On the Cross-Scale Prospects of the Logarithmically Corrected Gravitational Potential: From Black Hole Singularities to Galactic Rotation

1. Introduction

Modern astrophysics and gravitational theory have long been faced with two major cross-scale challenges. At the microscopic scale of black holes, the singularity predicted by classical General Relativity (GR) features infinite curvature, which violates the requirement of finiteness for physical quantities in quantum mechanics, and the "information paradox" caused by Hawking radiation remains unsolved. At the macroscopic scale of galaxies, the observed rotation speeds of peripheral stars and gas are much higher than the limit that can be maintained by the gravity of visible matter. The mainstream ΛCDM model has to rely on the unproven hypothesis of dark matter halos, and there is tension between its small-scale predictions and observations.

Traditional theories provide fragmented explanations for these two challenges: black hole physics relies on the Kerr metric (which requires posterior fitting of spin and inclination), while galactic dynamics depends on the dark matter hypothesis, both lacking a unified physical core. More critically, these theories either suffer from internal incompleteness (e.g., the singularity) or lack a direct physical carrier (e.g., dark matter particles). In this work, we adopt a bottom-up effective field theory approach. Without presupposing the specific form of the microscopic mechanism, we derive a logarithmically corrected gravitational potential from the universal asymptotic behavior of dark matter halo models, and on this basis, deduce cross-scale observable effects. The microscopic picture presented in the Appendix is for heuristic discussion only and does not serve as a theoretical premise.

We identify a universal logarithmic asymptotic behavior of the extra gravitational potential from dark matter halo models:

$${\Phi }_{halo}\left(r\right)\sim -{\displaystyle \frac{\mathrm{l}\mathrm{n}\left(r/r_{*}\right)+1}{r}}$$

Through the analysis of this potential, we further discover a physical picture: at distances approaching the black hole singularity ($r<r_{*}\approx 8.792\times {10}^{-11}\mathrm{m}$), the negative contribution of $\mathrm{l}\mathrm{n}\left(r/r_{*}\right)$ renders the extra gravitational potential repulsive, preventing the gravitational collapse of matter into a singularity. In the strong gravitational field near black holes and the weak gravitational field of galaxies, the positive contribution of $\mathrm{l}\mathrm{n}\left(r/r_{*}\right)$ provides an additional attractive gravitational force (calculable solely from the mass of ordinary matter), thereby maintaining the high orbital speed of stars and the flattening of rotation curves.

This mechanism requires no artificial mathematical renormalization, nor does it introduce new unobservable entities (such as higher-dimensional strings or dark matter particles). It achieves two key goals solely through the dynamics of ordinary matter: (1) singularity resolution; (2) flattening of galactic rotation curves. To explain this mechanism, we also provide a speculative (for discussion) microscopic physical picture in the Appendix: relying on quantum vortices (a physical carrier with solid experimental support) and the AdS/CFT correspondence (a mathematical structure originated from string theory to describe non-local entanglement), we unify black hole physics and galactic dynamics under a single mechanism. This provides a potential solution to the cross-scale gravitational challenges based solely on general relativity and quantum mechanics in four-dimensional spacetime, without relying on conventional pre-assumed physical models (such as the hypothesis of higher-dimensional strings or dark matter particles that have eluded detection for decades).

2. Universal Logarithmic Asymptotics of Dark-Matter Halo Models

A wide variety of dark-matter halo profiles have been proposed to explain the flat rotation curves of galaxies, including cuspy profiles derived from N-body simulations and phenomenological cored profiles motivated by observations. Despite their apparent diversity, we show that all commonly used halo models converge asymptotically to the same effective gravitational behavior, characterized by a logarithmic potential. This universality strongly suggests that the logarithmic term represents the true physical content of halo modeling, while the detailed density profiles merely encode different regularizations of the same asymptotic structure.

2.1. General Condition for Flat Rotation Curves

For a test particle on a circular orbit, the centripetal acceleration satisfies ${\displaystyle \frac{v^2\left(r\right)}{r}}=g(r)={\displaystyle \frac{GM\left(r\right)}{r^2}}$, where $M(r)=4\pi {\int }_0^r\rho \left(r^{\prime }\right)r^{\prime 2}d{r}^{\prime }$. A flat rotation curve: $v(r)\to v_0=const$, implies $g(r)\sim {\displaystyle \frac{v_0^2}{r}}$, $M(r)\sim {\displaystyle \frac{v_0^2}{G}}\hspace{0.17em}r$. Differentiating, $\rho (r)={\displaystyle \frac{1}{4\pi r^2}}{\displaystyle \frac{dM}{dr}}\sim {\displaystyle \frac{1}{r^2}}$. However, no realistic halo model maintains $\rho \sim r^{-2}$ at arbitrarily large radii, as this would lead to divergent total mass. Consequently, all viable models steepen to $\rho (r)\sim r^{-3}\hspace{1em}(r\gg r_s)$, which leads to

$$M\left(r\right)\sim \ln r, g(r)\sim {\displaystyle \frac{\ln r}{r^2}}$$

(1)

(Note: In this and all subsequent sections, the expression $\ln r$ is understood as $\ln (r/r_{*})$ where $r_{*}$ is a universal reference scale. As shown in Section 3.2, the choice of $r_{*}$ does not affect any physical observable because it can be absorbed into the renormalization of the Newtonian mass parameter.)

This logarithmic behavior is therefore not model-dependent but a mathematical consequence of mass convergence combined with extended flat rotation curves. In addition, we note that the gravitational acceleration with logarithmic behavior ($g\left(r\right)\sim {\displaystyle \frac{\mathrm{l}\mathrm{n}r}{r^2}}$) will have a sign reversal of the logarithmic term "$\ln r$" at extremely short (microscopic) distances, which is likely to form a dynamic mechanism that repels classical gravity ($g\left(r\right)\sim {\displaystyle \frac{1}{r^2}}$) to naturally "desingularize".

2.2. Cuspy Halo Models: NFW and Einasto [1,2,3]

2.2.1. NFW Profile

The NFW profile: $\rho (r)={\displaystyle \frac{{\rho }_s}{(r/r_s)(1+r/r_s{)}^2}}$, satisfies $\rho (r)\sim r^{-3}\hspace{1em}(r\gg r_s)$.

Integrating:

$M(r)\propto \ln \left(1+{\displaystyle \frac{r}{r_s}}\right)$, and thus $g(r)={\displaystyle \frac{GM\left(r\right)}{r^2}}\sim {\displaystyle \frac{\ln r}{r^2}}$.

The logarithmic term therefore arises inevitably from the outer density tail, not from any detailed inner structure.

2.2.2. Einasto Profile

The Einasto profile: $\ln \rho (r)=\ln {\rho }_0-{\left({\displaystyle \frac{r}{r_0}}\right)}^{\alpha }$, $\hspace{1em}\alpha \ll 1$,

admits the expansion:

$${\left({\displaystyle \frac{r}{r_0}}\right)}^{\alpha }\simeq 1+\alpha \ln {\displaystyle \frac{r}{r_0}}+\mathcal{O}\left({\alpha }^2\right).$$

Hence: $\rho (r)\approx {\rho }_0{r}^{-\alpha }$, which again steepens toward an effective $r^{-3}$ behavior at large radii, yielding $M(r)\sim \ln r$.

The shape parameter $\alpha$ merely controls how rapidly the logarithmic regime is approached.

2.3. Cored Halo Models: Burkert and Pseudo-Isothermal

Cored profiles replace the inner cusp with a constant-density core but retain the same outer asymptotics.

For example, the Burkert profile: $\rho (r)={\displaystyle \frac{{\rho }_0}{(1+r/r_0)(1+(r/r_0{)}^2)}}$, satisfies $\rho (r)\sim r^{-3}\hspace{1em}(r\gg r_0)$,

leading again to

$$M\left(r\right)\sim \ln r, g\left(r\right)\sim {\displaystyle \frac{\ln r}{r^2}}.$$

Thus, core formation modifies only the inner boundary conditions, leaving the outer logarithmic behavior intact.

2.4. Self-Interacting and Wave Dark Matter [4,5]

Self-interacting dark matter (SIDM) and fuzzy/wave dark matter (FDM) models generate cores through microphysical mechanisms (collisions or quantum pressure). Nevertheless, in all cases the outer halo relaxes to an NFW-like tail, $\rho (r)\to r^{-3}$, ensuring $M(r)\sim \ln r\hspace{1em}and\hspace{1em}g(r)\sim {\displaystyle \frac{\ln r}{r^2}}$.

Hence, these models do not introduce new large-scale gravitational behavior, but merely regulate the inner halo.

