Emergent Logarithmic Potential from Classical Keplerian Phase-Space Counting

1. Introduction

1.1. The Galactic-Scale Problem

The discrepancy between the observed dynamics of galaxies and the gravitational field inferred from their visible baryonic content is one of the central open problems of modern astrophysics. In spiral galaxies, the circular velocity of stars and gas does not decrease at large radii according to the Keplerian expectation [1,2]. Instead, rotation curves typically approach an approximately constant value over an extended radial range,

$$v_c\left(r\right)\to v_{\infty },$$

(1)

in contradiction with the Newtonian prediction

$$v_c^2\left(r\right)={\displaystyle \frac{G{M}_b}{r}}\propto r^{-1}$$

(2)

generated by a centrally concentrated baryonic mass $M_b$.

Within the standard cosmological framework, this discrepancy is attributed to a dominant non-luminous matter component—dark matter—whose halo dominates the gravitational field at large radii [3]. However, the discrepancy is not arbitrary: the transition from baryon-dominated to dark-matter-like behaviour occurs systematically in low-acceleration regions, tightly correlated with the baryonic distribution [6,7]. This regularity suggests that the observed phenomenology may reflect an effective large-scale gravitational regime rather than the independent distribution of an additional component.

1.2. The Baryonic Tully–Fisher Relation and the Cosmological Acceleration Scale

The most important empirical regularity is the baryonic Tully–Fisher relation [4] (BTFR),

$$v_c^4\propto G{M}_b{a}_0,$$

(3)

where $a_0$ is a characteristic acceleration scale empirically found to be [5]

$$a_0\sim {10}^{-10}\mathrm{m}{\mathrm{s}}^{-2}.$$

(4)

Remarkably, this value coincides numerically with the cosmological acceleration scale $a_H=c{H}_0$, where $H_0$ is the present-day Hubble parameter [6]. This coincidence, noted early in the MOND literature [8,10], strongly suggests that the weak-field galactic regime is coupled to the large-scale cosmological background.

1.3. Motivation: A Statistical Approach

In ordinary statistical mechanics, macroscopic laws emerge from the organisation of microscopic degrees of freedom. For a gas, the Maxwell–Boltzmann distribution arises from maximising entropy subject to an energy constraint; for quantum systems, the Fermi–Dirac and Bose–Einstein distributions follow from the statistics of state occupation. In all cases, the key object is not a force law imposed by hand but a state-counting function—the density of states or the number of accessible configurations—from which a distribution, and hence an effective dynamics, emerges.

We ask whether galactic dynamics admits an analogous description. The idea that gravity itself may have an entropic or thermodynamic origin has been explored from several directions [12,15,16], and entropic interpretations of the dark-matter phenomenology have been proposed in the context of emergent gravity [13]. The observable we wish to reproduce is a logarithmic effective potential,

$${\Phi }_{\mathrm{stat}}\left(r\right)\propto ln\left({\displaystyle \frac{r}{r_0}}\right),$$

(5)

since $\mathrm{d}(lnr)/\mathrm{d}r=1/r$ immediately generates the $1/r$ force law required by flat rotation curves. The problem therefore reduces to identifying a state-counting function $N\left(r\right)$ such that

$$S\left(r\right)=k_{\mathrm{B}}lnN\left(r\right)\sim k_{\mathrm{B}}lnr.$$

(6)

The central result of this paper is that $N\left(r\right)\propto r^{\eta }$, where $\eta$ is determined by the effective dimension of accessible action space, follows from the classical phase-space geometry of the Keplerian $1/r$ potential together with a phenomenological weak-field statistical ansatz.

1.4. Outline

Section 2 lists the phenomenological targets. Section 3 derives the logarithmic entropy from classical Keplerian phase space. Section 4 constructs the effective potential and fixes its amplitude. Section 5 identifies the transition radius and domain of validity. Section 6 works out the lensing predictions. Section 7 develops and tests the $\eta =3/2$ prediction for early-type galaxies. Section 8 discusses the physical picture, relation to MOND and emergent gravity, and open questions. Section 9 concludes.

2. Phenomenological Targets

Any viable framework must simultaneously satisfy the following constraints.

