Newtonian gravity and dark-matter phenomenology from an emergent scalar field sourced by ordinary matter

1 Introduction

General relativity describes gravity to high precision in the Solar System, binary pulsars, and gravitational-wave propagation (Will, 2014). At galactic and cluster scales, however, the observed dynamics exceed the Newtonian weak-field prediction computed from the directly observed baryonic matter (McConnachie, 2012; Lelli et al., 2016; McGaugh et al., 2016). The standard resolution, cold dark matter (CDM) within the cosmological concordance model, fits the cosmic microwave background, baryon acoustic oscillations, and large-scale structure with high precision (Aghanim et al., 2020). It does not, by itself, explain why the empirical transition in galaxy dynamics occurs at an acceleration of order cH ₀, where c is the speed of light and H ₀ is the present Hubble constant, and direct-detection experiments have not yet produced a confirmed non-gravitational dark-matter signal (Deffayet et al., 2010; Bertone and Tait, 2018; Vokrouhlický et al., 2024).

Modified Newtonian dynamics (MOND) takes the cH ₀-scale coincidence as basic phenomenology and reproduces many rotation curves with a single acceleration constant a ₀≈1.2 × 10^-10 m s⁻², but it does not derive that constant, underpredicts part of the cluster mass discrepancy, and is constrained by wide-binary tests whose interpretation is debated (Milgrom, 1983; Bekenstein and Milgrom, 1984; Bekenstein, 2004; Famaey and McGaugh, 2012; Banik et al., 2023; Chae, 2023). Related medium-based and thermodynamic proposals exist but have not yet produced a single quantitative model that spans galaxy, cluster, local, and cosmological tests without additional assumptions (Matsakos and Diaferio, 2016; Verlinde, 2017; Pizzuti et al., 2025). An earlier phenomenological study by Stålhammar (2026) showed that an empirically calibrated, environment-adaptive gravitational kernel can reconcile galaxy rotation curves, wide binaries, dwarf spheroidals, and strong-lensing residuals with one fixed parameter set. In the present study, we develop and evaluate a candidate physical mechanism for this empirical pattern.

The mechanism starts from a simple hypothesis: the microscopic constituents of ordinary matter continuously emit waves of a single additional scalar field, with phases that are random and uncorrelated between sources. We refer to the summed field as an emergent scalar field: it is generated collectively by all matter, no individual emitter is observable, and every quantitative element of the framework derives from the statistics of the summed field. The emergent object is the field, not gravity itself, which distinguishes the framework from entropic emergent-gravity proposals (Verlinde, 2017). The waves are not gravitational waves; the tensor sector of the theory is exactly that of general relativity. When the emitted waves from all sources are summed at a point and the intensity of the total field is averaged over time, three kinds of terms appear, and each maps onto one aspect of gravitation. The product of each source’s wave with itself averages to a stationary, positive term, and these terms add like masses; the framework identifies ordinary Newtonian gravity with this stationary part. The waves arriving from all other matter combine into a fluctuating background whose strength sets an acceleration scale of order cH ₀, and the interference between the locally generated field and this background, rectified by the material medium, yields the additional attraction observed at low accelerations. Newtonian attraction, the galactic acceleration scale, and the dark-matter-like correction are thus not separate ingredients: they are three statistical sectors of one wave field.

Tested against public galaxy and cluster data with object-level statistics and a strict separation of calibration and validation, the framework reproduces galaxy rotation-curve accelerations comparably to fitted MOND without using any galaxy-fitted parameters and outperforms both MOND at its standard acceleration constant and general relativity with baryons only. At the outer radii of massive clusters, it performs well under leave-one-cluster-out validation, with a calibrated plasma coefficient of the order predicted by a simple ionization argument. In cluster interiors and in low-mass dwarf spheroidals, however, the current model underpredicts substantially.

2 Materials and methods

The emergent-scalar-field model is a single framework based on the same physical content, described at three different levels. The first level is the microscopic picture: matter consists of localized oscillating emitters; it supplies the physical interpretation, and no numerical result depends on its microscopic details (Section 2.1). The second level is the covariant theory: a standard scalar-tensor action in which a massless scalar field couples to baryonic matter, supplemented by a classical stochastic source representing the random-phase emitter background; it fixes which interactions exist and how they scale (Sections 2.2–2.4). The third level is the quasi-static model equation: an explicit prediction equation with screening, transport, and plasma factors, which is what is tested against data (Sections 2.5, 2.6). Section 2.7 describes the data and the statistical methods, Table 1 presents the symbols used throughout, and Table 2 records, for every model quantity, its origin, its status, the data used to set it, and whether those data re-enter any validation.

TABLE 1. Symbols and definitions.

TABLE 2. Model constants: origin, status, and data reuse.

2.1 Conceptual picture and scope

The model developed in this section is, formally, a scalar-tensor theory of gravity of standard form: its complete covariant specification is the action of Equation 4 with the field equations of Equation 6, and readers who prefer the standard construction may read Section 2.4 first. The material preceding Section 2.4 specifies the stochastic-source hypothesis that distinguishes the framework from a generic scalar-tensor theory. This hypothesis is not decorative: it fixes the rectification coefficient π/2 (Section 2.3.4, verified in Section 3.1), constructs the acceleration scale g_w from measured cosmological quantities (Section 2.3.3), and yields the independent ionization estimate against which the calibrated plasma coefficient is compared (Sections 2.6.4 and 3.3). Without it, these quantities would be free parameters.

2.1.1 Emitters

Matter is modeled, at the heuristic level, as a collection of localized sources, known as emitters, each radiating a scalar wave. In neutral matter, a bound atom acts as one emitter because its internal binding energies vastly exceed the energy scales of the slow macroscopic dynamics considered here; in fully ionized plasma, each free charged particle acts as an independent emitter. This counting rule is part of Assumption A1, and its observable consequence, an enhanced emitter density per unit mass in ionized gas, is what the calibrated plasma coefficient γ_p quantifies (Section 2.6.4). The hypothesized wave is not gravitational radiation: it belongs to an additional scalar degree of freedom of the gravitational sector, exactly as in scalar-tensor theories, where scalar (breathing-mode) radiation is a standard, experimentally constrained possibility, and the tensor sector remains exactly that of general relativity (Section 2.4) (Will, 2014). Individual emitters and their frequencies are not observable and are not assumed observable; the emitters may have an arbitrary spectral distribution, and the construction uses only the statistics of the summed field, which are Gaussian by the central limit theorem for any spectrum with finite variance and independent phases (Section 2.3). The physical assumption is that the mean-square emitted amplitude per unit baryonic mass is universal for matter in a given binding state (Assumption A1).

2.1.2 Spacetime in this framework

The formal development in this study takes place on a standard general-relativistic background: spacetime is described by a metric g _*μν* obeying Einstein’s equations, and the scalar field propagates on this metric. The framework’s underlying interpretation reaches further: because the same wave field carries the local gravitational sector and the cosmic background, spacetime itself may ultimately be constituted by this field, with the metric as a coarse-grained, collective description of it, in the tradition of induced-gravity, thermodynamic, and entropic proposals (Sakharov, 1991; Jacobson, 1995). There is no contradiction in a picture in which the degrees of freedom that constitute an effective metric also deform it; this situation is realized concretely in analog-gravity systems, where an effective acoustic metric emerges from a fluid and is simultaneously perturbed by the fluid’s own excitations (Barceló et al., 2011). Because the present study neither derives nor evaluates metric emergence, the calculations assume ordinary spacetime throughout, and the emergent interpretation is carried as the framework’s long-term program rather than as an input to any result.

2.2 Assumptions

The physical content of the proposed model, which goes beyond standard scalar-tensor gravity, is based on three assumptions.

2.2.1 Assumption A1 (baryonic source field)

A real massless scalar field φ is generated by baryonic matter. The mean-square amplitude (the intensity) of the emitted field per unit baryonic mass is a universal value for neutral matter and is larger in fully ionized plasma, in proportion to the number of independent emitters per unit mass. The frequency spectrum of the emitters is arbitrary; their phases are mutually independent and uniformly distributed.