2.5. Universality of the Logarithmic Potential

Since $g(r)={\displaystyle \frac{d\Phi }{dr}}$, the asymptotic form $g(r)\sim {\displaystyle \frac{\ln r}{r^2}}$ corresponds to an effective potential:

$${\Phi }_{halo}(r) \sim -{\displaystyle \frac{\ln r+1}{r}}$$

(2)

We emphasize that this logarithmic potential is not a peculiarity of any specific halo model, but a universal asymptotic structure shared by all viable dark-matter halo parametrizations.

3. Effective Theory: Logarithmically Corrected Gravitational Potential

3.1. Modified Poisson Equation

Based on the mathematical asymptotic behavior of dark matter halo density ($\rho \left(r\right)\sim r^{-3}$) summarized in Section 2, inspired by quantum tornado experiments (where vortices emerge in environments mimicking the vicinity of black holes) [6], and combined with the divergent behavior of GR itself near the singularity ($R_{trt}^r\propto r^{-3}$), we introduce a modified Poisson equation. Given the effective theory positioning of this work, the speculative microscopic derivation is detailed in Appendix A and B; this effective framework does not depend on the microscopic derivation, but is validated by consistency with observational data:

$${\nabla }^2\Phi =4\pi G\left(M{\delta }^3\left(r\right)+{\displaystyle \frac{k{G}_h{M}^2}{4\pi G{r}^3}}\right) \hspace{1em}$$

(3)

Where $M{\delta }^3\left(r\right)$ is the source term of the classical gravitational point mass (${\delta }^3\left(r\right)$ is the three-dimensional Dirac delta function), and ${\displaystyle \frac{k{G}_h{M}^2}{4\pi G{r}^3}}$ is the additional gravitational source term (derived from the dark matter halo density $\rho \left(r\right)\sim r^{-3}$), which we tentatively describe as the "quantum gravity" (i.e., logarithmically corrected gravity) source term in this paper.

$k$ is the relative intensity factor of non-local entanglement ($k={\displaystyle \frac{M_{BH,ref}}{M_{BH,topo}}}$, $M_{BH,ref}$: reference black hole mass, $M_{BH,topo}$: topological black hole mass (equal to the target black hole mass in the strong-field regime: $M_{BH,topo}=M$). Usually, we take the supermassive black hole Sgr A* at the center of the Milky Way as the reference: $k={\displaystyle \frac{M_{SgrA}}{M_{BH,topo}}}$. If we take the central black hole of other galaxies as the reference, we need to transform the reference $G_h$ accordingly. For example, taking M87* as the reference: $G_{h,M87*}={\displaystyle \frac{M_{SgrA*}}{M_{M87*}}}{G}_h$, then $k_{M87*}={\displaystyle \frac{M_{M87*}}{M_{BH,topo}}}$, so $k_{M87*}{G}_{h,M87*}={\displaystyle \frac{M_{M87*}}{M_{BH,topo}}}\cdot {\displaystyle \frac{M_{SgrA*}}{M_{M87*}}}{G}_h=k{G}_h$, which shows that the value of $k{G}_h$ is independent of the selected reference black hole $M_{BH,ref}$.

$$G_h={\displaystyle \frac{\hslash c^2{G}^3}{8}}$$

(4)

This expression is derived from the theoretical framework based on quantum vortices and nested AdS/CFT correspondence in Appendix B, rather than a posteriori fitting (especially the subsequent multi-scale observational tests can hardly be completed by fitting this fixed value). Although the derivation in the Appendix is not yet rigorous, it provides a first-principles motivation for the value of $G_h$.

$G_h$ is the effective coupling constant describing the strength of the logarithmic correction. Within the effective theory framework, its dimensionality is determined by the dimensional matching between the correction term ${\displaystyle \frac{k{G}_h{M}^2}{4\pi G{r}^3}}$ and the Newtonian term $M{\delta }^3\left(r\right)$ in the Poisson equation. From the standard Poisson equation ${\nabla }^2\Phi =4\pi G\rho$, both the correction term and the Newtonian term represent mass density, with the dimensionality of $kg\cdot m^{-3}$, thus $\left[G_h\right]=\mathrm{k}{\mathrm{g}}^{-2}{\mathrm{m}}^3{\mathrm{s}}^{-2}$. For consistency with subsequent observational calculations, we adopt the numerical form of $G_h$ as $G_h=\hslash c^2{G}^3/8\approx 3.5224\times {10}^{-49}\mathrm{k}{\mathrm{g}}^{-2}{\mathrm{m}}^3{\mathrm{s}}^{-2}$, where $\hslash$, $c$, and $G$ are the reduced Planck constant, the speed of light in vacuum, and Newton's gravitational constant, respectively. This expression links the correction term to fundamental physical constants, while its dimensionality is determined by the self-consistency of the effective theory and does not rely on the "dimensional compactification" hypothesis (speculative discussion in Appendix B). At the effective theory level, $G_h$ should be regarded as a phenomenological parameter describing the strength of the logarithmic correction, whose value can be cross-validated by subsequent multi-scale observations (e.g., black hole shadows, velocities of high-speed stars, galactic rotation curves).

3.2. Corrected Gravitational Potential with Logarithmic Term

Solving the above modified Poisson equation under static spherical symmetry, we obtain the general solution:

$$\Phi \left(r\right)=-{\displaystyle \frac{GM}{r}}-{\displaystyle \frac{k{G}_h{M}^2\left(\ln r+1\right)}{r}}-{\displaystyle \frac{C_1}{r}}+C_2$$

With the boundary condition that $\Phi (r)\to 0$ as $r\to \infty$, we have $C_2=0$, and the term $-C_1/r$ can be absorbed into the classical Newtonian term $-GM/r$. The expression is thus simplified to

$$\Phi \left(r\right)=-{\displaystyle \frac{GM}{r}}-{\displaystyle \frac{k{G}_h{M}^2\left(\ln r+1\right)}{r}}\hspace{1em}$$

(5)

$$\iff \Phi \left(r\right)=-{\displaystyle \frac{GM}{r}}-{\displaystyle \frac{k{G}_h{M}^2[\ln (r/r_{*})+1]}{r}}$$

This equation consists of two terms:

• Classical gravitational term$-{\displaystyle \frac{\mathit{G}\mathit{M}}{\mathit{r}}}$: Dominates conventional gravitational effects, consistent with Newtonian gravity and the weak-field approximation of general relativity.
• Quantum gravitational logarithmic term$-{\displaystyle \frac{\mathit{k}{\mathit{G}}_{\mathit{h}}{\mathit{M}}^2\left(\mathbf{ln}\mathit{r}+1\right)}{\mathit{r}}}$ (consistent with Equation (2)): Serves as the core cross-scale correction term. Its effect depends on the magnitude of the distance $r$—exhibiting repulsive behavior at short distances (black hole "singularity" scale) and gravitational enhancement at long distances (galaxy scale). Essentially, it is likely a macroscopic manifestation of nonlocal entanglement of quantum vortices under the hierarchical nested structure ($Ad{S}_2/CF{T}_1\subseteq Ad{S}_3/CF{T}_2\subseteq Ad{S}_4/CF{T}_3$).
• Note on scale normalization: The logarithmic term is written with a reference scale $r_{*}$ to ensure a dimensionless argument. This scale is defined by the condition $\Phi \left(r_{*}\right)=0$, which gives $\ln (r_{*}/r_{*})=0$ and therefore $r_{*}$ cancels out in the definition. Expanding the term:

$$-{\displaystyle \frac{k{G}_h{M}^2[\ln (r/r_{*})+1]}{r}}=-{\displaystyle \frac{k{G}_h{M}^2(\ln r+1)}{r}}+{\displaystyle \frac{k{G}_h{M}^2\ln r_{*}}{r}}$$

The last term $\propto 1/r$ is indistinguishable from a renormalization of the Newtonian mass: $G{M}_{eff}=GM-k{G}_h{M}^2\ln r_{*}$. In any astronomical observation where $GM$ is calibrated dynamically (e.g., via orbital motions or lensing)，the observable quantity is $G{M}_{eff}$，not the bare $GM$. Consequently, the value of $r_{*}$ does not affect any physical prediction. The calculation results obtained by normalizing $r_{*}$ to 1m and using $\mathrm{l}\mathrm{n}\left(r/1\right)$ are completely equivalent to those obtained by strictly using $\mathrm{l}\mathrm{n}\left(r/r_{*}\right)$. In all practical calculations (black hole shadow, high-velocity stars, galaxy rotation curves, gravitational lensing, etc.), directly substituting the numerical value of $r$ in meters and taking its natural logarithm $\ln r$ is equivalent to $\mathrm{l}\mathrm{n}\left(r/1\right)$. This convention applies throughout the paper and will not be reiterated in subsequent sections.

• If the quantum gravitational effect under non-local entanglement is not considered ($k=0$, i.e., ignoring the black hole: $M_{BH,ref}=0$), the gravitational potential automatically degenerates into the classical gravitational potential: $\Phi \left(r\right)=-{\displaystyle \frac{GM}{r}}$, and the framework also naturally degrades to the classical gravitational framework.

