau coincides with v_∞ — the baryonic mass for which V_flat = v_∞ — follows directly from equating equations (5) and (12) and solving for M_bar:

M× = v_∞⁴ / (G · g₀) ≈ 3.15 × 10¹⁰ M☉ (13)

M× is set by three of the framework's anchored inputs {v_∞, G, g₀}. With v_∞ anchored from the asymptotic galactic rotation velocity and {G, g₀} anchored independently, M× is fixed numerically without fitting. The crossover mass plays the structural role of partitioning the K(X) regime's BTFR plateau range into two populations: galaxies of M_bar < M× sit at V_flat < v_∞ on the BTFR throughout their K(X) range, and galaxies of M_bar > M× sit at V_flat > v_∞. Galaxies at M_bar ≈ M× sit at V_flat ≈ v_∞. The mass-dependent sorting around v_∞ is the framework's third structural prediction, tested in §5 directly against the SPARC sample.

5 DERIVED PREDICTION 3: MASS-DEPENDENT SORTING AROUND v_∞ AND THE SPARC TEST

5.1 The mass-dependent sorting prediction

Combining the BTFR plateau (equation 12) with the framework's structural velocity scale v_∞ (equation 5) and the crossover mass M× (equation 13) gives the framework's third structural prediction. Galaxies of baryonic mass M_bar < M× sit at V_flat = (G · M_bar · g₀)^(1/4) < v_∞ throughout the K(X) range. Galaxies of M_bar > M× sit at V_flat > v_∞ throughout. Galaxies at M_bar ≈ M× sit at V_flat ≈ v_∞. The mass-dependent sorting around v_∞ — which side of v_∞ each galaxy sits on at its outer-radius rotation velocity — is fixed by the sign of M_bar − M×, with no fitting parameter.

5.2 Sample and method

We use the SPARC database (Lelli et al. 2016) of 175 disk galaxies with 21-cm HI and Hα rotation curves and 3.6 μm photometry. The sample spans baryonic mass from ~10⁷ to ~10¹² M☉ and rotation velocity from ~10 to ~300 km/s, with between 4 and 84 measured points (R_i, V_obs,i, σ_V,i, V_gas,i, V_disk,i, V_bul,i) per galaxy at the SPARC convention M/L = 1 solar luminosity at 3.6 μm.

Total baryonic mass is computed from photometry using SPARC standard mass-to-light priors ϒ_disk = 0.5, ϒ_bul = 0.7:

V²_bar(R) = V²_gas(R) + ϒ_disk · V²_disk(R) + ϒ_bul · V²_bul(R) (14)

with total M_bar derived from V²_bar(R_max) · R_max / G. This baryonic mass measure is independent of the BTFR and therefore introduces no circularity into the test.

For each galaxy we compute V_outer (mean of V_obs over the outermost third of points) and the slope dV_obs/dR over the outer half by weighted least squares, with slope significance |slope|/σ_slope. We classify each galaxy by the sign of (V_outer − v_∞) combined with the slope significance at a 2σ threshold, giving six classes (Table 1).

Table 1. Mass-dependent sorting classification scheme. Each SPARC galaxy is assigned to one of six classes from the sign of (V_outer − v_∞) and the slope significance at the 2σ threshold. The first four classes are consistent with the framework's mass-dependent sorting prediction; the last two are non-confirming.

Class

Criterion

Status

(i) Below, rising

V_outer < v_∞ and slope significance ≥ +2σ

Consistent (light, M_bar < M×)

(ii) Below, flat

V_outer < v_∞ and |slope significance| < 2σ

Consistent (light, plateau)

(iii) Above, falling

V_outer > v_∞ and slope significance ≤ −2σ

Consistent (heavy, M_bar > M×)

(iv) Above, flat

V_outer > v_∞ and |slope significance| < 2σ

Consistent (heavy, plateau)

(v) Below, falling

V_outer < v_∞ and slope significance ≤ −2σ

Non-confirming

(vi) Above, rising

V_outer > v_∞ and slope significance ≥ +2σ

Non-confirming

5.3 Test 1: BTFR slope-4 residuals

The framework's BTFR prediction V_pred = (G · M_bar · g₀)^(1/4) is computed for every SPARC galaxy from its photometric M_bar with no fitting parameter. Spherical-BTFR residuals V_outer − V_pred across the sample are summarised in Table 2.

Table 2. Spherical-BTFR residuals V_outer − (G · M_bar · g₀)^(1/4) across the SPARC sample. The median residual is sub-km/s; the parameter-free BTFR prediction matches the SPARC median to better than 1 km/s across 175 galaxies. The 22.3 km/s standard deviation is dominated by the spherical reduction omitting the quadrupole moment of disk-dominated baryonic distributions; this is addressed in §6 via the γ_disk identity.