2.2.2 Assumption A2 (rectified response; the central assumption)

The material medium at the observation point is immersed in a combined field with two components: the locally generated fluctuation δφloc, produced by the surrounding matter distribution, and the cosmic background fluctuation δφcos. The assumption states that the response of the medium to the interference between these components is determined by the energy of the combined field and is independent of its phase: the medium responds to the product of the two components' envelope amplitudes (their absolute magnitudes), not to the instantaneous signed product, and the resulting acceleration is directed along the local Newtonian field. Without this assumption, the interference term averages to zero, and the framework predicts no anomaly; A2 is therefore the central physical hypothesis of the framework, and its microscopic derivation is an open problem (Section 4.3). The numerical coefficient that follows from A2 and the consequences of alternative response laws are described in Section 2.6.1.

2.2.3 Assumption A3 (causal boundary condition)

The variance of the cosmic background component is accumulated over the causal past and is parameterized by a column of length L_H = c/H₀, the Hubble length, at the cosmic mean baryon density. The status of this simplification, along with what a full treatment would require, is discussed together with the derived constants in Section 2.6.2

Every derived or calibrated quantity in this study is conditional on these assumptions, and the text and Table 2 state the dependence explicitly wherever the quantities appear.

2.3 Statistical structure of the summed wave field

2.3.1 Notation and the double sum

At any point, the instantaneous field is a sum of enormously many independently phased contributions, of order 10⁵⁷ emitters in the Sun. Such random-phasor sums are studied exhaustively in statistical optics, and their envelope statistics are governed by the central limit theorem, regardless of the physical nature of the wave (Goodman, 2020). Only these standard mathematical results are imported below; every physical element of the framework enters through the explicit assumptions of Section 2.2.

Each emitter i at position x _*i* contributes a wave to the field at the observation point x (Equation 1):where A _*i* is the per-emitter amplitude (falling as one over distance from the emitter, the standard far-zone behavior of a radiated wave), ω _*i* is its angular frequency, k _*i* is its wave vector, and θ _*i* is its phase, drawn independently and uniformly from [0,2π). The total field is Φ _*tot* = ∑_*i* ϕ _*i* over all emitters in causal contact, and the time-averaged intensity is a double sum, as presented in Equation 2:

ϕ i x , t = A i ⁡ cos ω i t − k i · x − x i + θ i , (1)

Φ t o t 2 = ∑ i ϕ i 2 + ∑ i ≠ j ϕ i ϕ j . (2)

The i = j terms of this double sum are the self-terms: the square of each source’s own wave, non-negative, adding incoherently, and surviving phase averaging. The i≠j terms are the cross terms: pairwise products of waves from different sources, carrying random phase differences, with zero ensemble mean but finite variance.

For a narrow frequency band, the sum is conveniently written through the complex envelope ψ = ∑ i A i e i θ i e − i k i · x − x i, with Φ 2 = ψ 2 2. For a general spectrum, the same construction applies band by band, and the envelope in each band becomes, by the central limit theorem, a circular complex Gaussian variable with zero mean and variance σ 2 = 1 2 ∑ i A i 2 set by the local emitter population. The Gaussianity of the envelope is insensitive to the shape of the frequency spectrum; only variance depends on the source population, and Assumption A1 fixes this variance to be proportional to the baryonic mass column contributing at the observation point. This is the precise sense in which the framework is robust to per-species variation in emitter properties.

2.3.2 Self-terms and the Newtonian sector

Because the self-terms add positively and each emitter’s intensity falls as the inverse square of distance, the stationary part of the summed intensity from a mass distribution has the same spatial kernel as the Newtonian potential of that distribution. The framework identifies the ordinary Newtonian sector with these terms. Three properties delimit this identification. First, the value of Newton’s constant G is taken from measurement, and in the covariant implementation, the Einstein–Hilbert term is included as the standard low-energy gravitational sector. Second, mapping from summed wave intensity to metric potential is not constructed microscopically in this study. Third, the framework’s new, testable content therefore lies entirely in the cross-term statistics; all quantitative results use the standard Newtonian g _*N* computed from the observed baryons with the measured G.

2.3.3 Background variance and the acceleration scale

The cross-term variance contributed by the cosmic emitter population accumulates over the causal column of Assumption A3. The associated characteristic source column is Σ _*cos* = Ω _*b* ρ _*crit* L _*H*, the cosmic mean baryon density times the Hubble length, and the acceleration scale obtained by converting this column with the spherical geometric factor 3/(8π) (the mean inverse-square projection of an isotropic column onto a radial direction, adopted as part of the model definition) is presented in Equation 3:for H ₀ = 67.7 km s⁻¹ Mpc⁻¹ (Aghanim et al., 2020). A spatially uniform background acceleration cannot by itself produce flat rotation curves; the observable consequence comes from the interference between the local field and the background.

g w = 3 8 π c H 0 ≈ 7.85 × 10 − 11 m s − 2 , (3)

2.3.4 The local-cosmic cross-term

Writing the fluctuating part of the field at the observation point as δϕ = δϕ _*loc*+δϕ _*cos*, with the two parts generated by emitters inside and outside the system under study, the intensity contains δ ϕ l o c 2 (absorbed in the local sector), δ ϕ cos 2 (the background), and the cross-term 2δϕ _*loc* δϕ _*cos*. The signed ensemble mean of the cross-term is zero; its root-mean-square amplitude is the geometric mean of the two intensities. Under Assumption A2, the medium responds to the envelope-magnitude product, whose ensemble mean is (π/2)σ _*loc* σ _*cos* for independent circular-Gaussian envelopes (Section 2.6.1). Converting intensities to accelerations through the source-column correspondence of Assumption A1 (I _*loc* ∝ g _*N*; I _*cos* ∝ g _*w*; this constitutive correspondence is part of the model specification) yields a cross-term acceleration proportional to g N g w, aligned with the local field.

No sharp boundary radius between local and cosmic emitters enters the numerical implementation: the local intensity is set by the enclosed baryonic column of the system under study (the enclosed baryonic mass per unit area), the cosmic intensity is by the cosmic source column Σ _*cos*, and the predictions are independent of where a nominal boundary would be drawn, provided that it lies far outside the system and far inside the Hubble length. The statistical independence of the two components follows from the independence of their source phases.

2.4 Covariant implementation

2.4.1 Action

The covariant implementation places the scalar in the standard conformally coupled scalar-tensor form; couplings of this matter-metric type are a special case of the effective-metric couplings studied more generally by Damour and Esposito-Farese (1992) and Bekenstein (1993). In the Einstein frame, Equation 4 can be obtained:where M _*Pl* = (8πG)^-1/2 is the reduced Planck mass (natural units ℏ = c = 1 are used in this subsection; all empirical formulae below are in SI units), R isthe Ricci scalar, and ϕ is a massless scalar with no potential. Baryonic matter, denoted collectively by ψ _*b* and treated throughout as a classical fluid characterized by its stress-energy tensor, couples to the conformal metric g ∼ μ ν = A 2 ϕ g μ ν with coupling function (Equation 5):with dimensionless constants β (linear coupling) and η (quadratic coupling). Any non-baryonic energy content, including the cosmological constant, is collected in S _*nb* and couples only to g _*μν*. Restricting the conformal coupling to the baryonic sector encodes Assumption A1; an alternative coupling to all matter would multiply the source column of Section 2.6.2 by Ω _*m*/Ω _*b*≈6.5 and is disfavored by the data at that level. No ϕR or higher curvature-scalar operators are included, so the tensor sector is exactly that of general relativity and gravitational waves propagate at the speed of light, which is consistent with the binary-neutron-star constraint (Abbott et al., 2017a; Abbott et al., 2017b).