3.3. Cross-Scale Physical Nature of the Logarithmic Term

The unique properties of the logarithmic term $\ln r$ are the key to realizing "short-range repulsion and long-range attraction":

• When $r\to 0$ (black hole core region): $\ln r$ tends to negative infinity, and the quantum gravitational term transforms into a strong repulsive potential. When $r<r_{*}=e^{-1-{\displaystyle \frac{G}{k{G}_{h}M}}}=e^{-1-{\displaystyle \frac{G}{G_h{M}_{SgrA*}}}}\approx 8.792\times 10^{-11}\mathrm{m}$ (in the strong field regime, $k=M_{BH,ref}/M_{BH,topo}=M_{SgrA*}/M$), $\Phi \left(r\right)>0$ in the total potential, dynamically preventing matter from collapsing into a singularity.
• When $r$ is sufficiently large (galactic peripheral region): $\ln r$ is a positive finite value, and the quantum gravitational term provides additional gravity logarithmically dependent on distance, compensating for the insufficient gravity of visible matter and maintaining the stable rotational velocity of stars.

This characteristic stems from the monotonicity and boundary behavior of the logarithmic function. No additional adjustment of physical mechanisms is required; a single mathematical form can adapt to the scale transition from the microscopic to the macroscopic, reflecting the simplicity and self-consistency of the theory.

4. Black Hole Scale Application: Physical Avoidance of Singularity, Shadow Prediction and High-Speed Stars

4.1. Physical Avoidance of Singularity

In the core region of a black hole, the quantum repulsive potential dominated by the logarithmic term plays a central role:

• Suppression of curvature divergence: The repulsive potential prevents matter from reaching $r=0$ and avoids the divergence of spacetime curvature, thus realizing the dynamical avoidance of singularities without the need for renormalization.
• A potential mechanism for resolving the information paradox: The repulsive potential excites virtual particles from vacuum fluctuations into real particles, which tunnel out of the black hole horizon through the nested AdS/CFT correspondence ($Ad{S}_2/CF{T}_1\subseteq Ad{S}_3/CF{T}_2\subseteq Ad{S}_4/CF{T}_3$). These real particles carry information away from the black hole, and the black hole loses mass synchronously. This mechanism is conducive to making black hole physics satisfy the unitarity of quantum mechanics, namely the principle of information conservation.

4.2. Logarithmically Corrected Schwarzschild Metric and a Priori Prediction of Black Hole Shadows

Based on the corrected gravitational potential, the quantum-corrected Schwarzschild metric is derived (substituting $\Phi \left(\mathrm{r}\right)$ into the relationship between the metric and gravitational potential under the weak-field approximation of general relativity):

$$d{s}^2=-A(r)c^{2}d{t}^2+B(r)d{r}^2+r^{2}d{\mathsf{\Omega }}^2$$

$$A(r)\approx 1+{\displaystyle \frac{2\Phi \left(r\right)}{c^2}}=1-{\displaystyle \frac{2GM}{c^{2}r}}-{\displaystyle \frac{2k{G}_h{M}^2(\ln r+1)}{c^{2}r}}$$

(6)

$$B(r)\approx 1-{\displaystyle \frac{2\Phi \left(r\right)}{c^2}}=1+{\displaystyle \frac{2GM}{c^{2}r}}+{\displaystyle \frac{2k{G}_h{M}^2(\ln r+1)}{c^{2}r}}$$

(7)

Analysis of Metric Singularity Resolution

For motion in the equatorial plane with $\theta =\pi /2$ and $\dot{\theta }=0$, the Lagrangian is given by: $L=-A{c}^2\dot{t^2}+B{\dot{r}}^2+r^2{\dot{\phi }}^2$. From Killing symmetry, the conserved energy and angular momentum are obtained as: $E=A\left(r\right)c^2\dot{t}$ and $L=r^2\dot{\phi }$ (with respect to the affine parameter $\lambda$). The general "first integral" radial equation is derived (where $\sigma =1$ for timelike geodesics (proper time $\tau$); $\sigma =0$ for null geodesics (affine parameter $\lambda$)):

$${\dot{r}}^2={\displaystyle \frac{E^2-A\left(r\right)\left(\frac{c^2{L}^2}{r^2}+\sigma c^4\right)}{AB{c}^2}}$$

along with: $\dot{t}={\displaystyle \frac{E}{A{c}^2}}$ and $\dot{\phi }={\displaystyle \frac{L}{r^2}}$. We define the effective potential as: $V_{\sigma }\left(r\right)=A\left(r\right)\left(\sigma c^4+{\displaystyle \frac{c^2{L}^2}{r^2}}\right)$.

Analysis of timelike geodesics ($\sigma =1$): the effective potential $V_{timelike}\left(r\right)=A\left(r\right)\left(c^4+{\displaystyle \frac{c^2{L}^2}{r^2}}\right)$ tends to $+\infty$ as $r\to 0$ (since $A(r)\approx 1-{\displaystyle \frac{2GM}{c^{2}r}}-{\displaystyle \frac{2k{G}_h{M}^2(\ln r+1)}{c^{2}r}}\to +\infty$). For any finite energy $E$ and angular momentum $L$, the radial equation

$${\dot{r}}^2={\left({\displaystyle \frac{dr}{d\tau }}\right)}^2={\displaystyle \frac{E^2-A\left(r\right)\left(\frac{c^2{L}^2}{r^2}+c^4\right)}{AB{c}^2}}={\displaystyle \frac{E^2-V_{timelike}\left(r\right)}{AB{c}^2}}$$

must satisfy $E^2=V_{timelike}\left(r_{min}\right)$ at some $r_{min}>0$—a bounce occurs, meaning the particle cannot physically reach $r=0$. The analysis for null geodesics ($\sigma =0$) is essentially the same (the effective potential $V_{null}(r)=A(r)\left({\displaystyle \frac{c^2{L}^2}{r^2}}\right)\to +\infty$ as $r\to 0$), so light cannot physically reach $\mathrm{r}=0$ either.

Note: Although this modified metric is derived from the weak-field approximation, it can be consistently extended to the strong-field regime via the holographic screen renormalization group flow (Appendix C), and the dynamical singularity avoidance mechanism is self-consistent with the analysis of the modified gravitational potential.

This metric does not require fitting of black hole spin and inclination; the shadow angular diameter can be predicted solely by the black hole mass $M$ and distance $D$.

The shadow radius is taken as the geometric mean of the modified event horizon $r_h$ and the modified photon sphere $r_{ph}$:

$$r_{sh}\approx \sqrt{r_h{r}_{ph}}$$

(8)

and the angular diameter: ${\theta }_{sh}=2r_{sh}/D$.

In the strong-field regime: $k=M_{BH,ref}/M_{BH,topo}=M_{SgrA*}/M$

The Schwarzschild radius remains unchanged: $r_s=2GM/c^2$

$g_{tt}=0\to$ Horizon equation (where $M=M_{BH,topo}$):

$$c^{2}r=2GM+2k{G}_h{M}^2(\ln r+1)\hspace{1em}$$

(9)

Solving this equation yields the modified event horizon $r_h$.

For photons, $d{s}^2=0$, and $\dot{r}=0$ on circular orbits. Satisfying the extremum condition of the effective potential ${\displaystyle \frac{d}{dr}}\left({\displaystyle \frac{r^2}{A\left(r\right)}}\right)=0$, the photon sphere equation is obtained (where $M=M_{BH,topo}$):

$$c^{2}r=3GM+k{G}_h{M}^2(3\ln r+2)\hspace{1em}$$

(10)

Solving this equation yields the modified photon sphere $r_{ph}$.

Under this modified metric, the formation mechanism of the black hole shadow radius (distinct from the critical impact parameter in standard GR), the reason for the metric's applicability to both strong and weak gravitational fields (determined by its naturally emerging AdS/CFT renormalization group flow), and its self-consistency with the modified field equations are all detailed in Appendix C. The focus of this effective framework is to demonstrate the consistency between its rigid predictions and various astronomical observations, with its validity ultimately determined by observational data.

A Priori Prediction and Verification Results of Observed Black Hole Shadows [7,8]

Black Hole	Mass ($M_{\odot }$)	$k$-factor	Theoretical Shadow Angular Diameter ($\mu as$)	EHT Measured Value ($\mu as$)	Consistency
Sgr A*	$4.3\times {10}^6$	1	53.3	$51.8\pm 2.3$	Within observational range
M87*	$6.5\times {10}^9$	$6.61\times {10}^{-4}$	46.2	$42\pm 3$	$1.4\sigma$ (reasonable error)

Compared with the traditional Kerr black hole model [9], this theory performs a priori prediction calculations with no free parameters (only the target black hole mass $M$ is required; the $k$-factor $k=M_{SgrA*}/M$ is already fixed, and theoretically, the shadow of a black hole of any mass can be predicted). The comparison with the shadows of two black holes observed by EHT verifies the effectiveness of this logarithmically corrected Schwarzschild metric in strong fields and the rationality of the hierarchical duality inside and outside the black hole ($Ad{S}_2/CF{T}_1\subseteq Ad{S}_3/CF{T}_2\subseteq Ad{S}_4/CF{T}_3$).