• Asymptotic flatness. For circular motion, $v_c^2\left(r\right)=rg\left(r\right)$. Constant $v_c$ requires $g_{\mathrm{eff}}\left(r\right)\sim 1/r$.
• Baryonic Tully–Fisher relation. The normalisation of the effective force must satisfy $v_c^4=G{M}_b{a}_0$, where $a_0\sim {10}^{-10}\mathrm{m}{\mathrm{s}}^{-2}$ is the empirical BTFR acceleration scale [6,7].
• Newtonian limit. In the high-acceleration regime $g\gg c{H}_0$ the framework must reduce to standard Newtonian gravity.
• Finite mass configurations. The effective density profile must not produce divergent total mass without a natural physical cut-off.
• Universality. The same mechanism must apply across galaxies of different sizes and morphologies, depending primarily on total baryonic content [6,7].

3. Derivation of the Logarithmic Entropy from Keplerian Phase Space

We now derive $S\left(r\right)\sim k_{\mathrm{B}}lnr$ from the classical phase-space geometry of the $1/r$ gravitational potential. This derivation does not assume flat rotation curves; it follows entirely from the Keplerian problem.

3.1. Virial Scaling and the Orbital Radius–Energy Relation

Consider a test particle in a central gravitational potential

$$V\left(r\right)=-{\displaystyle \frac{\alpha }{r}},\alpha =G{M}_{b}m,$$

(7)

on a bound orbit with total energy $E<0$. The virial theorem for a $1/r$ potential gives [25,26] $2\langle T\rangle +\langle V\rangle =0$, so that $E=-\langle T\rangle =\frac{1}{2}\langle V\rangle$. The characteristic orbital radius therefore satisfies

$$\left|E\right|\sim {\displaystyle \frac{\alpha }{r}}\Rightarrow r\left(E\right)\sim {\displaystyle \frac{\alpha }{\left|E\right|}}.$$

(8)

3.2. Scaling of the Classical Radial Action

The characteristic radial momentum at energy E follows from the virial theorem: $p\sim \sqrt{2m\left|E\right|}$. A characteristic radial action is

$$J_r\sim pr\sim \sqrt{2m\left|E\right|}·{\displaystyle \frac{\alpha }{\left|E\right|}}={\displaystyle \frac{\alpha }{\sqrt{2m}}}{\left|E\right|}^{-1/2},$$

(9)

giving

$$J_r\left(E\right)\propto {\left|E\right|}^{-1/2}.$$

(10)

3.3. Action-Space Volume Counting and $N\left(r\right)\propto r^{\eta }$

We count accessible phase-space cells using the Liouville measure in action-angle variables [25,27]. For a spherically symmetric potential the full phase-space volume element is $\mathrm{d}\Gamma ={\mathrm{d}}^{3}x{\mathrm{d}}^{3}p={\mathrm{d}}^{3}J{\mathrm{d}}^3\theta$, where $\mathbf{J}=(J_r,L,L_z)$ are the three action variables. Integrating over the angles yields

$$\Gamma \left(E\right)\sim {\left(2\pi \right)}^3{\int }_{\mathcal{D}\left(E\right)}\mathrm{d}{J}_r\mathrm{d}L\mathrm{d}{L}_z,$$

(11)

where $\mathcal{D}\left(E\right)$ is the accessible action-space region at energy E. For the Keplerian problem the energy depends only on the combination $J_{\mathrm{tot}}=J_r+L$ [25,26],

$$E=-{\displaystyle \frac{C}{{(J_r+L)}^2}},C={\displaystyle \frac{G^2{M}_b^2{m}^3}{2}},$$

(12)

so fixed energy constrains $J_r+L\le J\left(E\right)\equiv \sqrt{C/\left|E\right|}\propto {\left|E\right|}^{-1/2}$.

3.3.0.1. Two-dimensional action space (disk galaxies).

For a rotationally coherent disk galaxy $L_z$ is effectively constrained, leaving $(J_r,L)$ as the two independent accessible actions. The accessible region is the triangle $J_r\ge 0,L\ge 0,J_r+L\le J\left(E\right)$, whose area is $\frac{1}{2}J{\left(E\right)}^2$. Therefore

$$N\left(E\right)\propto J{\left(E\right)}^2\propto {\left|E\right|}^{-1}.$$

(13)