S = ∫ d 4 x − g M P l 2 2 R − 1 2 g μ ν ∂ μ ϕ ∂ ν ϕ + S b ψ b , A 2 ϕ g μ ν + S n b ψ n b , g μ ν , (4)

ln ⁡ A ϕ = β ϕ M P l + η 2 ϕ 2 M P l 2 , (5)

2.4.2 Field equations and the Newtonian limit

Varying Equation 4 yields the scalar equation of motion (Equation 6):where T ∼ b is the trace of the baryonic stress-energy in the matter frame and ξ is a classical stochastic source with 〈ξ〉 = 0 representing the random-phase emitter background of Section 2.3; its correlator 〈ξ(x)ξ(x ^′)〉 is fixed by the emitter statistics, and the treatment is classical throughout, in structural analogy to the noise kernel of stochastic semiclassical gravity (Hu and Verdaguer, 2008). Test bodies made of baryons follow geodesics of g ∼ μ ν, which, in the weak-field limit, adds to the metric acceleration a fifth-force term -c ²∇lnA(ϕ).

□ ϕ = − α ϕ M P l T ∼ b + ξ , α ϕ ≡ M P l d ⁡ ln ⁡ A d ϕ = β + η ϕ M P l , (6)

For the static weak-field mean field, linearizing Equation 6 around ϕ = 0 with T ∼ b ≈ − ρ b c 2 yields ∇ 2 ϕ ¯ = β / M P l ρ b c 2 (SI restored), whose solution tracks the Newtonian potential. The resulting total attraction between baryonic bodies is the standard scalar-tensor result, expressed through the effective gravitational constant G _*eff* and the parametrized post-Newtonian (PPN) parameter γ _*PPN* (Equation 7):

G e f f = G 1 + 2 β 2 , γ P P N − 1 = − 2 β 2 1 + β 2 . (7)

The Cassini bound |γ _*PPN*-1| ≤ 2.3 × 10^-5 then requires β ² ≲ 1.2 × 10^-5, so the linear coupling produces at most a small correction to G that is absorbed into its measured value (Bertotti et al., 2003). The quadratic coupling η does not contribute to the mean field at this order; its role is to couple matter to the intensity of the field fluctuations, δϕ ², which contains the local-cosmic cross-term. This is the covariant location of the anomaly: expanding A(ϕ) to second order, the matter-frame energy of a baryonic element acquires a term proportional to η δ ϕ l o c δ ϕ cos / M P l 2, whose ensemble-averaged, rectified value (Assumption A2) is the model parameterized in Section 2.5. The chain from Equation 4 to the model equation is thus as follows: the action fixes which couplings exist (linear: a small correction to Newtonian gravity; quadratic: sensitivity to field intensity), the emitter statistics fix the magnitude of the intensity fluctuations, and Assumption A2 fixes how the medium responds to them. The dependence on the local matter distribution enters at the model level (Section 2.5).

2.4.3 Conservation and noise budget

The stochastic source injects energy-momentum at a rate whose ensemble mean vanishes with 〈ξ〉 = 0 and whose variance represents the noise of the random-phase background. An order-of-magnitude estimate with the background energy density bounded by the source column of Section 2.6.2 yields a stochastic heating rate for intergalactic baryons more than 20 orders of magnitude below the Lyman-alpha-forest bound of approximately 10^–20 K Gyr⁻¹; the noise budget is unobservably small, and the framework’s observable content resides entirely in the rectified mean of Section 2.5. Total stress-energy conservation holds in the ensemble-averaged sense standard for scalar-tensor theories with an external stochastic source.

2.4.4 Lensing and the equal-potential condition

In the conformal-Newtonian gauge with potentials Φ _*g* (time-time) and Ψ _*g* (space-space), the anisotropic stress determines the gravitational slip Φ _*g*–Ψ _*g*. Pressureless baryons carry none; the ensemble-averaged canonical scalar stress-energy carries none for an isotropic random-phase background because the signed cross-gradients average to zero; and the rectified cross-term acts as a scalar effective source aligned with the local field, again without anisotropic stress at leading order for the spherical systems analyzed here. Under these conditions Φ _*g* = Ψ _*g*, lensing and dynamics respond to the same effective mass profile M _*eff*(r). This equality is a target-level result conditional on the listed conditions, and it is falsifiable by comparing dynamical and lensing masses at matched radii (Section 3.8).

2.5 The prediction equation

2.5.1 Master equation

All empirical tests in this study use the following analytical prediction (Equation 8), with fixed parameters, for the total radial gravitational acceleration in a quasi-static, approximately spherical system:where g _*N*(r) is the Newtonian acceleration of the observed baryons, χ _*ϕ* = π/2 is the rectification coefficient (Section 2.6.1), g _*w* is the cosmic acceleration scale of Equation 3, and S(r) is the source-column screening factor (Equation 9):which suppresses the cross-term where the system’s own column density Σ _*eff* exceeds the cosmic source column Σ _*cos* (Section 2.6.2); M _*eff*(r) is the total effective dynamical mass enclosed within r, determined self-consistently (Section 2.5.3), and T(r) is the transport (saturation) factor (Equation 10):which describes the accumulation of the rectified response over a density-dependent length anchored at the Solar-neighborhood value l t r , ⊙ (Section 2.6.3). Because ρ _*loc*(r) = M _*eff*(r)/((4/3)πr ³), the exponent satisfies r / l t r r = 3 Σ e f f r / Σ t r with Σ t r ≡ l t r , ⊙ ρ ⊙, so the transport factor is a function of the effective column alone; the appropriate density variable for systems whose mass is not smoothly distributed is discussed in Section 3.6. E _*ion*(r) is the ionized-baryon mass fraction at radius r, and γ _*p* is the plasma coefficient (Section 2.6.4). The ionized-plasma term follows from the emitter-counting rule of Assumption A1: ionization multiplies the number of independent emitters per unit mass, so the enhancement applies wherever baryons are ionized, weighted by the local ionized mass fraction. In a fully ionized intracluster medium E _*ion* ≈ M _*gas*/M _*bar*, while in galaxy disks, dwarf spheroidals, the Solar System, and wide binaries, the ionized mass fraction is negligible for the present purpose, and E _*ion* = 0 is used.

g r = g N r + S r + γ p E i o n r χ ϕ T r g N r g w , (8)

S r = Σ cos Σ e f f r + Σ cos , Σ e f f r = M e f f r 4 π r 2 , (9)

T r = 1 − exp − r / l t r r , l t r r = l t r , ⊙ ρ ⊙ ρ l o c r , ρ l o c r = M e f f r 4 / 3 π r 3 , (10)

Any viable modification of gravity must suppress its new effects in dense regions such as the Solar System; in the present framework, this suppression is carried by the screening and transport factors defined above (Khoury and Weltman, 2004; Burrage and Sakstein, 2018). Two statements about the status of Equation 8 apply throughout. First, the functional forms of S and T are modeling choices constrained by their limits (S→1 for transparent, low-column environments and S→0 for high columns, motivated by a Wentzel–Kramers–Brillouin attenuation argument for wave transport through the perturbed metric; T→1 at large radius or high density); the sensitivity of the results to alternative functional forms is quantified in Section 3.2.5. Second, Equation 8 stands to the covariant theory as its quasi-static model equation, in the sense defined in Section 1: the action fixes the existence, sign, and g N g w scaling of the second term, while the factors S, T, and E _*ion* are specified at the model level.

2.5.2 Equivalent field formulation

For non-spherical applications, Equation 8 is equivalent to a modified Poisson equation, in which the divergence of the cross-term acceleration field, directed along the local Newtonian field with magnitude χ ϕ T S + γ p E i o n g w ∇ Ψ N, acts as an additional effective source density alongside ρ _*b*; the construction parallels quasi-linear formulations of MOND, with a square-root kernel and a density-dependent multiplier in place of an interpolation function (Bekenstein and Milgrom, 1984). In spherical symmetry, the effective enclosed mass is M e f f r = M b r + r 2 / G χ ϕ T r S r + γ p E i o n r g N r g w, which is the dark-matter-like mass profile generated by the model and the quantity compared with lensing masses in Section 3.8.