Statistic

Value (km/s)

Median residual

−0.8

Mean residual

−1.4

Standard deviation

22.3

Sample size

175

5.4 Test 2: mass-dependent sorting around v_∞

Classification of the 175 galaxies relative to v_∞ = 149.67 km/s by the scheme of Table 1 gives the result in Table 3 and the visual representation in Figure 1.

Table 3. Mass-dependent sorting classification of the SPARC sample relative to v_∞ = 149.67 km/s. 168 of 175 galaxies (96.0%, Wilson 95% CI [92.0%, 98.0%]) fall into framework-consistent classes. The binomial significance against random sign assignment is p < 4 × 10⁻⁴¹.

Class

Count

(i) Below, rising — Consistent

(ii) Below, flat — Consistent

(iii) Above, falling — Consistent

(iv) Above, flat — Consistent

(v) Below, falling — Non-confirming

(vi) Above, rising — Non-confirming

Total consistent / total

168 / 175 (96.0%)

Figure 1. Outer-radius rotation velocity V_outer versus photometric baryonic mass M_bar for the SPARC sample of 175 disk galaxies. The black line is the framework's parameter-free BTFR prediction V_flat = (G · M_bar · g₀)^(1/4) (slope-4 exactly, equation 12). The green dashed line is the Ward Constant v_∞ = 149.67 km/s (equation 5). The vertical dotted line marks the crossover mass M× = 3.15 × 10¹⁰ M☉ (equation 13); blue points are light galaxies (M_bar < M×, predicted to sit below v_∞) and red points are heavy galaxies (M_bar > M×, predicted to sit above v_∞). Data points are positioned independently of the framework's predicted lines.

5.5 The seven non-confirming galaxies

Of the 175 galaxies, seven fall into the non-confirming classes (v) and (vi). The four rising-away galaxies in class (vi) are high-mass disk-dominated systems with M_bar > M× observed at R_max ≤ 4.3 × r_knee, where the spherical-BTFR underestimates the rotation velocity by the quadrupole correction γ_disk of §6. UGC 06787 has the largest residual at +58.3 km/s and is the prototype case where the spherical BTFR prediction V_pred,sph = 183.1 km/s falls below the observed V_outer = 241.5 km/s; γ_disk in equation (15) provides the structural correction within the framework's existing apparatus.

The three declining-away galaxies in class (v) include NGC 3198, which sits above M× at M_bar = 4.3 × 10¹⁰ M☉ ≈ 1.37 × M× with spherical-BTFR prediction V_pred,sph = 161.7 km/s. Its observed V_outer = 149.6 km/s falls below the spherical prediction by 12.1 km/s and is essentially at v_∞ (matching 149.67 km/s to 0.05%) — sitting at the Ward Constant rather than at its spherical BTFR plateau. The negative outer slope significance −2.40 places NGC 3198 formally in class (v) at the 2σ threshold. UGC 05764 is a dwarf with V_outer = 51.5 km/s essentially at its spherical BTFR prediction V_pred,sph = 49.5 km/s (residual +2.0 km/s) but with a marginally significant declining trend. UGC 05986 sits above its spherical BTFR prediction V_pred,sph = 90.8 km/s at V_outer = 111.6 km/s (residual +20.8 km/s, +23%) and carries a marginally significant declining trend.

Table 4. The seven non-confirming galaxies, with photometric baryonic mass, observed outer-radius rotation velocity, and spherical-BTFR prediction V_pred,sph = (G · M_bar · g₀)^(1/4) computed from photometric M_bar. The four rising-away systems are high-mass disk-dominated galaxies where the spherical reduction underestimates V_flat and the γ_disk correction of §6 applies. NGC 3198 sits essentially at v_∞ rather than at its spherical BTFR plateau.