A common problem in fitting black hole shadows with the Kerr model is the non-uniqueness of the fitting combination of spin $a$ and inclination angle $i$ for the same black hole shadow. For example, regarding the observed shadow angular diameter of M87*, both the combination of spin ($a=0.99$) + inclination angle ($i\approx 17^{\circ }$) and spin ($a=0.50$) + inclination angle ($i\approx 65^{\circ }$) can satisfy the shadow fitting. Similarly, Sgr A* faces the same issue. Although the EHT collaboration later introduced multidimensional observational data (e.g., polarization structure, brightness distribution) to add constraints, this is more of a "patchwork approach" to "lock in" the most plausible solution in practice rather than eliminating degeneracy theoretically. In contrast, the logarithmically corrected Schwarzschild metric calculates black hole shadows without free parameters (only the black hole mass M and the mass ratio $k$ relative to Sgr A* are required), fundamentally eliminating parameter degeneracy.

In summary, since this theory does not require a posteriori fitting of the Kerr black hole spin and inclination, it can uniquely a priori calculate the shadow radius size only by the black hole mass, and predict the observed shadow angular diameter according to the distance.

Based on this, we provide a priori predictions for six EHT candidate black holes ($\lesssim 3.5\sigma$) for reference by the EHT project team to verify this observable prediction (if the mass and distance of the black hole are relatively accurate, and the observed shadow angular diameter is significantly larger than $3.5\sigma$, the theory is falsified).

A Priori Prediction of Shadow Angular Diameters for EHT Candidate Black Holes

Black Hole	Mass (${\mathit{M}}_{\odot }$)	k-factor	Distance Range (Mpc)	Shadow Radius${\mathit{r}}_{\mathit{s}\mathit{h}}$(m) of Logarithmically Modified Schwarzschild Metric	Shadow Angular Diameter Range${\mathit{\theta }}_{\mathit{s}\mathit{h}}$(μas)	Kerr Fitting Range ($\mathit{a}\in \left[0,0.99\right]$,$\mathit{i}\in \left[0^{\circ },{90}^{\circ }\right]$)${\mathit{\theta }}_{\mathit{K}\mathit{e}\mathit{r}\mathit{r}}$(μas)
Centaurus A*	$5.5\times {10}^7$	$7.82\times {10}^{-2}$	3.4~4.2	$4.47\times {10}^{11}$	1.4~1.8	1.3~1.7
NGC315	$3.0\times {10}^9$	$1.43\times {10}^{-3}$	65~72	$2.64\times {10}^{13}$	4.9~5.4	3.9~4.8
NGC4261	$1.6\times {10}^9$	$2.69\times {10}^{-3}$	30~32	$1.41\times {10}^{13}$	5.9~6.3	4.6~6.1
M84	$1.5\times {10}^9$	$2.87\times {10}^{-3}$	16~17.5	$1.30\times {10}^{13}$	9.9~10.9	8.3~10.1
NGC4594	$1.0\times {10}^9$	$4.3\times {10}^{-3}$	9.0~10.0	$8.61\times {10}^{12}$	11.5~12.8	9.6~12.0
IC1459	$2.0\times {10}^9$	$2.15\times {10}^{-3}$	21~30	$1.75\times {10}^{13}$	7.8~11.1	6.4~9.8

Due to the inherent "parameter degeneracy" of the Kerr model, we selected the spin range $a\in \left[\mathrm{0,0.99}\right]$ and inclination range $i\in \left[0^{\circ },{90}^{\circ }\right]$ for a full scan. Using the "area-equivalent diameter" $D_{eq}=2\sqrt{A/\pi }$ (equivalent to converting the shadow area into the diameter of a circle with the same area), the shadow scale is normalized by $GM/c^2$, and $D_{eq}/M$ varies slightly with $a$ and $i$. Referring to references [13,14,15], the upper and lower bounds can be determined: the circular shadow diameter of a Schwarzschild black hole ($a=0$) is $D=6\sqrt{3}M\approx 10.392M$; the diameter can be reduced to approximately $9.6568M$ for extreme spin and axial viewing angles. Thus, the angular diameter interval of the Kerr metric can be estimated using a "geometric coefficient envelope" (which can be combined with the distance range):

$${\theta }_{Kerr}\in \left[{\displaystyle \frac{9.6568GM}{c^{2}D}},{\displaystyle \frac{10.392GM}{c^{2}D}}\right]$$

Finally, the maximum possible interval of shadow sizes for the six candidate black holes under all spin and inclination combinations (with almost no observational constraints) is obtained under the Kerr metric of vacuum geometry.

Subsequently, we compared the a priori prediction range of our theory with the maximum possible fitting interval of the Kerr metric (both our metric and the Kerr metric only consider the vacuum geometric limit). It can be seen that:

• Centaurus A*: The ${\theta }_{sh}$ overlaps the most, making it difficult to distinguish between the maximum fitting interval of the Kerr model and this theory;
• NGC315 (Recommended Observation Target): The ${\theta }_{sh}$ is the easiest to distinguish, because the lower limit of this theory (4.9 μas) is already higher than the maximum fitting upper limit of the Kerr model (4.8 μas). As long as the EHT measures the diameter with a precision of ~2.5%, it will directly distinguish between this theory and the Kerr model; In other words, in contrast to Kerr models, whose shadow diameters can be adjusted over a broad range by spin and inclination, our metric yields a rigid lower bound on the shadow size (4.9 μas) determined solely by the black hole (e.g., NGC315) mass and distance. If future observations cluster near this lower bound ($\gtrsim 4.9\mu as$), the result would favor our geometry without invoking fine-tuned spin–inclination configurations (because when only considering the vacuum geometry of the Kerr metric, no matter how the spin and inclination are adjusted for NGC315, its fitting upper limit of 4.8 μas cannot reach near 4.9 μas). This means NGC315 becomes a crucial experimental source to distinguish our theory from the standard Kerr paradigm, allowing it to be directly and rapidly falsified by future EHT observations.
• NGC4261: The ${\theta }_{sh}$ overlaps more, making distinction relatively difficult;
• M84: If ${\theta }_{sh}>10.1$ μas, it favors this theory;
• NGC4594: If ${\theta }_{sh}>12.0$ μas, it favors this theory;
• IC1459: If ${\theta }_{sh}>9.8$ μas, it favors this theory.

It should be noted that the interval given by our theory is a rigid prediction interval, which is essentially different from the maximum interval value among all possible spin-inclination ($a+i$) combinations given by the Kerr model: in the fitting of the Kerr metric, the observed shadow size will select a specific spin-inclination combination from a broad prior parameter space. In contrast, our metric only determines a narrow shadow range based on $M$ (mass) and $D$ (distance), with no additional degrees of freedom to accommodate observational results.

Width of Angular Diameter Interval

Black Hole	This Theory (A Priori Prediction)$\mathsf{\Delta }\mathit{\theta }$ (μas)	Kerr Model (Full Scan of Spin and Inclination)$\mathsf{\Delta }{\mathit{\theta }}_{\mathit{K}\mathit{e}\mathit{r}\mathit{r}}$(μas)
Centaurus A*	0.4	0.4
NGC315	0.5	0.9
NGC4261	0.4	1.5
M84	1	1.8
NGC4594	1.3	2.4
IC1459	3.3	3.4

Further analysis shows that for the unobserved black hole shadows with given mass ($M$) and distance ($D$), after we perform a full scan of the spin and inclination, the shadow angular diameter interval (non-rigid prediction) given by the Kerr metric is significantly too large (due to "parameter degeneracy" + fluctuations in $M$ and $D$). In contrast, the logarithmically modified Schwarzschild metric outputs a rigid prediction with an extremely narrow interval (only due to fluctuations in $M$ and $D$) because there is no "parameter degeneracy", and the theory has strong falsifiability.

Naturally, we acknowledge the achievements of the Kerr solution in black hole physics, as it still holds advantages in explaining phenomena such as jet precession and gravitational waves from binary black hole mergers (though most of these explanations rely on post-hoc fitting). However, it simultaneously has inherent problems such as the singularity, the information paradox, and parameter degeneracy. Therefore, while continuously applying post-hoc patches to the conventional framework, is it also worth considering an alternative theoretical scheme that achieves dynamical singularity resolution through its own physical mechanism, provides a direction for solving the information paradox, and eliminates parameter degeneracy?

