Extra Gravity in SPARC and KiDS-1000 Without a Dark-Matter Particle or Changing Newtonian Gravity: The Connection to Electron Mass and the Fine-Structure Constant

Abstract

The extra gravity in SPARC disks and KiDS-1000 stacks is identified with organised mass in a physical matter substrate. The author names that substrate the Spaticle field. The Einstein cosmological constant with Λ ≈ 1.11 × 10⁻⁵² m⁻² converts to ρ_Λ = Λ c² / (8π G) = 5.9 × 10⁻²⁷ kg m⁻³, or u_Λ = 5.30 × 10⁻¹⁰ J m⁻³. That map is not adopted: it inherits H₀, and published H₀ values include ≈ 64, 67.4 and 73.50 km s⁻¹ Mpc⁻¹. The BFUT particle sector derives the equilibrium Spaticle-field density independently from the condensation functional and particle constants:

ρₛ = (G c² / 8π ħ_vss⁴) [m_e_vss / α_vss]⁶ = 7.3 × 10⁻²⁷ kg m⁻³.

The particle-sector result is approximately 23.8 percent above the cosmological reference value and is adopted throughout this paper as ρₛ = 7.3 × 10⁻²⁷ kg m⁻³. It follows from BFUT particle dynamics and is independent of H₀ and galaxy data. The factor 1/(8π) is fixed by the gravitational field-energy normalization and by the Einstein coupling of stress-energy to curvature. The Spaticle field is the physical matter substrate whose deformation is represented by spacetime curvature. Its finite density allows organised motion to entrain substrate mass, and that mass gravitates through the same 8πG coupling.

The substrate response scale is defined by the inverse-length parameter μₛ, with μₛ² = 3 G ρₛ / c². The corresponding mass scale is mₛ = ħ_vss μₛ / c. The organised-regime acceleration is

aₛ = c² μₛ / 3 = c (G ρₛ / 3)^{1/2} = 1.20840317 × 10⁻¹⁰ m s⁻².

The implied acceleration is aₛ = 1.20840317 × 10⁻¹⁰ m s⁻². Newton’s law is not changed. No dark-matter particle is introduced. For organised substrate entrainment, the effective law is g² = gᵦ² + Θ aₛ gᵦ, equivalently v² = vᵦ² [1 + Θ aₛ R / vᵦ²]^{1/2}, with 0 ≤ Θ ≤ 1. On 175 SPARC galaxies, Υ = 0.5 locked, the median outer relative residual is 0.096, shape agreement is 92.0 percent, and flat classification is 98.8 percent. The same algebra at MOND’s a₀ = 1.20 × 10⁻¹⁰ m s⁻² gives median residual 0.094. On KiDS-1000 Fig. 3, four mass bins and 60 points, the published 60×60 ESD covariance gives χ²/N = 9.78 for the DME law. Extra mass is therefore interpreted as organised or entrained substrate mass. In the fully organised limit Θ = 1, the deep-organised relation is v⁴ = G Mᵦ aₛ. Ultra-diffuse and disrupted systems provide the complementary regime in which coherent entrainment can be weak.

Keywords: galaxy rotation curves; SPARC; KiDS-1000; weak lensing; dark matter; Newton; fine-structure constant; electron mass; MOND; matter substrate

1. Introduction

This paper identifies the physical fabric of space as pervaded by a matter substrate. The author names that substrate the Spaticle field. Ordinary matter is embedded in it. Newtonian gravity and the Einstein field equation are retained. The effective relation connects organised motion to the substrate mass that becomes coherently entrained and contributes to the total gravitating mass.

The extra gravity in disks, stacks and clusters is an observational fact (Rubin & Ford 1970; Bosma 1981; Lelli, McGaugh & Schombert 2016; Brouwer et al. 2021). The Spitzer Photometry and Accurate Rotation Curves database is denoted SPARC, and the 1000-square-degree data release of the Kilo-Degree Survey is denoted KiDS-1000. Cold dark matter assigns the extra gravity to a collisionless particle halo fitted per system. Modified Newtonian Dynamics (MOND) changes the relation between force and acceleration below a scale read from rotation curves (Milgrom 1983; Famaey & McGaugh 2012). BFUT identifies the Spaticle field as a real, physical, gravitating and deformable matter substrate in which ordinary matter is embedded. Organised or entrained Spaticle-field mass supplies the additional gravitating component.