2.5.3 Self-consistency

Equation 8 is implicit: the screening variables Σ _*eff* and ρ _*loc* depend on M _*eff*, which includes the cross-term contribution itself. Each radial point is therefore solved by damped fixed-point iteration started from the baryonic mass alone, M ^(*k*+1) = (1-λ)M ^(*k*)+λr ² g(M ^(*k*))/G, with damping λ = 0.5, relative tolerance 10^–5, and a 500-iteration cap. Only baryonic observables enter; the observed dynamical mass is never used in the primary predictions. Convergence, uniqueness across widely separated starting points, and iteration counts are reported in Section 3.2.6

2.6 Model constants

Table 2 lists the four model constants with, for each, its value, origin, epistemic status, the data used to set it, its uncertainty, and whether those data re-enter any validation. Three are fixed before any fit to the test samples; one (γ _*p*) is calibrated on X-COP only.

2.6.1 Rectification coefficient χ _*ϕ* = π/2

For two independent circular-Gaussian envelopes with per-component standard deviations σ _*loc* and σ _*cos*, the envelope moduli are Rayleigh-distributed, and the mean of their product is (π/2)σ _*loc* σ _*cos*. Conditional on Assumption A2, the coefficient of the rectified cross-term is therefore exactly π/2; this is a theorem of the statistics, and the physical content resides in A2. A direct Monte Carlo of the random-phasor ensemble (3 × 10⁵ realizations; 200 local and 1,000 background emitters of unit amplitude) yields 1.5726, within 0.12% of π/2. Alternative response laws change the coefficient in a testable way: a signed-product response averages to zero (no anomaly), and a response to the product of rectified field components yields 2/π≈0.64. Substituting alternative coefficients, with everything else fixed, degrades the SPARC fit of Section 3.2: a coefficient of 1 increases the root-mean-square error (RMSE) from 0.228 to 0.245 dex, and 2/π increases it to 0.284 dex. The data therefore prefer the modulus-product response among the simple alternatives tested.

2.6.2 Cosmic source column *Σ* _***cos*** and acceleration scale g _*w*

Assumptions A1 and A3 fix the cosmic source column to Equation 11:using Ω _*b* = 0.0486, ρ _*crit* = 8.61 × 10^-27 kg m⁻³, and L _*H* = c/H ₀ = 1.37 × 10²⁶ m (Aghanim et al., 2020). There is no numerical freedom in Equation 11 once A1 and A3 are adopted; the quantity inherits the baryonic-coupling commitment (a total-matter coupling would yield 0.37 kg m⁻²) and the Hubble-length simplification of the causal column. The acceleration scale g _*w* in Equation 3 is the geometric conversion of this column and shares its status.

Σ cos = Ω b ρ c r i t L H = 0.057 kg m − 2 , (11)

The use of L _*H* in place of the full past-light-cone integral in an expanding background is an imposed simplification: the two differ by factors of order unity that depend on the source-population history, and the required integral of the source correlator over the Friedmann–Lemaître–Robertson–Walker light cone remains to be performed. The empirical proximity of the galactic transition acceleration to cH ₀, long noted in the MOND literature, motivates the choice (Milgrom, 1983; Famaey and McGaugh, 2012).

2.6.3 Transport anchor l t r , ⊙

The transport length in Equation 10 depends only on the product l t r , ⊙ ρ ⊙, an effective column Σ t r = l t r , ⊙ ρ ⊙. We adopt l t r , ⊙ = 0.37 kpc with reference midplane density ρ _⊙ = 0.10M _⊙ pc⁻³, that is, Σ _*tr* = 37M _⊙ pc⁻², of the order of the measured local Galactic disk surface density of approximately 47 M _⊙ pc⁻² (McMillan, 2016). The anchor is an externally measured, order-of-magnitude normalization, rather than a precision input: varying l t r , ⊙ by ±30% changes the SPARC fit by at most 0.007 dex (Section 3.2.5). No SPARC, X-COP, or Li information enters this anchor.

2.6.4 Plasma coefficient γ _*p*

Under the emitter-counting rule of A1, fully ionizing a primordial-composition gas (hydrogen mass fraction = 0.752; helium = 0.248) increases the number of independent emitters per unit mass by the factor 2.07, from 0.815/m _*H* to 1.690/m _*H* with m _*H* the hydrogen mass, predicting an enhancement coefficient γ p i o n ≈ 1.08 after the helium mass correction. This counting argument ignores the temperature dependence of the coupling, the relative weighting of electron and ion contributions, and any partial phase coherence in the plasma, so it is an order-of-magnitude estimate. The working value of γ _*p* is calibrated empirically by profile likelihood on the X-COP sample alone (Section 3.3) and then frozen for all other analyses, including the Li validation sample. The calibrated value is γ _*p* = 0.69 with cluster-bootstrap 95% interval 0.60–0.79, which is 36% below the counting estimate: consistent in order of magnitude, with the residual difference attributed to the physics the counting argument omits. γ _*p* is an empirically calibrated coefficient with a physically motivated ionization interpretation.

2.7 Data and statistical methods

2.7.1 Data

Galaxies: the SPARC compilation of 175 disk galaxies with homogeneous near-infrared photometry and rotation curves (Lelli et al., 2016). We use the published rotation-curve decompositions at face value, which corresponds to a stellar mass-to-light ratio of 1.0 (in solar units at 3.6 μm) for disk and bulge; the standard population-synthesis values of 0.5 (disk) and 0.7 (bulge) are analyzed as a robustness variant, and the mass-to-light treatment is applied identically to all models compared (Lelli et al., 2016; McGaugh et al., 2016). The gas contribution is used as published, with its sign convention. The primary selection keeps radial points with baryonic acceleration g _*N*<10^-9 m s⁻², the weak-field regime targeted by the framework; sensitivity to this threshold, up to and including no threshold, is reported in Section 3.2.5. The selection retains 3,187 points in all 175 galaxies. Reported velocity uncertainties are propagated into log-acceleration errors.

Clusters (calibration): the X-COP sample of 12 massive clusters with hydrostatic masses at R ₅₀₀ and R ₂₀₀ (the radii enclosing mean densities of 500 and 200 times the critical density) from combined X-ray and Sunyaev–Zel’dovich analysis, along with measured gas fractions at both radii, yielding 24 cluster-radius points (Eckert et al., 2019; Ettori et al., 2019). Baryonic mass is modeled as M _*bar* = 1.12M _*gas*, where the factor 1.12 represents the stellar contribution at these radii; hydrostatic masses serve as the observed dynamical masses.

Clusters (validation): the 22-cluster eROSITA sample of Li et al. with hydrostatic and gas mass profiles of more than 50 kpc to approximately 2 Mpc, 3,916 radial points in total, disjoint from X-COP. A central-galaxy stellar component is added as a Hernquist profile with total mass 5 × 10¹¹ M _⊙ and effective radius 25 kpc; the results are also reported for central-galaxy masses of 0 and 10¹² M _⊙, and the conclusions do not depend on the choice.

Dwarf spheroidals: five local group satellites (Crater II, Draco, Fornax, Sculptor, and Sextans) with literature stellar masses, half-light radii, and line-of-sight velocity dispersions (McConnachie, 2012). Solar System and wide-binary calculations use standard orbital parameters and Gaia DR3-era bounds (Banik et al., 2023).