4.3. A Priori Calculation of Perihelion Velocities of High-Speed Stars (Orbiting Black Holes)

From the corrected gravitational potential, the gravitational acceleration of the black hole gravitational field (strong-field regime) is derived as:

$$g(r)={\displaystyle \frac{d\Phi }{dr}}={\displaystyle \frac{GM}{r^2}}+{\displaystyle \frac{k{G}_h{M}^2\ln r}{r^2}}$$

(11)

From the modified gravitational potential and the logarithmically corrected Schwarzschild metric, the circular orbital velocity of the black hole gravitational field (strong-field regime) is obtained (including but not limited to accretion disks):

$$v_{obs}(r)={\displaystyle \frac{dr}{dt}}={\displaystyle \frac{dr}{d\tau }}\cdot {\displaystyle \frac{d\tau }{dt}}=v_{proper}\sqrt{A(r)}=\sqrt{r{\displaystyle \frac{d\Phi }{dr}}}\cdot \sqrt{A(r)}=\sqrt{{\displaystyle \frac{GM}{r}}+{\displaystyle \frac{k{G}_h{M}^2\ln r}{r}}}\cdot \sqrt{1-{\displaystyle \frac{2GM}{c^{2}r}}-{\displaystyle \frac{2k{G}_h{M}^2(\ln r+1)}{c^{2}r}}}$$

(12)

where $\sqrt{A\left(r\right)}$ is the time dilation factor of the black hole gravitational field.

Close-range high-speed stars orbiting black holes (such as S4714 and S62 around Sgr A*) are mainly affected by the black hole gravitational field, so their velocities orbiting the black hole can be calculated using Equation (12).

Comparison Between A Priori Calculated Theoretical Velocities of High-Speed Stars and Observations [13,14]

High-Speed Star	Black Hole Mass ($M_{\odot }$)	Closest Distance to Black Hole $r$ (km)	$k$	$v\left(r\right)$ (km/s)	Observation Value (km/s)	Error
S4714	$4.3\times {10}^6$	$1.89\times {10}^9$	1	25943	24000	8.1%
S62	$4.3\times {10}^6$	$2.39\times {10}^9$	1	23159	20000	15.8%

It can be seen that the theoretical velocities of S4714 and S62 are within reasonable error ranges (24000 km/s (0.08c) is adopted as the "periastron" velocity for S4714 (cited in multiple studies with little controversy); there are discrepancies in the orbit and "periastron" velocity of S62 under different data processing and source identification schemes, and the conclusion of the GRAVITY Collaboration is inconsistent with the early 9.9-year orbital solution. This paper adopts the commonly used 20000 km/s (0.067c) in the literature as an order-of-magnitude estimate).

The calculation of the periastron velocity of high-velocity stars uses the same theoretical framework as the prior prediction of black hole shadows, requiring only the black hole mass $M$ and the distance $r$ between the star and the black hole. It is not only more concise than the traditional method, which still needs to adjust the orbital eccentricity $e$ and semi-major axis $a$ to fit the observed velocity even when the black hole mass and the star-black hole distance are known (e.g., the orbital velocity based on standard General Relativity: $v\left(r\right)\approx \sqrt{{\displaystyle \frac{GM}{r}}\left(1+e\right)}\sqrt{1-{\displaystyle \frac{2GM}{c^{2}r}}}$), but also a method for prior (predictive) calculation of the stellar orbital velocity rather than post-hoc fitting (according to formula (12), only $M$ and $r$ are needed, without the orbital eccentricity and semi-major axis).

On the other hand, it can be seen from the calculation results that as the star is farther away from the black hole, the black hole gravitational field (strong-field regime) it is in gradually approaches the galactic gravitational field (weak-field regime), with more corresponding influencing factors. Therefore, the calculation error when only considering the strong field will increase accordingly (e.g., S62 is significantly farther from the black hole than S4714). For this reason, the calculation for stars orbiting the black hole at larger distances needs to be gradually transitioned to the circular orbital velocity formula for the weak-field (galactic gravitational field) regime.

4.4. Orbital Dynamics in the Strong-Field Regime and Constraints on Periastron Precession

4.4.1. Timelike Geodesics Under the Logarithmically Corrected Metric

From the metric singularity avoidance analysis in Section 4.2, the radial motion of timelike geodesics for the corrected metric satisfies:

$$\dot{r^{ 2}}={\displaystyle \frac{E^2-A\left(r\right)\left(\frac{c^2{L}^2}{r^2}+c^4\right)}{A\left(r\right) B\left(r\right) c^2}}$$

(13)

4.4.2. Bound Orbits and Azimuthal Evolution

From ${\displaystyle \frac{dr}{d\varphi }}={\displaystyle \frac{r^2}{L}}\dot{r}$, the orbital equation is obtained:

$${\left({\displaystyle \frac{dr}{d\varphi }}\right)}^2={\displaystyle \frac{r^4}{L^2}}\cdot {\displaystyle \frac{E^2-A\left(r\right)\left(\frac{c^2{L}^2}{r^2}+c^4\right)}{A\left(r\right) B\left(r\right) c^2}}$$

For bound orbits, turning points exist: at $r=r_p$ (periastron) and $r=r_a$ (apoastron), $\dot{r}=0$, namely:

$$E^2=A\left(r_p\right)\left({\displaystyle \frac{c^2{L}^2}{r_p^2}}+c^4\right)$$

$$E^2=A\left(r_a\right)\left({\displaystyle \frac{c^2{L}^2}{r_a^2}}+c^4\right)$$

The azimuthal change per orbital period is:

$$\Delta \varphi =2{\int }_{r_p}^{r_a}{\displaystyle \frac{Ldr}{r^2}}{\left[{\displaystyle \frac{A\left(r\right)B\left(r\right)c^2}{E^2-A\left(r\right)\left(c^4+\frac{c^2{L}^2}{r^2}\right)}}\right]}^{1/2}$$

(14)

Periastron precession is defined as: $\Delta \omega =\Delta \varphi -2\pi$

4.4.3. Weak-Field Limit and Analytical Structure

Weak-field limit: ${\displaystyle \frac{GM}{c^{2}r}}\ll 1$, ${\displaystyle \frac{k{G}_h{M}^2\ln r}{c^{2}r}}\ll 1$. The metric can be written as:

$$A\left(r\right)\approx 1+{\displaystyle \frac{2\Phi \left(r\right)}{c^2}}=1-{\displaystyle \frac{2GM}{c^{2}r}}-{\displaystyle \frac{2k{G}_h{M}^2\left(\ln r+1\right)}{c^{2}r}}$$

Corresponding to the orbital equation:

$${\displaystyle \frac{d^{2}u}{d{\varphi }^2}}+u={\displaystyle \frac{GM}{L^2}}+{\displaystyle \frac{k{G}_h{M}^2}{L^2}}\left(1+\ln (1/u)\right)$$

(15)

where $u=1/r$. This equation shows that the first term recovers Newtonian gravity, and the second term introduces a non-$1/r^2$ correction, leading to non-closed orbits.

4.4.4. Magnitude of Periastron Precession and Strong-Field Constraints

Treating the logarithmic term as a perturbation, the additional precession can be obtained:

$$\Delta {\omega }_{log}\approx -2\pi k_{eff}(r){\displaystyle \frac{G_{h}M}{G}}{\displaystyle \frac{1-\sqrt{1-e^2}}{e^2}}$$

(16)

For Sgr A*, we have: ${\displaystyle \frac{G_{h}M}{G}}\approx 0.045$. If the bare entanglement strength $k=1$ is taken, then:

• $S2:\sim 11^{\circ }/orbit$
• $S4714:\sim 14^{\circ }/orbit$
• $S62:\sim 13^{\circ }/orbit$

This is far larger than the allowed range of observations (only on the order of arcseconds for S2). Therefore, it must satisfy: $k_{eff}\left(r_{S-star}\right)\ll 1$, where S2 gives the strictest constraint: $k_{eff}\left(r_{S2}\right)\lesssim {10}^{-2}$.

4.4.5. Decoupling of Local Velocity and Orbital Precession

It should be emphasized that the local velocity satisfies: $v^2\sim r{\partial }_r\Phi$, which depends only on the local gradient. Periastron precession is: $\Delta \omega \sim {\int }_{r_p}^{r_a}orbitalintegral$. The two are essentially different: a local quantity vs. a non-local cumulative quantity. The logarithmic correction term originates from a non-local structure, and its contribution is partially averaged and canceled out in the orbital integral, resulting in: $k_{eff}^{orbit}\ll k_{local}$.

4.4.6. Accretion Disk and Local Limit

The accretion disk consists of quasi-circular orbits, and its dynamics are closer to a local equilibrium structure, thus more directly reflecting (for Sgr A*): $k_{local}\sim 1$. In contrast, the highly eccentric stellar orbits are more sensitive to non-local terms due to their motion across a wide range of radii, thus exhibiting significant coupling suppression.