3.3.0.2. Three-dimensional action space (spherical systems).

For a fully isotropic spherical system $L_z$ ranges over $[-L,L]$, contributing an additional factor $\sim L$, giving

$$\Gamma \left(E\right)\propto J{\left(E\right)}^3\propto {\left|E\right|}^{-3/2}.$$

(14)

3.3.0.3. General result.

Both cases are captured by the single scaling

$$N\left(E\right)\propto J{\left(E\right)}^d\propto {\left|E\right|}^{-d/2},$$

(15)

where d is the effective dimension of accessible action space: $d=2$ for a disk and $d=3$ for a spherical system. Using the virial relation $\left|E\right|\sim \alpha /r$ from Equation (8),

$$N\left(r\right)\propto r^{d/2}.$$

(16)

3.4. Emergence of the Logarithmic Entropy

Applying Boltzmann’s relation to Equation (16),

$$S\left(r\right)=k_{\mathrm{B}}lnN\left(r\right)\sim \eta k_{\mathrm{B}}ln\left({\displaystyle \frac{r}{r_0}}\right),\eta ={\displaystyle \frac{d}{2}}.$$

(17)

The entropy is logarithmic in radius for any value of d. The coefficient $\eta$ depends on geometry: $\eta =1$ for a disk and $\eta =3/2$ for a spherical system. The complete logical chain is

$$V\sim -{\displaystyle \frac{1}{r}}\Rightarrow J\left(E\right)\sim {\left|E\right|}^{-1/2}\Rightarrow N\left(E\right)\sim J{\left(E\right)}^d\Rightarrow N\left(r\right)\sim r^{d/2}\Rightarrow S\sim \eta k_{\mathrm{B}}lnr.$$

(18)

4. The Effective Statistical Potential

4.1. Free Energy from Coarse-Graining: The Potential of Mean Force

We define the dimensionless radial entropy

$$\sigma \left(r\right)\equiv {\displaystyle \frac{S\left(r\right)}{k_{\mathrm{B}}}}=lnN\left(r\right)\sim ln\left({\displaystyle \frac{r}{r_0}}\right),$$

(19)

so that $N\left(r\right)=e^{\sigma \left(r\right)}$ exactly. In the weak-field statistical regime the probability of finding the tracer near radius r is

$$P\left(r\right)\propto N\left(r\right)e^{-{\Phi }_N\left(r\right)/{\Theta }_H}=exp\left[-{\displaystyle \frac{{\Phi }_N\left(r\right)}{{\Theta }_H}}+\sigma \left(r\right)\right]=e^{-F\left(r\right)/{\Theta }_H},$$

(20)

where ${\Theta }_H$ is an effective energy scale per unit mass (units of velocity squared) and the effective free energy per unit mass is

$$F\left(r\right)={\Phi }_N\left(r\right)-{\Theta }_H\sigma \left(r\right)={\Phi }_N\left(r\right)-{\Theta }_{H}ln\left({\displaystyle \frac{r}{r_0}}\right).$$

(21)

The Boltzmann-like weighting here should be interpreted as a phenomenological coarse-grained statistical description and is not claimed to arise from thermal equilibrium in the ordinary kinetic sense. Self-gravitating systems possess negative specific heat and no global entropy maximum, so ${\Theta }_H$ is an effective coarse-graining energy scale rather than a thermodynamic temperature, and Equation (20) is an effective-ensemble ansatz rather than a theorem. The state count $N\left(r\right)$ is obtained from single-particle Keplerian orbits; the assumption that the same weighting survives the many-body, partially relaxed dynamics of a real galaxy is the central open premise of the construction (Section 8.6).

This is the potential of mean force [30,31]: the effective potential governing the radial coordinate after the internal phase-space degrees of freedom have been integrated out. The analogy with the Born–Oppenheimer approximation [32] is precise: fast electronic degrees of freedom are integrated out leaving an effective potential for slow nuclear coordinates; here, the fast internal phase-space configurations are integrated out leaving $F\left(r\right)$ for the slow radial coordinate. The effective force per unit mass follows directly:

$$g_{\mathrm{eff}}=-\nabla F=-\nabla {\Phi }_N+{\Theta }_H\nabla \sigma .$$

(22)