The derivation has three parts. First, the BFUT condensation functional generates the particle constants and the substrate density. Second, the carrier dynamics generate the acceleration scale. Third, the Dark Matter Effects relation connects organised motion to entrained substrate mass and is applied to SPARC and KiDS-1000. Newtonian gravity remains unchanged throughout the calculation.

The extra gravitating component in these data is organised or entrained mass in the physical Spaticle substrate at the derived equilibrium density. That is the content of the law.

2. Substrate density

This paper derives the equilibrium density of the matter substrate, which the author names the Spaticle field. That density is written ρₛ. Two constructions are given. Section 2.1 converts the Einstein cosmological constant into a mass density. Section 2.2 derives ρₛ from the condensation functional and the particle constants. Only the second construction is adopted.

2.1 Cosmological-constant map

The Einstein cosmological constant Λ is converted to a mass density by ρ_Λ = Λ c² / (8π G). The observational value of that constant is Λ ≈ 1.11 × 10⁻⁵² m⁻². Substitution gives ρ_Λ = 5.9 × 10⁻²⁷ kg m⁻³. The corresponding energy density is u_Λ = ρ_Λ c² = 5.30 × 10⁻¹⁰ J m⁻³. This is the cosmological-constant mapped value.

That conversion is not adopted here. The value of Λ used in ΛCDM is linked to the adopted cosmological parameter set, including H₀, and H₀ is not unique. Published values include ≈ 64 km s⁻¹ Mpc⁻¹ (Faucher, Benisty & Mota 2026, A&A 705, A112), 67.4 ± 0.5 km s⁻¹ Mpc⁻¹ (Planck Collaboration 2020), and 73.50 ± 0.81 km s⁻¹ Mpc⁻¹ (H0DN Collaboration 2026). Because the mapped density moves when H₀ moves, 5.9 × 10⁻²⁷ kg m⁻³ is not the working density. The working density is the particle-sector result of Section 2.2, ρₛ = 7.3 × 10⁻²⁷ kg m⁻³, which contains no H₀.

2.2 Particle-sector derivation

The four-term free-energy functional E(R) = A/R² + BR² + CR + D/R describes the condensation scale. The adopted dimensionless coefficients are A = 1/2, B = 0.56308, C = −1/3 and D = 1. A follows from the normalized quantum kinetic term. B follows from void-filling geometry. C follows from three-fold symmetry. D follows from one complete topological phase winding.

A is the quantum kinetic energy cost of confining the condensate. In SI, A_SI = ħ_vss²/(2 m_eff). Convert to model units with E_unit = m_eff c² and ℓ_model = ħ_vss/(m_eff c). Then A_model = A_SI / (E_unit × ℓ_model²) = [ħ_vss²/(2 m_eff)] / [m_eff c² × ħ_vss²/(m_eff² c²)] = [ħ_vss²/(2 m_eff)] × [m_eff/ħ_vss²] = 1/2. Every factor of ħ and m_eff cancels. This identity does not depend on the value of R₀ or on any measured constant. The factor 1/2 is the same factor that appears in the Schrödinger kinetic term.

D is the energy of one complete topological phase winding of the condensate. In SI, D_SI = ħ_vss·c. With the same definition m_eff = ħ_vss/(c·ℓ_model), D_model = ħ_vss·c / (m_eff c² ℓ_model) = 1 exactly. D = 2A exactly, reflecting their common origin.

C is the surface term. When the central interstice is expelled as the electron precursor, three co-rotating condensates compress toward a common centre and energy is released at the boundary, so the surface term is negative. The magnitude 1/3 follows from the fact that there are exactly three condensates and they are identical: same substrate, same density, same orbital radius, each facing the void across the same 60-degree arc. There is no physical distinction between them. One expelled centre shared equally among three identical sectors gives C = −1/3 exactly, by C3v symmetry.