2.7.2 Statistics

The primary goodness-of-fit statistic is the RMSE of log ₁₀ g _*pred*-log ₁₀ g _*obs*, reported in dex. Because radial points within one galaxy or cluster share distance, inclination, mass-model, and calibration systematics, they are not independent, and all uncertainty statements treat the object, not the radial point, as the independent unit: confidence intervals for RMSEs and for RMSE differences between models are obtained by block bootstrap over objects (2,000 resamples of the 175 galaxies, or of the 12 clusters, with replacement; models with fitted constants are re-fitted within every resample), and out-of-sample performance is quantified by leave-one-object-out cross-validation, in which any fitted constant is re-estimated with the object excluded and the object is then predicted. Per-object RMSE distributions are reported alongside pooled values. As secondary, bin-level statistics, we report the Akaike and Bayesian information criteria computed from a Gaussian likelihood in log-acceleration with the propagated observational error plus a fitted intrinsic-scatter term per model; these carry the explicit caveat that bin-level likelihoods overstate the effective sample size. Residual diagnostics use Spearman rank correlations, with two-sided P-values, of residuals against acceleration, radius, and galaxy properties. The calibration-validation design is as follows: γ _*p* calibrated on X-COP only; SPARC and all other samples analyzed with no fitted parameters; Li analyzed only with all constants frozen. All analyses were run using R version 4.3.3 (R Core Team, Vienna, Austria); the complete pipeline script that reproduces every statistical result, table, and data figure (Figures 1–5) is provided as Supplementary Material; its input data are assembled from the published sources listed in Section 2.7.1.

3 Results

3.1 Verification of the rectification coefficient

We simulated 3 × 10⁵ realizations of the random-phasor ensemble (200 local and 1,000 background emitters with independent, uniformly distributed phases) and computed the product of the local and background envelope moduli. The simulated mean is 1.5726, in agreement with the Rayleigh value π/2 ≈ 1.5708 to 0.12% (Figure 1), so the coefficient in Equation 8 is exactly the value required by circular-Gaussian statistics, conditional on Assumption A2, and is not adjustable. The alternative response laws remain numerically well separated (zero and 2/π; Section 2.6.1). Figure 1 also shows the model functions S and T with the alternative forms (panels a and b; tested in Section 3.2.5), the profile likelihood from which γ_p is calibrated (panel c; Section 3.3), and the Monte Carlo distribution itself (panel d).

3.2 Galaxy rotation curves (SPARC)

3.2.1 Primary result

With all model constants fixed in advance (γ _*p* does not enter because E _*ion* = 0 in disks), the self-consistent prediction of Equation 8 applied to the 3,187 weak-field SPARC points yields an RMSE of 0.228 dex, with a galaxy-block bootstrap 95% confidence interval of 0.204 – 0.253 dex. The fixed-point iteration converged for 100% of points (median 13 iterations). A diagnostic variant that substitutes the observed dynamical mass into the screening variables, instead of solving for M _*eff*, yields 0.217 dex, confirming that the self-consistent solution from baryons does not degrade the prediction appreciably. In Figure 2a, the predictions track the equality line over the full weak-field acceleration range, and Figure 2b shows the residual histogram whose width defines the 0.228 dex result. The median M _*eff*/M _*obs* is 1.17.

3.2.2 Model comparison

Table 3 reports the comparison, on identical data with identical error treatment, against general relativity with baryons only, MOND with the standard radial-acceleration-relation form g = g N / 1 − exp − g N / a 0 at the canonical a ₀ = 1.2 × 10^-10 m s⁻², and the same MOND form with a ₀ re-fitted to the selected points (McGaugh et al., 2016). The results are as follows: general relativity 0.398 dex (bootstrap interval 0.367 – 0.432); MOND at canonical a ₀ 0.270 (0.244 – 0.298); MOND with fitted a ₀ = 4.0 × 10^-11 m s⁻² 0.213 (0.192 – 0.235); the emergent-scalar-field framework 0.228 (0.204 – 0.253). Bootstrap distributions of the pairwise RMSE differences place the framework 0.043 dex below canonical-a ₀ MOND (95% interval of the difference 0.030 – 0.054) and 0.170 dex below general relativity (0.136 – 0.204), along with 0.015 dex above fitted-a ₀ MOND (0.009 – 0.022). Leave-one-galaxy-out cross-validation yields median per-galaxy RMSEs of 0.191 dex for the framework (which has nothing to re-fit, so its cross-validated and in-sample predictions coincide) and 0.177 dex for MOND with re-fitted a ₀. The bin-level information criteria in Table 3 yield the same ordering.

TABLE 3. Model comparison on SPARC weak-field points (identical data, mass model, and error treatment; n = 3,187 points in 175 galaxies; bootstrap CIs from 2,000 galaxy-block resamples).

Bin-level AIC/BIC use a Gaussian likelihood with propagated observational errors plus a fitted intrinsic scatter (one additional parameter per model, included in the counts); because radial points within a galaxy are correlated, these criteria overstate effective sample size and the bootstrap/cross-validation columns are primary. Published MOND fits with per-galaxy mass-to-light nuisance parameters (roughly 175 additional parameters, different selection) reach 0.11 – 0.13 dex and are quoted for context only.

The framework, with no quantity fitted to SPARC, therefore performs between the two MOND variants: better than MOND at its standard acceleration constant, slightly worse than MOND when a ₀ is optimized for this selection. For completeness, published MOND fits with per-galaxy mass-to-light ratios as nuisance parameters reach 0.11 – 0.13 dex on the full SPARC sample; this comparison involves approximately 175 additional fitted parameters and a different selection and is quoted for context only (McGaugh et al., 2016; Chae et al., 2020). The fitted a ₀ strongly degenerate with the mass-to-light choice (it moves from 4.0 × 10^-11 to 9.5 × 10^-11 m s⁻² between the face-value and population-synthesis variants), whereas the framework’s prediction moves by only 0.005 dex between the same variants (Section 3.2.5).

3.2.3 Object-level performance

The per-galaxy RMSE distribution for the framework has a median of 0.191 dex and an interquartile range of 0.117 – 0.257 dex, with a 95th percentile of 0.540 dex. In Figure 2c, the two per-galaxy distributions overlap substantially, with the framework’s distribution shifted slightly toward larger RMSE. The corresponding fitted-MOND distribution is similar (median 0.177 dex). The pooled fit is therefore not driven by a small number of well-fitted systems, and conversely, a tail of poorly fitted galaxies exists under both frameworks. The per-galaxy median signed residual is +0.07 dex, representing a mild median over-prediction.

3.2.4 Residual structure

Residuals correlate with baryonic acceleration (Spearman ρ = +0.39, P = 2.6 × 10⁻¹¹⁴ at bin level) and, at the galaxy level, with gas fraction (ρ = − 0.40, P = 4.5 × 10⁻⁸) and central disk surface brightness (ρ = +0.20, P = 0.008), but not with distance (ρ = +0.08, P = 0.31) or outer rotation velocity (ρ = −0.03, P = 0.72). Figure 2d shows the binned median residual against g _*N* for the framework and fitted MOND: both frameworks share a downturn at the lowest accelerations, and the framework’s residuals increase at the high-g _*N* end of the weak-field window. These correlations indicate that the universal model interpolates imperfectly between gas-rich, low-surface-brightness systems and star-dominated, high-surface-brightness systems. Any refinement introduced to absorb them (for example, separate emitter amplitudes per baryonic phase) would add parameters and must be validated out of sample; no such refinement is included in this study.

3.2.5 Robustness

The SPARC result is insensitive to the discretionary choices in the model and in the sample. The galaxy-block bootstrap distributions of the SPARC RMSE for the framework and both MOND variants, which underlie the confidence intervals of Table 3, are shown in Figure 3a. Replacing the exponential transport form by a rational saturation x/(1 + x) leaves the RMSE unchanged (0.2276 versus 0.2275 dex), tanh saturation yields 0.2303 dex, and only a sharper-onset Gaussian form degrades the fit appreciably (0.286 dex); replacing the rational screening form by an exponential one even improves it slightly (0.215 dex). In Figure 3b, the four transport forms appear as bars against the dashed canonical result. The result is therefore not an artifact of one specific saturating shape, and the canonical forms were not tuned to be optimal, while the spread across forms, up to 0.06 dex for the least similar shape, quantifies the freedom carried by the model functions. The stellar mass-to-light treatment matters equally little for the framework: with the population-synthesis values (0.5 for the disk and 0.7 for the bulge), the selection increases to 3,279 points and the framework’s RMSE moves only from 0.228 to 0.232 dex, whereas general relativity with baryons degrades from 0.398 to 0.538 dex and fitted MOND improves from 0.213 to 0.195 dex, with a₀ shifting by a factor of 2.4. The weak-field threshold is likewise uncritical: widening the cut from 10^–9 m s⁻² to no cut at all changes the framework’s RMSE from 0.2275 to 0.2292 dex and fitted MOND from 0.2128 to 0.2154 dex. Finally, varying the transport anchor l t r , ⊙ by ±30% changes the RMSE by at most +0.007 dex.