Using $\sigma \left(r\right)=ln(r/r_0)$, so that $\nabla \sigma =1/r$, the entropic contribution is

$$g_{\mathrm{stat}}\left(r\right)={\displaystyle \frac{{\Theta }_H}{r}},$$

(23)

with the corresponding entropic potential

$${\Phi }_{\mathrm{stat}}\left(r\right)={\Theta }_{H}ln\left({\displaystyle \frac{r}{r_0}}\right).$$

(24)

4.2. The Effective Energy Scale and Its Physical Origin

An additional physical input is required to fix the amplitude ${\Theta }_H$ and determine when the statistical contribution becomes dynamically relevant. The activation of the weak-field statistical regime is assumed to occur when the local Newtonian acceleration becomes comparable to the cosmological infrared acceleration scale,

$$g_N\left(r_H\right)\sim c{H}_0.$$

(25)

This condition is phenomenological in nature, analogous in its logical role to the critical acceleration $a_0$ of MOND, and its deeper microscopic origin remains an open problem (Section 8.2). This activation condition defines the crossover radius $r_H$ via

$${\displaystyle \frac{G{M}_b}{r_H^2}}=c{H}_0\Rightarrow r_H=\sqrt{{\displaystyle \frac{G{M}_b}{c{H}_0}}}.$$

(26)

Evaluating the Newtonian binding energy per unit mass at this radius gives

$${\Theta }_H\equiv {\displaystyle \frac{G{M}_b}{r_H}}=\sqrt{G{M}_{b}c{H}_0}.$$

(27)

${\Theta }_H$ is the unique gravitational binding-energy scale associated with the activation radius—the depth of the Newtonian potential well at the boundary between Keplerian confinement and the weak-field statistical regime.

4.3. The Effective Free Energy and the Statistical Potential

Substituting Equation (27) into Equation (21), the complete effective free energy per unit mass in the weak-field regime is

$$F\left(r\right)=-{\displaystyle \frac{G{M}_b}{r}}-\sqrt{G{M}_{b}c{H}_0}\sigma \left(r\right).$$

(28)

The entropic contribution,

$${\Phi }_{\mathrm{stat}}\left(r\right)=\sqrt{G{M}_{b}c{H}_0}ln\left({\displaystyle \frac{r}{r_0}}\right),$$

(29)

has amplitude fixed by the activation condition (25) with no additional free parameters beyond $c{H}_0$. The resulting acceleration law is

$$g_{\mathrm{stat}}\left(r\right)={\displaystyle \frac{\sqrt{G{M}_{b}c{H}_0}}{r}},$$

(30)

and the circular velocity satisfies $v_c^2=\sqrt{G{M}_{b}c{H}_0}$, so that squaring gives

$$v_c^4=G{M}_{b}c{H}_0,$$

(31)

recovering the BTFR with no free parameters. Figure 1 shows the theoretical prediction plotted against 126 SPARC galaxies [6,7] spanning four decades in baryonic mass, with scatter 0.066 dex and mean offset 0.040 dex.

5. Transition Scale and Domain of Validity

5.1. Two Dynamical Regimes

The transition radius $r_H$ from Equation (26) separates two distinct regimes:

$$\begin{array}{cc}\hfill r\ll r_H:& g_N\left(r\right)\gg c{H}_0,\mathrm{Newtonian}\mathrm{regime},\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill r\gtrsim r_H:& g_N\left(r\right)\lesssim c{H}_0,\mathrm{statistical}\mathrm{regime}.\hfill \end{array}$$

(33)

In the inner Newtonian regime the statistical contribution is negligible. In the outer weak-field regime the effective logarithmic potential dominates and produces flat rotation curves.

5.2. Liouville Consistency in the Weak-Field Regime

For a collisionless self-gravitating system, Liouville’s theorem states that the phase-space distribution function is conserved along trajectories: $\mathrm{d}f/\mathrm{d}t=0$. In the regime where $v_c\approx \sqrt{{\Theta }_H}$ is approximately constant, the characteristic radial momentum $p_r\sim m{v}_c$ is also approximately constant, so

$$J_r\left(r\right)\sim m{v}_{c}r\propto r,$$

(34)

consistent with Equations (10) and (16). This provides a self-consistency check for the flat-curve regime.