B is the filling-deficit ratio of the normalized three-condensate cluster. Three mutually touching unit-radius condensates sit at orbital radius d = 2/√3 and enclose A_void = √3 − π/2. The d condensate faces the void on-axis with component 1. Each u condensate faces it at 60° with component 1/2, giving δ_d = 2δ_u. The void-filling constraint gives the d share A_void/2 and the symmetric share A_void/3, so the d extra is A_void/6. Therefore B = [π(d + 1)² − 3π + A_void] / [3π + A_void/6] = 0.56308208, which gives the adopted coefficient B = 0.56308.

Using A = 0.5, B = 0.56308, C = −0.333333 and D = 1, stationarity gives 2BR₀⁴ + CR₀³ − DR₀ − 2A = 0. Its positive real root is R₀ = 1.27348221, and the second derivative is positive at this root.

The measured proton charge radius rp = 0.8414 × 10⁻¹⁵ m fixes the physical length represented by one model-radius unit: ℓ_model = rp/R₀ = 6.60708091 × 10⁻¹⁶ m.

With the measured proton mass m_p = 1.67262192369 × 10⁻²⁷ kg, this length gives the BFUT-derived reduced Planck constant: ħ_vss = m_p c ℓ_model/π = 1.05457687 × 10⁻³⁴ J s.

The BFUT-derived electron mass and fine-structure constant are m_e_vss = mp/(6π⁵) = 9.10955518 × 10⁻³¹ kg and α_vss = e²/(4π ε₀ ħ_vss c) = 0.00729731763044, with α_vss⁻¹ = 137.036655199.

The corresponding dimensional consistency check is R₀ = rp mp c/(π ħ_CODATA) = 1.27348831, within 0.00048 percent of the stationary root.

The characteristic particle-sector mass is m*_vss = m_e_vss/α_vss = 1.24834297 × 10⁻²⁸ kg.

Its reduced Compton wavelength equals the classical electron radius: λ* = ħ_vss/(m*_vss c) = α_vss ħ_vss/[m_e_vss c] = rₑ_vss = 2.81788728 × 10⁻¹⁵ m.

The weak-field gravitational energy-density magnitude is u_g = g*²/(8πG). At radius rₑ_vss, the gravitational field of m*_vss is g* = Gm*_vss/rₑ_vss². Substitution gives u_g = Gm*_vss²/(8πrₑ_vss⁴).

u_g = G[m_e_vss/α_vss]²/(8πrₑ_vss⁴) = 6.56355667 × 10⁻¹⁰ J m⁻³.

The factor 8π is fixed at both levels of the gravitational description. In the weak-field energy density it appears in u_g = g²/(8πG). In the covariant description the Spaticle field supplies physical stress-energy to G_μν = (8πG/c⁴)T_μν. The Spaticle field is the matter substrate whose bending, compression and waves are represented by spacetime geometry. Because it has the definite density ρₛ, organised motion entrains a definite amount of substrate mass. The gravitational contribution of that entrained mass carries the same Einstein coupling. This fixes 8πG as the physical normalization of the density derivation.

Dividing the particle-sector gravitational energy density by c² gives ρₛ = u_g/c² = G[m_e_vss/α_vss]²/(8πrₑ_vss⁴c²) = (Gc²/8πħ_vss⁴)[m_e_vss/α_vss]⁶.

ρₛ = 7.3 × 10⁻²⁷ kg m⁻³. The adopted working value is ρₛ = 7.3 × 10⁻²⁷ kg m⁻³, equivalent to u_g = 6.56355667 × 10⁻¹⁰ J m⁻³ or 4.096 GeV m⁻³.

3. Acceleration from the carrier mass term

The local gravitational carrier is the deformation of the Spaticle substrate. The carrier perturbation δψ_vss obeys the covariant carrier equation

g^{μν} ∇_μ ∇_ν (δψ_vss) − μₛ² δψ_vss = (1/c²) g^{μν} ∇_μ ∇_ν Ψ_matter.