3.2.6 Fixed-point diagnostics and the deepest weak-field points

Starting the iteration from 0.25 times, four times the baryonic mass, or from the observed dynamical mass changes, no converged prediction by more than 1.5 × 10⁻⁵ dex, direct numerical evidence that the fixed point is unique and stable over the relevant range; no point failed to converge (median 13; maximum 39 iterations). Figure 3c shows the full distribution of iteration counts, concentrated near the median with a short tail to the maximum. The framework, however, fails systematically at the deepest weak-field points: for the 103 points with observed acceleration below 10^–11 m s⁻² (3.2% of the selection, dominated by a few extreme dwarfs and ultra-diffuse disks), the median residual increases to +0.55 dex, that is, the model over-predicts. The transport factor saturates toward its limits in this regime, and the model loses discriminating structure; this regime lies outside the present model’s domain of validity.

3.3 Cluster calibration (X-COP)

For the X-COP points, the ionized fraction is E _*ion* = M _*gas*/M _*bar* = 1/1.12 = 0.89. The single free coefficient of the framework, γ _*p*, is calibrated here and nowhere else. The profile likelihood has its minimum at γ _*p* = 0.69, with a cluster-bootstrap 95% interval of 0.60 – 0.79; the leave-one-cluster-out re-calibrations span only 0.65 – 0.72, so no single cluster drives the value. Figure 3d shows the bootstrap distribution of γ_p about the calibrated value, with the ionization estimate 1.08 marked in red. At the calibrated value, the in-sample RMSE over the 24 points is 0.068 dex (bootstrap interval 0.048 – 0.088), the median residual is −0.012 dex, and 21 of 24 points lie within 0.1 dex. The primary statistic is the out-of-sample statistic: leave-one-cluster-out validation, in which each cluster is predicted with γ _*p* re-calibrated on the other 11, yields a pooled RMSE of 0.072 dex. Figure 4a plots predicted against observed accelerations at both radii; Figure 4b shows the per-cluster residuals from the leave-one-cluster-out procedure. Radial balance is good (median residual −0.034 dex at R ₅₀₀, +0.048 at R ₂₀₀), and the median converged-to-observed mass ratio is 0.97. Excluding A2319, whose exceptional non-thermal pressure support is documented, moves the calibration to 0.72 and the RMSE to 0.060 dex (Eckert et al., 2019).

On identical data (Table 4), general relativity with baryons yields 0.80 dex, and MOND at canonical a ₀ yields 0.138 dex with systematic under-prediction, the long-known residual cluster problem of MOND. Giving MOND the same freedom exercised here, one coefficient fitted to this sample, yields a ₀ = 2.2 × 10^-10 m s⁻² and 0.070 dex (leave-one-cluster-out 0.073), statistically indistinguishable from the framework’s result at equal fitted-parameter count. The difference between the frameworks on this sample is therefore not fit quality but internal consistency: the MOND coefficient required by the clusters is 5.5 times that fitted to the galaxies in Section 3.2, whereas the framework fits both regimes with the same three fixed constants plus a plasma coefficient that applies only where baryons are ionized and lies within a factor 1.6 of an independent ionization estimate. This consistency argument, not any single RMSE, is the substantive cluster-scale result, and it is falsifiable: γ _*p* re-measured on independent cluster samples must remain compatible with 0.69 (0.60 – 0.79).

TABLE 4. Cluster results.

3.4 Cluster validation (Li sample)

The 22-cluster eROSITA sample is analyzed with every quantity frozen (γ _*p* = 0.69 from X-COP; nothing tuned to this sample), so it constitutes an external validation, and it defines the framework’s principal empirical limitation. Over all 3,916 radial points, the RMSE is 0.435 dex with median residual −0.35 dex (underprediction; Table 4). The residuals are strongly radially structured (Table 5): the median residual is −0.61 dex at 20–100 kpc, −0.54 at 100–200 kpc, −0.40 at 200–400 kpc, −0.22 at 400–700 kpc, −0.05 at 700–1,100 kpc, and +0.14 dex at 1,100–2,100 kpc. Figure 4c shows the frozen predictions against the observations for all 3,916 points, and Figure 4d presents the median residual as a function of radius for all three frameworks. The framework thus works at the radii probed by X-COP (roughly 1 to 2.3 Mpc) and underpredicts increasingly toward cluster centers, with a mild over-prediction beyond 1.1 Mpc. For calibration of what the discrepancy means, the same table reports the identically selected baselines: general relativity with baryons yields 0.995 dex overall (−1.09 dex median at 20 – 100 kpc), and canonical MOND yields 0.496 dex (−0.62 in the same bin). The framework therefore roughly halves the central log-discrepancy of the baryons-only prediction but shares the qualitative central failure of MOND, and its per-cluster RMSE distribution (median 0.42, interquartile range 0.35 – 0.48) shows that the discrepancy is systemic rather than driven by outlier clusters. Adding or removing the central-galaxy component moves the overall RMSE only between 0.413 (10¹² M _⊙) and 0.463 (none), so the discrepancy is not an artifact of the stellar model. Unmodeled intracluster stellar mass and a genuinely radius-dependent plasma response are candidate explanations, but both would add freedom and must be specified in advance and validated on further independent samples before being adopted.

TABLE 5. Li-sample median residuals, log ₁₀(g _*pred*/g _*obs*), by radial range (central-galaxy mass 5 × 10¹¹ M _⊙; γ _*p* = 0.69 frozen from X-COP).

3.5 Solar system

The framework must, and does, suppress the cross-term locally, and the suppression follows from the calibrated screening, so the correct statement is conditional: given the calibrated model, the predicted anomalies are unobservably small. The linear-coupling sector is constrained as in any scalar-tensor theory: Equation 7 with the Cassini bound yields β ² ≲ 1.2 × 10^-5, and the MICROSCOPE equivalence-principle bound tightens this by several orders of magnitude for the composition-dependent part (an order-of-magnitude estimate yields β ≲ 6 × 10^-7, entirely compatible with the framework, whose observable content resides in the quadratic sector) (Bertotti et al., 2003; Touboul et al., 2022). For the cross-term, the Sun’s own column dominates the screening: the predicted fractional accelerations Δg/g _*N* are 8.5 × 10⁻¹⁴ at Mercury, 1.5 × 10⁻¹² at Earth, 1.3 × 10⁻⁹ at Saturn, and 4.0 × 10⁻⁸ at Neptune, between approximately three (Neptune) and more than eight (Mercury) orders of magnitude below the Cassini-scale sensitivity 2.3 × 10⁻⁵. The steep radial increase (approximately as r ³ from the combined column and transport factors) is itself a signature: ephemeris constraints at Neptune-to-Kuiper distances at the 10⁻⁷ level would begin to probe the model directly (Vokrouhlický et al., 2024). Figure 5a shows this predicted anomaly increasing steeply with orbital radius toward, but remaining well below, the Cassini-scale bound.