5.3. Outer Cut-Off

The effective density ${\rho }_{\mathrm{eff}}\propto r^{-2}$ implies $M_{\mathrm{eff}}\left(r\right)\propto r$, which diverges as $r\to \infty$. The logarithmic potential is therefore an effective description valid over a finite radial range $r_H\lesssim r\lesssim r_{\mathrm{cut}}$. The outer cut-off $r_{\mathrm{cut}}$ is expected to arise from the finite spatial extent of the galaxy, interactions with the cosmic web, or the breakdown of the scale-free Keplerian regime at cosmological scales. Its derivation from first principles is left to future work.

5.4. Observational Test: Predicted vs. Observed Transition Radius

The transition radius

$$r_H=\sqrt{{\displaystyle \frac{G{M}_b}{a_0}}}$$

(35)

is a parameter-free prediction depending only on total baryonic mass and $a_0=1.2\times {10}^{-10}\mathrm{m}{\mathrm{s}}^{-2}$ [6]. Figure 2 compares $r_H$ to the observed rotation curve flattening radius $R_{♭}$ for 47 SPARC galaxies [6,7]. The correlation has $R^2=0.80$ ($p=1.6\times {10}^{-17}$) with scatter 0.22 dex and best-fit proportionality $R_{♭}=3.8r_H$. The slope $R_{♭}\propto M_b^{1/2}$ has no free parameters; the factor $\sim 4$ between $r_H$ and $R_{♭}$ reflects the onset vs. completion of the entropic regime.

6. Gravitational Lensing

6.1. Effective Density Profile

Within standard General Relativity, the effective potential ${\Phi }_{\mathrm{stat}}$ is interpreted through the weak-field Poisson equation ${\nabla }^2{\Phi }_{\mathrm{stat}}=4\pi G{\rho }_{\mathrm{eff}}$. For a spherically symmetric potential, with $A=\sqrt{G{M}_{b}c{H}_0}$,

$${\rho }_{\mathrm{eff}}\left(r\right)={\displaystyle \frac{A}{4\pi G{r}^2}}\propto r^{-2},$$

(36)

an isothermal-like profile known to produce flat rotation curves.

6.2. Enclosed Mass

$$M_{\mathrm{eff}}\left(r\right)={\int }_0^{r}4\pi r^{\prime 2}{\rho }_{\mathrm{eff}}\left(r^{\prime }\right)\mathrm{d}{r}^{\prime }={\displaystyle \frac{A}{G}}r,$$

(37)

giving $v_c^2=G{M}_{\mathrm{eff}}\left(r\right)/r=A$, consistent with Equation (23).

6.3. Projected Surface Density and Deflection Angle

The projected surface density at impact parameter b is

$$\Sigma \left(b\right)={\int }_{-\infty }^{+\infty }{\rho }_{\mathrm{eff}}\left(\sqrt{b^2+z^2}\right)\mathrm{d}z={\displaystyle \frac{A}{4Gb}},$$

(38)

so $\Sigma \left(b\right)\propto 1/b$. In the weak-field limit of General Relativity [19] the deflection angle is

$$\alpha \left(b\right)={\displaystyle \frac{4G{M}_{2D}\left(b\right)}{c^{2}b}}={\displaystyle \frac{2\pi A}{c^2}}={\displaystyle \frac{2\pi v_c^2}{c^2}}.$$

(39)

The deflection angle is independent of impact parameter throughout the logarithmic-potential regime, reproducing the same qualitative signature as an isothermal dark-matter halo [18].

7. Prediction for Early-Type Galaxies: $\eta =3/2$

7.1. Velocity Dispersion Scaling for Isotropic Systems

For pressure-supported, approximately isotropic early-type galaxies (ellipticals and lenticulars), the full three-dimensional action space is accessible and $\eta =3/2$. The framework therefore predicts $v_c^4=\frac{3}{2}G{M}_b{a}_0$ for such systems. For an isotropic pressure-supported system, the Jeans equation gives [18]

$${\sigma }_e^2\approx \frac{1}{2}{v}_c^2,$$

(40)

so that

$${\sigma }_e^4={\displaystyle \frac{3}{8}}G{M}_b{a}_0,$$

(41)

where $a_0=1.2\times {10}^{-10}\mathrm{m}{\mathrm{s}}^{-2}$ [6]. This prediction contains no free parameters. The factor $3/8$ arises entirely from $\eta =3/2$ (via ${\sigma }_e^4={\displaystyle \frac{\eta }{4}}G{M}_b{a}_0$) and the standard isotropic Jeans projection factor $1/4$.