Here μₛ is the inverse-length response scale of the carrier. The BFUT carrier mass term is μₛ² = 3Gρₛ/c², so μₛ = √(3Gρₛ/c²). The organised acceleration associated with this response scale is aₛ = c²μₛ/3. Substitution gives aₛ = c√(Gρₛ/3) = 1.20840317 × 10⁻¹⁰ m s⁻². Thus the dependency chain is ρₛ → μₛ² → μₛ → aₛ, with no galaxy quantity entering the derivation.

The carrier dynamics therefore derive the acceleration scale directly from the particle-sector Spaticle-field density.

The associated coherence length is Lₛ = 1/μₛ = c/(3Gρₛ)¹ᐟ² ≈ 26.2 Gly. Its scale is far larger than a galaxy and leaves local inverse-square behaviour unchanged. SPARC and KiDS-1000 use the organised-gravity relation for the entrained mass.

3.1 Physical interpretation of the substrate scale

The Spaticle field is a real, gravitating and deformable substrate of space. Its equilibrium density is derived in Section 2. Ordinary matter is embedded in this substrate. Organised motion entrains a portion of the substrate, and that entrained mass is the extra gravitating component.

Ordinary matter condenses from the Spaticle field and remains embedded within it. The substrate bends, compresses and waves. Gravitational effects are deformations of this physical matter substrate. Physical disturbances and forces propagate through the same substrate.

The Spaticle field is not the nineteenth-century luminiferous ether. The luminiferous ether was introduced primarily as a carrier for light. The Spaticle field has a broader physical role: it is the matter substrate from which ordinary matter condenses, in which matter remains embedded, whose organised mass contributes gravitationally through entrainment, and through which electromagnetic and gravitational disturbances propagate.

The microscopic length in the density construction is the classical electron radius, rₑ = α ħ/(mₑ c). The equality λ_* = ħ/(m_* c) = rₑ follows from m_* = mₑ/α. That is the substrate reference length. The electron is not a constituent of the field. The length is the electromagnetic matching scale already fixed by mₑ and α.

The factor 8π is fixed by the gravitational field-energy normalization and the Einstein coupling G_μν = (8πG/c⁴)T_μν. The Spaticle field carries physical stress-energy, and its organised, entrained mass contributes through this coupling. The factor 3 in μₛ² = 3Gρₛ/c² is the BFUT carrier mass coefficient and propagates directly into aₛ = c²μₛ/3.

The inverse-length parameter μₛ is distinct from a particle mass. Defining μₛ² = 3Gρₛ/c² gives dimensions of length⁻². If a particle-like mass notation is desired, it is mₛ = ħ_vss μₛ/c. Algebraically, consistency with this definition requires aₛ = c² μₛ / 3 = c√(Gρₛ/3). This dimensional distinction avoids treating the inverse-length parameter as a mass-squared quantity.

3.2 Time, propagation capacity, and the unification of time dilation

In the BFUT framework, time is a physical consequence of propagation within the Spaticle field. The field has a finite propagation capacity. The universal speed c is the full propagation budget available to a massless excitation. For a massive object, spatial motion uses part of this finite budget, leaving the remainder available to internal propagation.

c² = vₛₚₐₜᵢₐₗ² + vᵢₙₜₑᵣₙₐₗ²

so that

vᵢₙₜₑᵣₙₐₗ = √(c² − vₛₚₐₜᵢₐₗ²).

If dτ is proper time and dt is coordinate time, the internal propagation fraction gives

dτ/dt = vᵢₙₜₑᵣₙₐₗ/c = √(1 − vₛₚₐₜᵢₐₗ²/c²),

and therefore

dt/dτ = 1/√(1 − vₛₚₐₜᵢₐₗ²/c²) = γ.

Thus kinematic time dilation follows from the finite propagation budget. With vₛₚₐₜᵢₐₗ = 0, the complete budget is available internally, so vᵢₙₜₑᵣₙₐₗ = c and dτ = dt.

Gravity produces the corresponding effect through deformation of the Spaticle field, which changes the local propagation conditions experienced by a clock. For a static spherical gravitational field, the time relation is

dτ = dt √(1 − 2GM/(rc²)),

or

dt/dτ = 1/√(1 − 2GM/(rc²)).

For a weak gravitational field,

dτ/dt ≈ 1 − GM/(rc²).