3.6 Wide binaries

Wide stellar binaries in the solar neighborhood probe weak internal accelerations inside the Galactic disk environment, where the pair’s own column sets the screening and the ambient disk medium sets the transport factor. For a 2 M _⊙ pair, the predicted fractional velocity excess Δv/v _*K* over the Keplerian expectation is 3.4 × 10⁻⁹ at a separation of 1 kAU, 1.9 × 10⁻⁶ at 5 kAU, 2.4 × 10⁻⁵ at 10 kAU, 1.9 × 10⁻³ at 50 kAU, 8.1 × 10⁻³ at 100 kAU, and 3.3 × 10⁻² at 200 kAU. These values are computed with the transport factor referred to the ambient disk medium, T = 1-exp(-ρ_⊙s/Σ_tr) with Σ tr ≡ l tr , ⊙ ρ ⊙ and s the pair separation, rather than to the enclosed-mass variables in Equation 10, which, for two point masses separated by vacuum, would smear the pair into a fictitious local density M/((4/3)πs³) corresponding to no physical matter column. The choice is stated explicitly because the two readings are not equivalent for point-pair systems: under the enclosed-mass reading, the transport factor saturates (the pair’s effective column at 10 kAU exceeds that of any dwarf spheroidal in Section 3.7), and the predicted excess would instead be 0.17 at 10 kAU and 0.48 at 20 kAU of the magnitude currently disputed between Gaia DR3 analyses (Banik et al., 2023; Chae, 2023). The ambient reading is adopted here as the physically motivated convention for point-pair systems, and Gaia DR4 will constrain the choice directly. Figure 5b shows the predicted excess against separation, with the Gaia-constrained 5–50 kAU band marked. Current Gaia DR3 analyses constrain excesses at roughly the percent level over 5–50 kAU, where the prediction lies below sensitivity by a factor of approximately five at 50 kAU and by nearly four orders of magnitude at 5 kAU; under this convention, the framework is untested using present wide-binary data, in contrast to MOND, whose predicted excess in this range is orders of magnitude larger and is the subject of ongoing dispute (Banik et al., 2023; Pittordis et al., 2025). The regime above 100 kAU is the discriminating regime for this convention: a confirmed excess bounded below 8 × 10⁻³ at 100 kAU or above 3.3 × 10⁻² at 200 kAU would falsify the model as implemented, while a confirmed MOND-scale excess at 5–50 kAU would instead support the enclosed-mass reading of the transport factor, with the practical caveats (Galactic tides, unresolved companions, and selection) that apply to all ultra-wide-binary tests.

3.7 Dwarf spheroidals

For each dwarf, the baryon-only prediction of Equation 8 (E _*ion* = 0) is converted into a line-of-sight velocity dispersion through spherical, isotropic Jeans scaling, σ l o s = g r 1 / 2 r 1 / 2 / 3 at the half-light radius r _1/2 (Wolf et al., 2010). The predictions (observations in parentheses) are, in km s⁻¹, as follows: Crater II 0.5 (2.7), Draco 2.1 (9.1), Fornax 11.9 (11.7), Sculptor 6.5 (9.2), Sextans 1.1 (7.9); RMSE 4.6 km s⁻¹. In Figure 5c, four of the five systems lie below the equality line. Only the densest, most massive system (Fornax) is reproduced; the others are under-predicted by factors of 1.4 – 7. The underprediction is substantial, and three considerations frame it. First, the comparison itself is simplified: one global dispersion per galaxy, isotropic equilibrium, no tidal modeling, no membership or binary corrections; a definitive dwarf test requires Jeans modeling of resolved dispersion profiles with anisotropy freedom, which is beyond this study. Second, the framework currently contains no external-field response: the dwarfs orbit inside the Milky Way’s field, and in MOND, the analogous external-field effect is essential precisely for these systems. The counterpart derivable within the present statistics enters through the background variance in quadrature, g N g w + g e x t, and is at the percent level for Milky Way satellites (g _*ext*≪g _*w*), so it cannot materially change the dwarf predictions; the discrepancy is instead tied to the low-density behavior of the transport factor. Third, the screening variables are computed from baryonic quantities alone; substituting the observed dynamical mass would import the observable being predicted and is therefore not done. The dwarf regime, together with the deep weak-field SPARC points of Section 3.2.6, where the model errs in the opposite direction, brackets the current model’s domain of validity at low baryonic surface density.

3.8 Lensing (exploratory)

Under the conditional equal-potential result in Section 2.4.4, lensing and dynamical masses coincide. As an exploratory check, the effective masses M _*eff*(R ₅₀₀) predicted for the four X-COP clusters with published weak-lensing masses (A1644, A1795, A2255, and A2319) agree with the lensing values to 〈log ₁₀ M _*lens*/M _*eff*〉 = +0.04 ± 0.07 dex (Eckert et al., 2019). Four clusters at one radius cannot examine the equal-potential prediction, and a lens-by-lens reanalysis of strong-lens samples such as the Sloan Lens ACS survey has not been performed (Auger et al., 2009). This subsection is retained to state the prediction: a systematic dynamical-versus-lensing mass discrepancy at matched radii would falsify Section 2.4.4.

3.9 Background cosmology (conditional)

The framework’s cosmological sector is an open problem, and the single quantitative statement made here is conditional. Suppose that the homogeneous energy density of the stochastic scalar background evolves with expansion like that of non-relativistic matter, Ω _*ϕ*(z) = Ω _*c*(1+z)³ with Ω _*c* = 0.265. The standard sound-horizon integral with Planck baryon and radiation densities then yields r _*s* = 150.7 Mpc at the drag epoch (z _*d* = 1021, in the Eisenstein–Hu approximation), which lies 2.4% above the observed 147.09 Mpc. This residual is methodological rather than physical: the scenario-A background expansion history is identical to that of ΛCDM by construction, and the offset reflects the Eisenstein–Hu drag-epoch value (z _*d* = 1021) against the Boltzmann-code determination (z _*d* ≈ 1060) that defines the observed sound horizon—the same integral started from z _*d* = 1060 yields r _*s* = 147.0 Mpc, within 0.05% of the observed value. Matter-radiation equality falls at z _*eq* = 3408 against the observed 3402 (Aghanim et al., 2020; Eisenstein and Hu, 1998). Figure 5d compares the sound horizon in the two scenarios with the observed value. If the scalar background instead contributes nothing to the homogeneous energy budget, the sound horizon comes out at r _*s* = 205.7 Mpc, discrepant by 40%, and equality falls at z _*eq* = 534, so this second scenario is excluded. Two routes could produce the required matter-like behavior. In the first, the background energy density is controlled by the emitter number density, which is tied to baryon number and therefore falls as (1+z)³. In the second, more conventional route, the scalar carries a small mass—relaxing the massless idealization of Assumption A1, adding a term not written in Equation 4, and entering no result of this study. The causally accumulated background then redshifts toward the mass gap, becomes non-relativistic, and thereafter behaves as ultralight scalar dark matter: its energy density falls as (1+z)³, and it can cluster gravitationally (Hu et al., 2000; Hui et al., 2017). The open question in that case becomes sharper: cosmological evolution must populate the non-relativistic modes at the observed abundance without forming standard halos, which would double-count the galactic phenomenology in Section 3.2. Until the effective stress-energy, equation of state, sound speed, and perturbation behavior are derived from the source correlator, the sound-horizon agreement remains a conditional consistency requirement that the eventual derivation must meet. The same reservation applies to the cosmic microwave background, structure growth, and the lensing of large-scale structure, none of which are computed here. Because the framework depends non-analytically on g _*N*, a linear modified-growth parameterization cannot be applied, and a dedicated simulation program of the kind performed for MOND would be required (Llinares et al., 2008; Candlish et al., 2015).

4 Discussion

4.1 What is established, what is assumed, and where the model fails

Within the stated assumptions, the results establish the following. The rectified interference between a system’s own wave field and the causally accumulated cosmic background is a single mechanism that spans the Newtonian sector, galaxy rotation curves, and massive-cluster outskirts, with three constants fixed in advance and one coefficient calibrated on clusters. It reproduces galaxy accelerations at 0.23 dex with nothing fitted to galaxies, reproduces cluster outskirts at 0.07 dex under leave-one-cluster-out validation, and does both with the same g N g w kernel, whose scale g _*w* is built from measured cosmological quantities rather than fitted. The rectification coefficient π/2 is a theorem given Assumption A2, verified numerically; the calibrated plasma coefficient is within a factor 1.6 of a parameter-free ionization estimate; and the model passes the Solar System and wide-binary consistency requirements through its own screening structure.