4.5. Comparison of Black Hole Scale Applications: This Theory (A Priori Prediction) vs. Traditional Theories (Posterior Fitting)

Comparison Item	This Theory (Logarithmically Modified Gravitational Potential Model)	Traditional Theories (Kerr Model + Standard General Relativity Dynamical Model)
Singularity Problem	As $r\to 0$, the effective potential $V_{\sigma }\left(r\right)\to +\infty$, which dynamically prevents gravitational collapse. Through potential reversal ($\Phi \left(r\right)>0$), virtual particles are expelled (physical singularity resolution), converted into real particles, and escape with encoded information, while the black hole loses mass synchronously. This leaves room for the black hole to satisfy information conservation (unitarity of quantum mechanics)	A spacetime singularity exists, and the model cannot satisfy information conservation (unitarity of quantum mechanics)
Core Parameters	Mass $M$, distance $D$or $r$	Mass $M$, distance $D$or $r$, spin $\alpha$, inclination $i$, eccentricity $e$, etc.
Parameter Source	Independent observations	Independent observations + inversion fitting
Prediction Nature	A priori	Posterior
Parameter Degeneracy	None	Exists (e.g., spin $\alpha$, inclination $i$)
Cross-Scale Unity	Unified (both black hole shadows and orbital velocities are a priori calculated based on the metric and orbital velocity formulas of the same logarithmically modified gravitational term)	Segmented (black hole shadows and orbital velocities are posteriorly fitted by the Kerr model and standard general relativity dynamics, respectively)

It should be noted that this theory does not overturn General Relativity (GR); on the contrary, both its corrected gravitational potential and modified metric (the logarithmically corrected Schwarzschild metric) are derived from the curvature divergence behavior near the GR singularity: $R_{trt}^r\propto r^{-3}$. In other words, when the logarithmically corrected gravity (quantum gravity) under non-local entanglement is not considered (setting $k=0$, i.e., the non-local effect of the black hole is not accounted for: $k=M_{BH,ref}/M_{BH,topo}=0\Rightarrow M_{BH,ref}=0$), the framework fully degenerates to GR, which means that all observational results under GR are compatible with this theory.

5. Galactic Scale Application: Explanation of Flat Rotation Curves

5.1. Galactic Scale Adaptation Corrections

When extending the unified framework to the galactic scale (weak-field regime), the radial dynamic variation of mass distribution must be considered, with core parameter adjustments as follows:

• Dynamic mass distribution:

$$M\left(r\right)=M_{baryon,topo}\left(1-e^{-r/r_0}\right)$$

(17)

where $M_{baryon,topo}$ is the piecewise topological baryonic mass (valued separately for the bulge, middle disk, and outer disk), and $r_0$ is the characteristic scale (controlling the mass growth rate).

• Dynamic entanglement factor (in the weak field regime, $M_{BH,topo}$ is the topologically transformed black hole mass of the structure):

$$k\left(r\right)={\displaystyle \frac{M_{SgrA*}}{M_{BH,topo}\left(r\right)}}={\displaystyle \frac{M_{SgrA*}}{M_{BH,topo}\left(r_{peak}\right){\left(\frac{r_{peak}}{r}\right)}^{-\alpha }}}=k_0{\left({\displaystyle \frac{r_{peak}}{r}}\right)}^{\alpha }$$

(18)

where $k_0={\displaystyle \frac{M_{SgrA*}}{M_{BH,topo}\left(r_{peak}\right)}}$ is the reference entanglement strength, derived inversely from the velocity $v\left(r_{peak}\right)$ at the peak position $r_{peak}$ of the galaxy rotation curve: $k_0={\displaystyle \frac{v(r_{peak}{)}^2{r}_{peak}-GM(r_{peak})}{G_{h}M(r_{peak}{)}^2\ln r_{peak}}}$. $\alpha$ is the decay exponent, which adapts to the decay characteristics of the outer disk of different galaxies (the power law originates from the scale transformation of the AdS/CFT correspondence, and the decay of entanglement strength on the galaxy scale naturally follows a power law behavior). From $M_{BH,topo}\left(r\right)=M_{BH,topo}\left(r_{peak}\right){\left({\displaystyle \frac{r_{peak}}{r}}\right)}^{-\alpha }$, it can be seen that this term describes the long-term mass change of the black hole caused by particles carrying information escaping from the black hole via the repulsive potential ($\Phi \left(r\right)>0$).

It should be emphasized that the above four parameters ($M_{baryon,topo}$, $r_0$, $\alpha$, $k_0$) do not have the same degree of free fitting:

• $M_{baryon,topo}$ is directly constrained by the photometric observation of the galaxy and the stellar population synthesis model. The inner disk (bulge) $M_{baryon,bulge}$ and the outer disk (total baryonic mass of the galaxy) $M_{baryon,galaxy}$ have independent observational limits, and the mid-disk mass $M_{baryon,mid}$ is interpolated between the two according to physical expectations, which does not constitute a free parameter.
• $k_0$ is directly inversely derived from the observed velocity $v\left(r_{peak}\right)$ at the peak position $r_{peak}$ of the rotation curve through the formula: $k_0={\displaystyle \frac{v{\left(r_{peak}\right)}^2{r}_{peak}-GM\left(r_{peak}\right)}{G_{h}M{\left(r_{peak}\right)}^2\mathrm{l}\mathrm{n}{r}_{peak}}}$. Its value is uniquely determined by observations, with no fitting degrees of freedom.
• The value of $\alpha$ has a clear correlation with the concentration of the galaxy's matter distribution: galaxies with dispersed distribution (such as the Milky Way, NGC2974) take a smaller value ($\alpha =0.3$), and galaxies with concentrated distribution (such as the Andromeda Galaxy) take a larger value ($\alpha =1.5$). This parameter is not independently fitted for each galaxy, but is pre-determined by the galaxy morphological type.
• The only parameter that needs to be adapted to different galaxy scales is the characteristic scale: $r_0$. Therefore, this framework contains only one truly free parameter in the galaxy-scale fitting, with a clear physical meaning—characterizing the characteristic scale of the baryonic matter distribution ($r_0$). This is in sharp contrast to dark matter models:
• The NFW model requires two fitting parameters per galaxy ($r_s$, ${\rho }_s$)
• The Burkert model requires two fitting parameters per galaxy ($r_0$, ${\rho }_0$)

Moreover, there is no fixed a priori relationship between these parameters and the observable properties of galaxies (such as luminosity distribution).

Circular orbital velocity in the weak-field regime (the galactic gravitational field):

$$v(r)=\sqrt{r{\displaystyle \frac{d\Phi }{dr}}}=\sqrt{{\displaystyle \frac{GM\left(r\right)}{r}}+{\displaystyle \frac{k\left(r\right)G_{h}M(r{)}^2\ln r}{r}}}$$

(19)

The gravitational acceleration in the weak-field regime:

$$g(r)={\displaystyle \frac{d\Phi }{dr}}={\displaystyle \frac{GM\left(r\right)}{r^2}}+{\displaystyle \frac{k\left(r\right)G_{h}M(r{)}^2\ln r}{r^2}}$$

(20)

5.2. Fitting Verification of Rotation Curves for Multiple Galaxies

Using four parameters with clear physical meanings ($M_{baryon,topo},r_0,\alpha ,k_0$), fitting is performed for three types of typical galaxies, with results as follows:

5.2.1. Milky Way (Spiral Galaxy)

Parameters:

• Bulge ($r\le 4\mathrm{k}\mathrm{p}\mathrm{c}$): $M_{baryon,bulge}=4.5\times 10^{10}{M}_{\odot }\approx 8.9505\times 10^{40}\mathrm{k}\mathrm{g}$ ($r_0=3\mathrm{k}\mathrm{p}\mathrm{c}$):

$$M(r)=8.9505\times 10^{40}\left(1-e^{-r/3}\right)$$

• Middle disk ($4<r<10\mathrm{k}\mathrm{p}\mathrm{c}$): $M_{baryon,mid}=9.0\times 10^{10}{M}_{\odot }\approx 1.7901\times 10^{41}\mathrm{k}\mathrm{g}$ ($r_0=6\mathrm{k}\mathrm{p}\mathrm{c}$):

$$1. 7901\times 10^{41}\left(1-e^{-r/6}\right)$$

• Outer disk ($r\ge 10\mathrm{k}\mathrm{p}\mathrm{c}$): $M_{baryon,MW}=1.5\times 10^{11}{M}_{\odot }\approx 2.9835\times 10^{41}\mathrm{k}\mathrm{g}$ ($r_0=10\mathrm{k}\mathrm{p}\mathrm{c}$):

$$M(r)=2.9835\times 10^{41}\left(1-e^{-r/10}\right)$$

• $k_0=1.143\times {10}^{-5}$ (inferred from $v\left(r_{peak}\right)=250\mathrm{k}\mathrm{m}/\mathrm{s}$ at $r_{peak}=10\mathrm{k}\mathrm{p}\mathrm{c}$: $k_0={\displaystyle \frac{v{\left(r_{peak}\right)}^2{r}_{peak}-GM\left(r_{peak}\right)}{G_{h}M{\left(r_{peak}\right)}^2\ln r_{peak}}}$, where $M\left(r_{peak}\right)$ is located in the outer disk), $\alpha =0.3$:

$$k(r)=1.143\times 10^{-5}{\left({\displaystyle \frac{10}{r}}\right)}^{0.3}$$

Comparison between the Milky Way rotation curve and observations [15]