The special-relativistic and gravitational expressions therefore have a common physical interpretation in BFUT: time is governed by the propagation budget of the Spaticle field. Spatial motion uses part of that budget, while gravitational deformation changes the local propagation state of the substrate. For weak gravity and low spatial velocity, the combined relation is

dτ/dt ≈ 1 − GM/(rc²) − vₛₚₐₜᵢₐₗ²/(2c²).

This unifies gravitational and kinematic time dilation as two manifestations of the same underlying propagation dynamics.

The massless limit follows from the substrate propagation capacity itself. In the absence of a massive object's spatial-motion demand, the full propagation capacity is c. A massless disturbance therefore propagates through the Spaticle substrate at c. Electromagnetic and gravitational disturbances are disturbances of the same physical Spaticle substrate and consequently share this propagation speed.

4. The Dark Matter Effects relation

Let vᵦ(R) be the Newtonian circular speed of the observed baryons. The Dark Matter Effects (DME) relation is

v²(R) = vᵦ²(R) [1 + Θ(R) aₛ R / vᵦ²(R)]¹ᐟ².

Equivalently,

g²(R) = gᵦ²(R) + Θ(R) aₛ gᵦ(R),

where gᵦ(R) = vᵦ²(R)/R = GMᵦ(R)/R².

For an approximately constant Θ in the deep-organised regime, v⁴ = Θ G Mᵦ aₛ. The fully organised limit Θ = 1 therefore gives v⁴ = G Mᵦ aₛ.

On a component i of a merging system the extra mass is

Mₑₓₜᵣₐ,ᵢ = Mᵦ,ᵢ { [1 + Θᵢ aₛ Rᵢ² / (G Mᵦ,ᵢ)]¹ᐟ² − 1 },

where 0 ≤ Θᵢ ≤ 1 measures the degree of coherent entrainment. Θᵢ = 1 is the fully organised limit and Θᵢ → 0 is the unorganised limit.

For the two-component merger application, Θ_gal = 1 and Θ_gas = 0: entrained extra mass remains with the organised galaxies and drops from the shocked gas.

5. Comparison with MOND

MOND uses the galaxy-calibrated scale a₀ = 1.20 × 10⁻¹⁰ m s⁻² (Milgrom 1983; Famaey & McGaugh 2012; McGaugh, Lelli & Schombert 2016). The BFUT particle-sector derivation gives aₛ = 1.20840317 × 10⁻¹⁰ m s⁻², a relative difference of 0.7 percent. This independent agreement provides a numerical consistency check on the derived acceleration scale. BFUT retains Newtonian gravity and identifies the additional gravitating component as organised or entrained Spaticle-field mass governed by Θ. SPARC and KiDS-1000 test this particle-derived acceleration scale and its organised-gravity relation.

6. SPARC

Sample: 175 disks with Spitzer photometry and resolved rotation curves (Lelli, McGaugh & Schombert 2016; see also McGaugh, Lelli & Schombert 2016). Stellar mass-to-light ratio Υ = 0.5 at 3.6 μm, locked for the whole sample (the SPARC 3.6 μm convention, not fitted here). Baryonic speed:

vᵦ² = v_gas |v_gas| + Υ v_disk² + Υ v_bulge².

Outer-half points have R at least half the last measured radius. A curve is classed flat when outer-half standard deviation over mean is below 0.12. The galaxy residual is the median of |V_pred − V_obs| / V_obs on the outer half.

The SPARC calculations use the fully organised value Θ = 1 and aₛ = 1.20840317 × 10⁻¹⁰ m s⁻². Shape agreement is 92.0 percent (161/175). Flat classification is correct for 98.8 percent (159/161), and non-flat classification is correct for 14.3 percent (2/14). The median outer residual is 0.096. The residual is below 0.20 for 132 galaxies and below 0.25 for 147 galaxies. The same relation at a₀ = 1.20 × 10⁻¹⁰ gives median residual 0.094. Newtonian baryons at the same Υ give median residual 0.495. Figure 1 shows the three residual histograms. The galaxy table is Appendix A.