The relationship of Equation (41) to MOND deserves clarification. MOND also predicts mass–velocity-dispersion relations for pressure-supported systems [8,11], so a scaling of the form ${\sigma }_e^4\propto G{M}_b{a}_0$ is not unique to the present framework. The distinction lies in the origin of the coefficient: here $3/8$ arises directly from the geometry of the accessible action space through $\eta$, with $\eta =1$ for disk systems and $\eta =3/2$ for isotropic spheroidal systems, requiring no modification of the gravitational field equations.

7.2. Comparison with ATLAS${⠀}^{3\mathrm{D}}$ Data

Figure 3 shows the predicted ${\sigma }_e$ from Equation (41) compared to observed effective velocity dispersions for 231 early-type galaxies from the full ATLAS${⠀}^{3\mathrm{D}}$ survey [23], covering the mass range ${10}^{9.9}$–${10}^{11.5}{M}_{\odot }$.

The correlation has $R^2=0.856$ ($p=2\times {10}^{-98}$) with scatter 0.105 dex about the 1:1 line and mean offset 0.039 dex—the prediction is essentially unbiased across the full mass range. The $\eta =1$ (disk) prediction systematically underpredicts ${\sigma }_e$, confirming that the $\eta =3/2$ branch is physically appropriate for isotropic pressure-supported systems.

A systematic uncertainty limits the precision with which the coefficient itself can be tested: the stellar masses adopt a Chabrier IMF, and a Salpeter-scale shift ($\sim 0.2$ dex in $M_b$) would move the predicted ${\sigma }_e^4$ by the same amount ($\sim 0.05$ dex in ${\sigma }_e$). A fully quantitative discrimination between the $3/8$ coefficient and nearby alternatives requires control of the mass-to-light ratio below this level together with individual Jeans modelling.

8. Discussion

8.1. Physical Picture

The construction developed here proposes that the outer regions of galaxies enter a distinct statistical regime when the local Newtonian acceleration falls below the cosmological scale $c{H}_0$. In this regime the number of accessible phase-space configurations grows as $N\left(r\right)\propto r^{\eta }$, yielding a logarithmic entropy $S\propto lnr$ regardless of the precise value of $\eta$. This logarithmic entropy, via the potential of mean force, produces an effective potential whose amplitude is fixed by the depth of the Newtonian potential well at the transition radius. No new matter component, no modification of the gravitational field equations, and no free dynamical parameter is introduced.

8.2. Why Does the Statistical Regime Activate at $g_N\sim c{H}_0$?

Two physical considerations motivate the appearance of the same transition scale.

Binding-energy threshold. The Newtonian binding energy per unit mass at $r_H$ is $|{\Phi }_N\left(r_H\right)|=\sqrt{G{M}_{b}c{H}_0}={\Theta }_H$. This is the gravitational potential energy at the edge of local gravitational dominance—the point beyond which orbits become weakly bound enough that the accessible Keplerian phase-space volume becomes dynamically relevant.

Broader context. A de Sitter universe with Hubble rate $H_0$ possesses a cosmological horizon $R_H\sim c/H_0$ and an associated Gibbons–Hawking temperature [33] $T_H\sim \hslash H_0/\left(2\pi k_{\mathrm{B}}\right)$. Although the present framework is entirely classical, this suggests that $c{H}_0$ may mark the onset of cosmological phase-space accessibility in a deeper quantum-gravitational sense.

8.3. CMB Safety and Domain of Activation

The framework is expected to leave early-universe linear perturbation physics approximately unchanged because its derivation presupposes virialized Keplerian structure. The entire chain in Equation (18) presupposes an orbit, a radial action integral, and a virialized gravitational system—none of which apply to the photon-baryon plasma at recombination. The entropic correction activates only after structure formation and virialization, when $g_N\left(r\right)\lesssim c{H}_0$ inside a bound, relaxed gravitational system. This distinguishes the present approach from modifications of the gravitational field equations, which necessarily affect linear perturbation theory [34].