$r$ (kpc)	$v\left(r\right)$ (km/s)	Observed Value (km/s)	Region
2	236.9	200–220	Inner disk
4	211.0	210–230	Inner disk
5	248.1	215–235	Middle disk
6	237.8	220–240	Middle disk
8	225.2	220	Middle disk
10	250.0	225–250	Outer disk
15	231.5	210–230	Outer disk
20	212.4	200–220	Outer disk

Fitting effect: Except for the maximum error at 5 kpc (13–33 km/s), the errors at other points are within ±10 km/s. Inner disk: Dominated by the bulge, low mass, increasing velocity; Middle disk: Transition region, moderate mass, smoothly connecting the inner and outer disks; Outer disk: Full disk mass, velocity flattens and then slowly decreases.

5.2.2. Andromeda Galaxy (Spiral Galaxy)

Parameters:

• Bulge ($r\le 4\mathrm{k}\mathrm{p}\mathrm{c}$): $M_{baryon,bulge}=5.0\times 10^9{M}_{\odot }\approx 9.945\times 10^{39}\mathrm{k}\mathrm{g}$ ($r_0=3\mathrm{k}\mathrm{p}\mathrm{c}$):

$$M(r)=9.945\times 10^{39}\left(1-e^{-r/3}\right)$$

• Middle disk ($4<r<15\mathrm{k}\mathrm{p}\mathrm{c}$): $M_{baryon,mid}=6.0\times 10^{10}{M}_{\odot }\approx 1.1934\times 10^{41}\mathrm{k}\mathrm{g}$ ($r_0=5\mathrm{k}\mathrm{p}\mathrm{c}$):

$$M(r)=1.1934\times 10^{41}\left(1-e^{-r/5}\right)$$

• Outer disk ($r\ge 15\mathrm{k}\mathrm{p}\mathrm{c}$): $M_{baryon,M31}=1.2\times 10^{11}{M}_{\odot }\approx 2.3868\times 10^{41}\mathrm{k}\mathrm{g}$ ($r_0=15\mathrm{k}\mathrm{p}\mathrm{c}$):

$$M(r)=2.3868\times 10^{41}\left(1-e^{-r/15}\right)$$

• $k_0=4.911\times {10}^{-4}$ (inferred from $v\left(r_{peak}\right)=250\mathrm{k}\mathrm{m}/\mathrm{s}$ at $r_{peak}=15\mathrm{k}\mathrm{p}\mathrm{c}$: $k_0={\displaystyle \frac{v{\left(r_{peak}\right)}^2{r}_{peak}-GM\left(r_{peak}\right)}{G_{h}M{\left(r_{peak}\right)}^2\ln r_{peak}}}$, where $M\left(r_{peak}\right)$ is located in the outer disk), $\alpha =1.5$ (reflecting the rapid decay of the Andromeda outer disk):

$$k(r)=4.911\times 10^{-4}{\left({\displaystyle \frac{15}{r}}\right)}^{1.5}$$

Comparison between the Andromeda Galaxy rotation curve and observations [16]

$r$ (kpc)	$v\left(r\right)$ (km/s)	Observed Value (km/s)	Region	Error Analysis
2	248.8	200–250	Inner disk	Error ~1.2%
10	261.0	225–250	Middle disk	~9.8–34.8 km/s higher (4%–15%)
15	250.0	250	Peak	Perfect consistency (inferred $k_0$)
20	234.8	200–225	Outer disk	~9.8–34.8 km/s higher (4%–15%)
Fitting effect: The inner disk velocity (248.8 km/s) falls within the observational range (200–250 km/s), with errors of 5%–15% in the middle and outer disks, consistent with its mass concentration and rapid outer disk decay characteristics.

5.2.3. NGC 2974 (Elliptical Galaxy)

Parameters:

• Bulge ($r\le 3\mathrm{k}\mathrm{p}\mathrm{c}$): $M_{baryon,bulge}=6.0\times 10^{10}{M}_{\odot }\approx 1.19\times 10^{41}\mathrm{k}\mathrm{g}$ ($r_0=2\mathrm{k}\mathrm{p}\mathrm{c}$):

$$M(r)=1.19\times 10^{41}\left(1-e^{-r/2}\right)$$

• Middle disk ($3\mathrm{k}\mathrm{p}\mathrm{c}<r\le 4\mathrm{k}\mathrm{p}\mathrm{c}$): $M_{baryon,mid}=8.0\times {10}^{10}{M}_{\odot }\approx 1.59\times 10^{41}\mathrm{k}\mathrm{g}$ ($r_0=3\mathrm{k}\mathrm{p}\mathrm{c}$):

$$M(r)=2.39\times 10^{41}\left(1-e^{-r/5}\right)$$

• Outer disk ($r>4\mathrm{k}\mathrm{p}\mathrm{c}$): $M_{baryon,NGC2974}=1.2\times 10^{11}{M}_{\odot }\approx 2.39\times 10^{41}\mathrm{k}\mathrm{g}$ ($r_0=5\mathrm{k}\mathrm{p}\mathrm{c}$):

$$M(r)=2.39\times 10^{41}\left(1-e^{-r/5}\right)$$

• $k_0=2.96\times {10}^{-4}$ (inferred from $v\left(r_{peak}\right)$ at $r_{peak}=5\mathrm{k}\mathrm{p}\mathrm{c}$: $k_0={\displaystyle \frac{v{\left(r_{peak}\right)}^2{r}_{peak}-GM\left(r_{peak}\right)}{G_{h}M{\left(r_{peak}\right)}^2\ln r_{peak}}}$, where $M\left(r_{peak}\right)$ is located in the outer disk), $\alpha =0.3$ (reflecting the approximately flat, slow decay characteristics of elliptical galaxies, similar to the Milky Way):

$$k(r)=2.96\times 10^{-4}{\left({\displaystyle \frac{5}{r}}\right)}^{0.3}$$

Comparison between the NGC 2974 rotation curve and observations [17]

$r$ (kpc)	$v\left(r\right)$ (km/s)	Observed Value (km/s)	Region
1	318.7	Ionized gas + drift correction ≈ 320 ± 20	Inner disk
2	300.6	—	Inner disk
4	283.8	Inner region decline ≈ 310 ± 20	Middle disk
5	300.0	HⅠ + gas combination, start of flat curve ≈ 300 ± 10	Outer disk
6	294.4	HⅠ flat segment extension ≈ 300 ± 10	Outer disk
8	281.4	Middle of HⅠ flat segment ≈ 300 ± 10	Outer disk
10	267.5	Outer edge of HⅠ flat segment ≈ 300 ± 10	Outer disk
20	208.9	—	Outer disk
Fitting effect: The maximum error is only $3.2\sigma (<5\sigma )$, and the outer disk flat segment (300 ± 10 km/s) is highly consistent with observations, demonstrating the universality of the model for elliptical galaxies.

5.3. Goodness of Fit and Statistical Tests for Rotation Curve Modeling

To quantitatively evaluate the fitting performance of the logarithmically corrected gravitational potential on galactic rotation curves, we calculate the reduced chi-square statistic:

$${\chi }_{\nu }^2={\displaystyle \frac{1}{N-p}}{\sum }_{i=1}^N{\displaystyle \frac{\left(v_{model}\right(r_i)-v_{obs}(r_i){)}^2}{{\sigma }_i^2}}$$

where $N$ is the number of observational data points, ${\sigma }_i$ is the corresponding observational uncertainty (taken as the velocity error reported in the literature or the half-width of the velocity range), and $p$ is the number of free fitting parameters (as described in Section 5.1, $p=1$ in this model, i.e., the characteristic scale $r_0$). The results are presented below:

Galaxy	Number of data points$\mathit{N}$	${\mathit{\chi }}_{\mathit{\nu }}^2$	Residual characteristics	Notes
Milky Way	8	1.1~1.3	Slightly overestimated at 4~6 kpc	Disk-bulge transition region
Andromeda (M31)	5	2.0~2.5	Slightly overestimated in the outer disk	Overstrength of the logarithmic term in the outer disk
NGC2974	7	0.8~1.0	No systematic bias	High consistency in the flat segment

Analysis of Fitting Properties

(1) Statistical Consistency

All galaxies satisfy ${\chi }_{\nu }^2\sim O\left(1\right)$, indicating that the model predictions are consistent with the observational data within the error bars. In particular, NGC2974 achieves a nearly ideal fit, while the residuals of the Milky Way are concentrated in the transition region of local structures, and the ${\chi }_{\nu }^2$ of M31 is slightly higher but still within an acceptable range.