Most observed-non-flat, predicted-flat cases already have a flat baryonic curve under the same outer-scatter rule. Non-negative extra mass cannot force a decline when V_bar is already flat. The same locked-Υ set was compared with NFW. A zero-parameter halo uses Moster et al. (2013) abundance matching for M200(Mᵦ) and the Dutton & Macciò (2014) c(M200) relation. Median outer residual: Newton 0.495, abundance-matching NFW 0.308, DME 0.096. Full-curve χ² on 3391 points with published Verr: Newton 1.49×10^6, NFW 1.85×10^6, DME 2.05×10⁵. Inner points with 1–2 km s^{-1} errors dominate χ² in every model. The designed metric remains the outer-half residual. A two-parameter NFW fit per galaxy (350 extra numbers) can push that residual to 0.029. That is a fit. Figure 2 is the zero-parameter comparison.

Figure 1. SPARC galaxy-by-galaxy median outer relative residual. Newton at locked Υ = 0.5 (grey), DME at aₛ = 1.20840317 × 10⁻¹⁰ (ink), same algebra at a₀ = 1.20 × 10⁻¹⁰ (gold step).

Figure 2. SPARC median outer residual at locked Υ = 0.5. Newton; zero-parameter abundance-matching NFW using Moster et al. (2013) with the Dutton & Macciò (2014) concentration relation; DME at aₛ = 1.20840317 × 10⁻¹⁰ m s⁻².

7. KiDS-1000

The public Figure 3 excess surface density (ESD) files of Brouwer et al. (2021) contain four isolated-lens stellar-mass bins. Their Equation 23 converts ESD to equivalent circular speed, v_circ = √(4GΔΣR). The baryonic contribution is represented as a point mass, vᵦ² = GM_gal/R, with log₁₀(M_gal/M_⊙) = {10.14, 10.57, 10.78, 10.96}. The data contain 60 radial points from 0.035 Mpc to 2.60 Mpc.

The KiDS-1000 calculations use the fully organised value Θ = 1. The published 60 × 60 ESD covariance of Brouwer et al. is the primary estimator. The reported DME result is χ²/N = 9.78 for the 60 measurements. The equivalent-speed profiles are flat and rank as M¹ᐟ⁴, as required by v⁴ = G M aₛ. Figure 3 shows the four bins. Appendix B gives the point table.

Figure 3. KiDS-1000 Fig. 3 equivalent circular speed. Points: ESD converted by Brouwer Eq. 23. Gold: DME at aₛ = 1.20840317 × 10⁻¹⁰. Dashed: Newton point-mass baryons.

8. Mergers and low-rotation systems

The two-component entrainment relation applies directly to a cluster collision. Before the merger, the gas shares organised motion with the galaxies, with Θ_gas = Θ_gal = 1. During the collision, the galaxies retain their organised motion while the gas is stripped and shock-heated, giving Θ_gal = 1 and Θ_gas = 0. Lensing follows the galaxies (Clowe et al. 2006; Paraficz et al. 2012). Figure 5 is the official JWST and Chandra composite (Cha et al. 2025; NASA/STScI/CXC). Intracluster plasma outweighs the stars by about six to seven (Clowe et al. 2006). The JWST aperture masses inside 250 kpc are centred on the brightest cluster galaxies (BCGs), at approximately 1.59 × 10¹⁴ and 0.87 × 10¹⁴ M_⊙ (Cha et al. 2025). The entrainment relation predicts this offset from the shocked X-ray gas.

NGC 1052-DF2 has a published dynamical mass consistent with the stars alone (van Dokkum et al. 2018; Emsellem et al. 2019). That mass is consistent with Θ = 0 and strongly disfavours the fully organised Θ = 1 limit. Figure 4 shows that comparison. NGC 1052-DF4 and FCC 224 lie on the same no-halo line (Buzzo et al. 2025).

Figure 4. NGC 1052-DF2 inside 2.7 kpc. Gold band: published M_dyn. Ink bar: DME with Θ = 0. Grey bar: DME forced at Θ = 1.

Figure 5. Official JWST NIRCam and Chandra composite of 1E 0657-56 (NASA/ESA/CSA/STScI/CXC; processing J. DePasquale; scientific analysis Cha et al. 2025). Yellow and white show galaxies and intracluster light (ICL), with Θ = 1. Pink shows X-ray gas, with Θ = 0. Blue shows the published lensing mass centred on the galaxies. The entrained Spaticle-field mass follows the organised galaxy component.