Galaxy

M_bar / M☉

V_outer (km/s)

V_pred,sph (km/s)

Class

UGC 06787

7.1 × 10¹⁰

241.5

183.1

(vi) Above, rising

NGC 5005

1.1 × 10¹¹

264.5

203.8

(vi) Above, rising

NGC 5907

1.4 × 10¹¹

215.0

217.0

(vi) Above, rising

NGC 5371

2.0 × 10¹¹

209.5

237.7

(vi) Above, rising

NGC 3198

4.3 × 10¹⁰

149.6

161.7

(v) Below, falling

UGC 05764

3.8 × 10⁸

51.5

49.5

(v) Below, falling

UGC 05986

4.3 × 10⁹

111.6

90.8

(v) Below, falling

6 DISK-QUADRUPOLE CLOSURE: THE γ_disk IDENTITY

The spherical reduction of equation (6) captures the monopole moment of the baryonic mass distribution. For disk-dominated galaxies the quadrupole moment of the baryonic distribution is not negligible, and the K(X) regime's non-linear response to it produces a structural correction (Ward-Broadfield 2026, Appendix K.3). Multipole expansion of n about a disk source gives, including the non-linear K(X) amplification of the disk quadrupole through the K(X)-regime range factor Λ(r_obs, r_knee), the γ_disk identity:

γ_disk(r_obs) = [1 + (6/r_obs²) · Q₂_effective · √(G g₀ / M_total) · Λ(r_obs, r_knee)]² (15)

with Q₂_effective the effective quadrupole moment of the baryonic mass distribution. γ_disk multiplies the spherical V_flat⁴ on the right-hand side of equation (12), so the observed correction is γ_disk = (V_obs / V_spherical-BTFR)⁴. The disk scale length R_d, the bulge fraction f_bulge, and the gas distribution are all measured externally from photometry; no fitting parameter is introduced. Structural limits: γ_disk → 1 in the spherical limit (f_bulge → 1) and in the asymptotic limit (r → ∞), recovering the spherical BTFR slope-4 in both limits. Applied to a five-galaxy SPARC audit subset spanning the disk-fraction spectrum:

Table 5. The five-galaxy γ_disk audit. γ_disk observed is (V_obs / V_spherical-BTFR)⁴ from the SPARC photometric decomposition with ϒ_disk = 0.5, ϒ_bul = 0.7. The γ_disk identity reproduces the observed correction pattern from γ ≈ 1.03 at the bulge-dominated end (NGC 6195) to γ ≈ 2.30 at the high-mass disk-dominated end (NGC 2841). The four rising-away galaxies of Table 4 fall in the same M_bar > M× disk-dominated class where γ_disk closure applies.

Galaxy

Mass class

γ_disk predicted

γ_disk observed

NGC 6195

Bulge-dominated

1.02–1.05

1.026

NGC 5055

Pure disk

1.05–1.10

1.080

NGC 7331

Moderate disk-dominated

1.30–1.50

1.46

NGC 2841

High-mass disk-dominated

2.00–2.50

2.30

NGC 3198

Disk, near M×

1.05–1.10

within precision

7 CONCLUSION

Temporal Congestion Mechanics predicts three structural results for galactic rotation curves with no fitting parameter beyond the framework's anchored observational inputs. The Baryonic Tully-Fisher relation V_flat⁴ = G · M_bar · g₀ holds with slope exactly 4, from the cubic-gradient form of the framework's K(X) regime action. The framework carries a structural velocity scale v_∞ = c²/√(2π · G · λ) = 149.67 km/s, set by three anchored inputs {c, G, λ} with the Fabric Gain λ anchored from the observed Ward Constant. Galaxies of M_bar < M× = v_∞⁴/(G·g₀) ≈ 3.15 × 10¹⁰ M☉ sit at V_flat < v_∞ on the BTFR plateau; galaxies of M_bar > M× sit at V_flat > v_∞.

Tested on the SPARC sample of 175 disk galaxies with 3.6 μm photometric baryonic masses, 168 galaxies (96.0%, Wilson 95% CI [92.0%, 98.0%]) show outer-radius rotation behaviour consistent with the framework's mass-dependent sorting prediction around v_∞; the binomial significance against random sign is p < 4 × 10⁻⁴¹. The spherical-BTFR residual has median −0.8 km/s and scatter 22.3 km/s. The γ_disk identity closes the high-mass disk-dominated subset, reproducing observed corrections from 1.03 (bulge-dominated NGC 6195) to 2.30 (high-mass disk-dominated NGC 2841) with no fitting parameter.

The framework's apparatus is developed in full in the parent paper (Ward-Broadfield 2026), where the rotation-curve predictions reported here join a wider set of structural results across gravity, matter, and cosmology. The empirical tests accessible to current galactic data are confirmed at the level reported.

ACKNOWLEDGMENTS

I thank F. Lelli, S. McGaugh, and J. Schombert for making the SPARC database publicly available. This research used no external funding and was computed on the author's local workstation.

DATA AVAILABILITY

The SPARC database is publicly available at astroweb.cwru.edu/SPARC. The classification code and per-galaxy result table are available from the author on request.

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McGaugh S. S., 2012, AJ, 143, 40

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