The assumptions and the unresolved regimes are equally explicit. The rectified response (A2) is the central physical hypothesis: no microscopic equation for it is derived here, and with a different response law, the framework predicts a different coefficient or no anomaly. The Hubble-length causal column (A3) is an imposed simplification. The model functions S and T are phenomenological, with demonstrated functional-form freedom at the 0.01 to 0.06 dex level on SPARC. Empirically, the model underpredicts cluster interiors in the Li validation sample, underpredicts four of five dwarf spheroidals, overpredicts the deepest weak-field galaxy points, and leaves residual correlations with gas fraction and surface brightness. These define the model’s current domain of validity: a single mechanism spans the tested regimes from galaxy disks to cluster outskirts, and its present model fails, in defined and falsifiable ways, in cluster interiors and at the lowest baryonic densities.

4.2 Comparison with MOND and CDM

The comparison with MOND, on identical data with identical error treatment (Table 3), is as follows: comparable on galaxies (between the canonical and fitted variants of MOND); statistically indistinguishable on X-COP at equal fitted-parameter count; and similarly limited in cluster interiors. What distinguishes the frameworks is structure rather than RMSE: MOND requires an acceleration constant that differs by a factor 5.5 between the galaxy and cluster fits performed here, whereas the framework spans both regimes with one fixed kernel plus an ionization-motivated coefficient active only in ionized environments. In addition, the framework’s prediction is nearly insensitive to the stellar mass-to-light choice that dominates the MOND fit’s freedom. Against CDM, the comparison is not head-to-head on these data: per-object CDM halo fits with two or more free parameters per system will fit any individual rotation curve or cluster profile better than any zero-per-object-parameter framework; the relevant CDM comparison is at the population level and in cosmology, where CDM is quantitatively successful and the present framework is undeveloped (Section 3.9).

4.3 Limitations

The principal limitations are the following. The rectification hypothesis lacks a microscopic derivation, and its energy and momentum balance is verified only at the level of ensemble means and an order-of-magnitude estimate of the associated noise. The calibrated γ _*p* sits 36% below the ionization estimate; treating that estimate as a prediction, the difference is significant relative to the calibration interval and points to physics missing from the counting argument (temperature dependence and electron–ion weighting) or from the model. The Li-sample validation shows that the model’s radial structure inside 700 kpc is incomplete in ionized environments, and the dwarf underprediction, together with the deep weak-field overprediction, brackets the model’s breakdown at low baryonic density from both sides. The dwarf comparison itself is methodologically minimal (global dispersions, isotropy, no tides, and no external-field response). The lensing check is exploratory, the cosmological statement is conditional, and no cosmological perturbation theory exists for the framework yet. Merging-cluster morphology is likewise not addressed: the effective source in Section 2.5.2 is built on the baryons and up-weights the ionized gas, whereas in dissociative mergers such as 1E 0657–56, the lensing peaks coincide with the collisionless galaxies rather than the gas; these systems therefore constrain the sourcing rule directly, and a quantitative treatment through the equivalent field formulation lies beyond this study (Clowe et al., 2006). Finally, although the number of fitted coefficients is one, the framework’s effective flexibility is larger than that count suggests, residing in the choice among alternative functional forms and anchor values; the released code makes every such choice explicit and auditable.

4.4 Falsification tests

The framework makes the following concrete, model-level predictions. (1) Wide binaries: under the ambient transport convention of Section 3.6, predicted fractional velocity excess of 8 × 10⁻³ at 100 kAU and 3.3 × 10⁻² at 200 kAU; a confirmed measurement inconsistent with these values in either direction falsifies the model as implemented, and a confirmed MOND-scale excess at 5–50 kAU would instead support the enclosed-mass reading of the transport factor. (2) Plasma coefficient: γ _*p* re-calibrated on any independent, well-measured cluster sample at comparable radii must lie within 0.60–0.79; a robustly discrepant value falsifies the ionization interpretation and, if far outside, the model itself. (3) Solar System: anomalous radial accelerations must follow the screening curve of Figure 5a, increasing roughly as r ³ to approximately 4 × 10⁻⁸ of the Newtonian value at Neptune; ephemeris bounds significantly below that curve at large radii would falsify the model. (4) Lensing versus dynamics: matched-radius lensing and dynamical masses must agree; a systematic discrepancy falsifies the equal-potential sector (Section 2.4.4). (5) Background cosmology: a derivation of the scalar background stress-energy that fails to make the background’s energy density evolve like that of non-relativistic matter through the drag epoch and beyond, or fails to allow the background to cluster gravitationally on large scales, falsifies the framework's current routes to consistency with the acoustic scale. (6) Rectification: constraints on the response of matter to stochastic scalar backgrounds that exclude an energy-driven response to the envelope amplitude at the relevant couplings would remove Assumption A2 and, with it, the framework’s mechanism.

4.5 Relation to stochastic semiclassical gravity

The construction shares structure with stochastic semiclassical gravity in the Hu–Verdaguer formulation: both couple classical geometry to a stochastic source with a specified two-point function (Hu and Verdaguer, 2008; Hu and Verdaguer, 2020). The relationship is structural analogy, not equivalence: the present source is a classical random-phase emitter ensemble rather than a quantum noise kernel, the stochastic term enters the scalar’s equation of motion rather than an Einstein–Langevin equation, and no demonstration exists that the two formalisms coincide in any limit.

4.6 Conclusions

The emergent-scalar-field framework describes the Newtonian sector, the galactic acceleration scale, and the dark-matter-like corrections in galaxies and galaxy clusters as three statistical aspects of one matter-sourced scalar wave field. The stationary part of the summed field intensity carries ordinary Newtonian attraction; the variance of the causally accumulated background sets an acceleration scale built from measured cosmological quantities; and the rectified interference between the local field and the background supplies the additional attraction otherwise attributed to dark matter. A single mechanism, three fixed model constants, and one calibrated plasma coefficient span every regime analyzed in this study.

If gravitation and the dynamics of spacetime are generated in the manner developed here, the resulting description accounts for the wide range of gravitational phenomena examined in this study with less adjustable freedom than the principal alternatives. The framework predicts galaxy rotation-curve accelerations with no galaxy-fitted parameter at an accuracy comparable to alternatives that use fitted constants (Section 3.2); it predicts massive-cluster outskirts under out-of-sample validation with a plasma coefficient that is stable under cluster-by-cluster re-calibration and lies close to an independent ionization estimate (Section 3.3); the same model passes Solar System and wide-binary constraints through its own screening structure (Sections 3.5 and 3.6); and its predictions are nearly insensitive to the stellar mass-to-light assumptions that dominate the freedom of competing fits (Section 3.2.5). By contrast, the standard dark-matter description requires per-object halo parameters for the same data, and modified Newtonian dynamics requires an acceleration constant that differs between the galactic and cluster fits performed here (Section 3.3).

The remaining tasks are equally well defined: a microscopic derivation of the rectified response, a transport theory for the model functions S and T, an external-field treatment for satellite systems, the model’s behavior in cluster interiors and at the lowest baryonic densities, and the cosmological evolution of the scalar background. Each is tied to a falsifiable prediction (Section 4.4), so the framework will stand or fall on measurement: independent cluster re-calibrations of the plasma coefficient, ultra-wide-binary velocities beyond approximately 100 kAU, matched-radius comparisons of lensing and dynamical masses, and the derived background evolution confronted with the acoustic scale.

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Supplementary material

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fspas.2026.1904894/full#supplementary-material

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Footnotes

Edited by: Mehrab Momennia, Michoacana University of San Nicolás de Hidalgo, Mexico

Reviewed by: Vladimir Yershov, University College London, United Kingdom

Ruben Cornelius Siagian, PT Siagian Global Research, Indonesia