9. The same architecture in any rotating medium

A rotating galaxy is an organised flow in a physical medium. The vortex analogy describes its kinematics. The Spaticle field retains the constitutive dynamics defined by BFUT.

Standard core identities: V_c = Ω_c r_c, Γ = 2π Ω_c r_c², Re_c = Ω_c r_c² / ν. Dimensionless controls include Π_R = r_c / R, Π_h = h / r_c, Π_ρ = ρ / ρ_ref, and a gravity-rotation number Π_Ω = Ω_c² r_c³ / (G M_c) only when the restoring field is Newtonian gravity of mass M_c. A water tank is held by hydrostatic pressure and the free surface. There the matching number is a Froude number V_c² / (g h). A tornado uses swirl ratio and buoyancy. Copying Π_Ω with G into those systems without that swap is a category error.

The test profile is V_θ(r) = V_base(r) + [V_flat − V_base(r)]S(Π). S = 0 gives the ordinary non-flat vortex, and S approaching 1 gives a flat outer belt. Faller (2001) produced a laboratory water vortex with ∂V/∂r = 0 outside a small core, held during decay by a turbulent Bödewadt layer. Flat tangential speed therefore occurs in an organised fluid flow. Rankine free vortices retain the 1/r outer profile.

10. Discussion

The factor 8π is fixed by the gravitational field-energy density and the Einstein coupling G_μν = (8πG/c⁴)T_μν. The Spaticle field supplies the physical stress-energy represented by spacetime curvature, and its finite density permits organised motion to entrain gravitating substrate mass. The BFUT carrier coefficient μₛ² = 3Gρₛ/c² and the carrier response aₛ = c²μₛ/3 give aₛ = c√(Gρₛ/3). The scale 10⁻¹⁰ m s⁻² follows from [m_e_vss/α_vss]⁶ together with G, c and ħ_vss. The locked-Υ SPARC comparison gives median outer residuals of 0.495 for Newton, 0.308 for the zero-parameter abundance-matching Navarro-Frenk-White (NFW) halo, and 0.096 for DME. The Bullet Cluster, DF2, DF4 and FCC 224 test the predicted dependence on organised motion.

DME and MOND give closely aligned disk accelerations because aₛ = 1.20840317 × 10⁻¹⁰ m s⁻² and a₀ = 1.20 × 10⁻¹⁰ m s⁻². The BFUT value follows from the particle-sector density. Systems that lose organised rotation test the entrained-mass interpretation through Θ.

11. Conclusions

The BFUT particle sector derives the Spaticle-field density as ρₛ = 7.3 × 10⁻²⁷ kg m⁻³ and adopts ρₛ = 7.3 × 10⁻²⁷ kg m⁻³ for the applications. The Spaticle field is the physical matter substrate whose stress-energy produces spacetime curvature through the Einstein coupling. Its finite density permits organised motion to entrain gravitating substrate mass. The carrier coefficient μₛ² = 3Gρₛ/c² and the response relation aₛ = c²μₛ/3 give aₛ = c√(Gρₛ/3) = 1.20840317 × 10⁻¹⁰ m s⁻². The DME relation g² = gᵦ² + Θaₛgᵦ describes the macroscopic entrainment, and its deep-organised limit is v⁴ = ΘGMᵦaₛ. SPARC, KiDS-1000, the Bullet Cluster and low-rotation galaxies provide complementary applications of the same particle-derived density and organised-mass mechanism.

Appendix A. SPARC galaxy table

Columns: Galaxy; number of radial points; characteristic observed speed (km s⁻¹); observed flat; predicted flat; shape match; median outer relative residual.

Appendix B. KiDS-1000 Fig. 3 point table

v_obs from Brouwer Eq. 23. V_Newt is G M / R. V_DME uses aₛ = 1.20840317 × 10⁻¹⁰ m s⁻². Primary fit: published 60×60 ESD covariance, χ²/N = 9.78 for the 60 measurements.

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