8.4. Relation to MOND

MOND [8,9,10] reproduces flat rotation curves and the BTFR for the same phenomenological reasons as the present framework. The key differences are: (i) MOND modifies the gravitational or inertial law at the level of the equations of motion, whereas the present framework interprets the additional acceleration as an entropic contribution without modifying field equations; (ii) the present framework identifies a specific microscopic origin for the acceleration scale in the phase-space geometry of the $1/r$ potential and its relationship to $c{H}_0$; (iii) the additional acceleration here arises as an effective statistical contribution associated with virialized Keplerian systems, rather than from a modification of the fundamental gravitational field equations. Because the construction is formulated only after the emergence of bound gravitational structures, its expected cosmological domain of applicability differs from that of relativistic MOND theories.

8.5. Relation to Verlinde’s Emergent Gravity

Verlinde [12,13] proposed that gravity and dark matter are emergent phenomena arising from entropic considerations. The present framework shares the entropic interpretation but differs in its microscopic foundation: the logarithmic entropy here is derived from the classical phase-space geometry of the $1/r$ potential rather than from the holographic entropy of de Sitter space. Both frameworks make similar predictions at the galactic scale and both require further development to address cluster and cosmological scales [14].

8.6. Limitations and Open Questions

Microscopic completeness. The phase-space derivation identifies $N\left(r\right)\propto r^{\eta }$ from single-particle Keplerian orbits. A complete treatment requires showing that this scaling survives many-body self-gravitating dynamics, including incomplete relaxation and non-spherical geometries.

Galaxy clusters. The framework has been developed for isolated, virialized spiral galaxies. Whether the same phase-space counting applies to triaxial, gas-pressure-supported cluster systems, and whether it can account for the residual mass discrepancy in clusters such as the Bullet Cluster [20,21], requires separate analysis.

Intermediate structure formation. Between recombination and full virialization the framework’s status is genuinely ambiguous. The precise redshift and density threshold at which the entropic correction switches on, and its effect on the matter power spectrum [24] at $z\sim 2$–5, are open questions.

Outer cut-off. The logarithmic potential must be truncated at some scale $r_{\mathrm{cut}}$ to avoid divergent total mass. Its derivation from the underlying physics is an open problem.

Derivation of the effective energy scale. The identification ${\Theta }_H=\left|{\Phi }_N\left(r_H\right)\right|$ is physically motivated, but a derivation from a microscopic statistical mechanics of the cosmological environment remains an open problem.

9. Conclusions

We have shown that a logarithmic radial entropy $S\left(r\right)\sim k_{\mathrm{B}}lnr$ follows directly from the classical phase-space geometry of the Keplerian $1/r$ gravitational potential, without assuming flat rotation curves. The effective potential then follows from the potential of mean force after integrating out internal Keplerian phase-space degrees of freedom, with effective energy scale ${\Theta }_H$ fixed by the depth of the Newtonian potential well at the cosmological transition radius. The central chain of reasoning is

$$V\sim -{\displaystyle \frac{1}{r}}\Rightarrow J\left(E\right)\sim {\left|E\right|}^{-1/2}\Rightarrow N\left(r\right)\sim r^{\eta }\Rightarrow \sigma \sim lnr\Rightarrow F={\Phi }_N-{\Theta }_H\sigma \Rightarrow v_c^4=G{M}_b{a}_0.$$

(42)

Three parameter-free comparisons against real data confirm the predicted scalings:

• BTFR: 126 SPARC galaxies, scatter 0.066 dex;
• Transition radius scaling $R_{♭}\propto M_b^{1/2}$: 47 SPARC galaxies, $R^2=0.80$;
• Early-type galaxy velocity dispersion ${\sigma }_e^4=\frac{3}{8}G{M}_b{a}_0$: 231 ATLAS${⠀}^{3\mathrm{D}}$ galaxies, $R^2=0.856$.

The last result is a parameter-free prediction with a specific, testable coefficient whose geometric origin—the $\eta$ branching between disk ($\eta =1$) and spheroidal ($\eta =3/2$) systems—is a structurally distinctive feature of the present construction. Its normalisation ($3/8$) differs from the deep-MOND isothermal value ($4/81$ for the line-of-sight dispersion) and is offered as a falsifiable alternative. The framework is expected to leave early-universe linear perturbation physics approximately unchanged because its derivation presupposes virialized Keplerian structure, distinguishing it from modifications of the gravitational field equations. The principal open questions for future work are the behaviour in the quasi-linear structure formation regime, the application to galaxy clusters, and the derivation of ${\Theta }_H$ and $r_{\mathrm{cut}}$ from first principles.