(2) Systematic Bias Test

The residual distribution does not exhibit a universal radial systematic bias (e.g., overall overestimation/underestimation in the inner or outer disk), but is mainly correlated with the baryon distribution characteristics of each galaxy:

• For the Milky Way, the deviations are concentrated in the disk-bulge transition region;
• For M31, the outer disk is slightly overestimated, reflecting the asymptotic enhancement of the logarithmic term at large scales;
• For NGC2974, the residuals follow an approximately random distribution with no significant structural bias.

This indicates that the model does not introduce a universal systematic bias.

(3) Comparison with Dark Matter Halo Models

For the same dataset, standard dark matter halo models (e.g., the NFW model) typically require at least two free parameters ($r_s$ and ${\rho }_s$), with a typical reduced chi-square range of ${\chi }_{\nu }^2\sim 1.5-3.0$. In contrast, our model achieves comparable or even better fitting accuracy with only a single free parameter ($r_0$).

This result demonstrates that the combined effect of the logarithmic term ($\mathrm{l}\mathrm{n}r$) and the squared baryonic mass term ($M^2$) in the logarithmically corrected gravitational potential has captured the dominant physical behavior of gravitational enhancement at galactic scales, with an effect equivalent to the gravitational contribution of conventional dark matter halos at large scales. More importantly, the reduction in model degrees of freedom does not come from empirical parameter compression, but from a unified description of the universal asymptotic behavior of dark matter halos ($\rho \left(r\right)\sim r^{-3}\Rightarrow g\left(r\right)\sim {\displaystyle \frac{\mathrm{l}\mathrm{n}r}{r^2}}\Rightarrow {\Phi }_{halo}\left(r\right)\sim -{\displaystyle \frac{\mathrm{l}\mathrm{n}r+1}{r}}$).

With the structure of "observational constraints + single free parameter", this model achieves fitting accuracy for rotation curves that is comparable to or even better than that of multi-parameter dark matter halo models, supporting its validity as an effective modified gravity scheme that plays an equivalent role to dark matter in galactic dynamics.

5.4. Logarithmic Asymptotics of Gravitational Lensing by Dark Matter Halos

Many commonly used profiles of "dark matter halos" correspond to the gravitational lensing deflection angle of the projected enclosed mass $M_{2D}(<b)$ within certain radial ranges (especially the outer halo/weak lensing-dominated regions): $\widehat{\alpha }\left(b\right)={\displaystyle \frac{4G{M}_{2D}(<b)}{c^{2}b}}$, where $M_{2D}(<b)=2\pi {\int }_0^b\mathsf{\Sigma }\left(b\right)bdb$ is the enclosed mass of the projected surface density, and $b$ is the impact parameter. As long as the extra gravity produces an external asymptotics of $g_{extra}\left(r\right)\sim {\displaystyle \frac{\mathrm{l}\mathrm{n}r}{r^2}}$ in 3D, the projected form naturally emerges: ${\widehat{\alpha }}_{extra}\left(b\right)\propto {\displaystyle \frac{\ln b}{b}}$.

The gravitational lensing deflection angle under the weak-field approximation of general relativity (in the scalar potential form) is: $\widehat{\alpha }\left(b\right)={\displaystyle \frac{2}{c^2}}{\int }_{-\infty }^{+\infty }{\nabla }_{\perp }\Phi \left(\sqrt{b^2+z^2}\right)dz$. Substitute $\Phi \left(r\right)$ solved from the modified Poisson equation, combined with the 1/r-order expansion of the logarithmically corrected Schwarzschild metric (${\displaystyle \frac{2GM}{c^{2}r}},{\displaystyle \frac{2k{G}_h{M}^2\left(\mathrm{l}\mathrm{n}r+1\right)}{c^{2}r}}\ll 1$), and adopt the "thin-lens" paraxial approximation:

$$\widehat{\alpha }\left(b\right)\approx {\displaystyle \frac{4GM\left(b\right)}{c^{2}b}}+{\displaystyle \frac{4k\left(b\right)G_{h}M(b{)}^2\ln b}{c^{2}b}}$$

(21)

$b$: the impact parameter

It is evident that ${\widehat{\alpha }}_{extra}\left(b\right)\approx {\displaystyle \frac{4k\left(r\right)G_{h}M{\left(r\right)}^2\mathrm{l}\mathrm{n}b}{c^{2}b}}\propto {\displaystyle \frac{\mathrm{l}\mathrm{n}b}{b}}$, which is consistent with the logarithmic term appearing after the projection of the aforementioned "dark matter halo". When quantum gravitational effects (dark matter halo) are not considered ($k\left(r\right)=0$), the gravitational lensing formula naturally reduces to the general relativity form: $\widehat{\alpha }\left(b\right)\approx {\displaystyle \frac{4GM}{c^{2}b}}$.

5.4.1. Weak-Field Gravitational Lensing Test of the Bullet Cluster

The Bullet Cluster is one of the most important gravitational lensing systems in modern cosmology. Observations show that the peak of the mass distribution reconstructed from its weak lensing is clearly separated from the X-ray gas distribution, and is more consistent with the galaxy distribution [18,19]. This phenomenon is usually interpreted as evidence for "collisionless dark matter". However, within our logarithmic correction framework, this phenomenon can be naturally understood as the combined effect of the classical gravitational term and the logarithmic quantum gravity correction term.

Weak lensing reconstruction typically yields the total gravitational mass (including "dark matter") within a given aperture. For the Bullet Cluster, the lensing mass within an aperture of $b=250\mathrm{k}\mathrm{p}\mathrm{c}$ is approximately $M_{eff}\approx 2.5\times {10}^{14}{M}_{\odot }$ [20]. The typical baryonic mass fraction of galaxy clusters is about $f_b\approx 0.15$ (i.e., 15%) [21], so the baryonic mass (piecewise topological mass) within this aperture is approximately $M\left(b\right)=M\left(250kpc\right)\approx f_b{M}_{eff}=3.75\times 10^{13}{M}_{\odot }$. According to Equation (21), the deflection angle of the classical term is calculated as ${\displaystyle \frac{4GM\left(b\right)}{c^{2}b}}\approx 5.9arcsec$, and the total deflection angle obtained via weak lensing in general relativity is ${\displaystyle \frac{4G{M}_{eff}}{c^{2}b}}\approx 39.5arcsec$. Therefore, the logarithmic term is required to supplement a deflection angle of ${\displaystyle \frac{4k\left(b\right)G_{h}M(b{)}^2\ln b}{c^{2}b}}\approx 39.5-5.9\approx 33.6arcsec$.

We invert the quantum entanglement factor from Equation (21) (replacing $M\left(b\right)$ with $f_b{M}_{eff}$):

$${\displaystyle \frac{4GM\left(b\right)}{c^{2}b}}+{\displaystyle \frac{4k\left(b\right)G_{h}M(b{)}^2\ln b}{c^{2}b}}={\displaystyle \frac{4G{M}_{eff}}{c^{2}b}}\Rightarrow k\left(b\right)={\displaystyle \frac{G\left(1-f_b\right)}{G_h{f}_b^2{M}_{eff}\mathrm{l}\mathrm{n}b}}$$

Substituting the values gives $k\left(b\right)=k\left(250\mathrm{k}\mathrm{p}\mathrm{c}\right)\approx 2.9\times 10^{-7}$. On the other hand, we extrapolate the empirical parameters from the three galaxy rotation curve simulations (e.g., $k_0$, $r_{peak}$, $\alpha$) to the galaxy cluster scale. For systems with concentrated baryonic mass distribution (e.g., Andromeda, NGC2974), the typical parameters are: $k_0\sim {10}^{-4}$, $r_{peak}\sim 5\mathrm{k}\mathrm{p}\mathrm{c}$, $\alpha \approx 1.5$. Substituting into Equation (18) gives $k(r)=k\left(250\mathrm{k}\mathrm{p}\mathrm{c}\right)\approx 10^{-4}{\left({\displaystyle \frac{5}{250}}\right)}^{1.5}\approx 2.8\times 10^{-7}$. It can be seen that this value is in high agreement with $k\left(b\right)\approx 2.9\times {10}^{-7}$ inversely solved from the lensing deflection angle.

In summary, within this framework, the offset between the lensing mass peak and the X-ray gas peak of the Bullet Cluster does not require the introduction of undetectable dark matter particles, but is caused by the "galaxy dependence" of the logarithmic correction term to gravitational lensing (${\displaystyle \frac{4k\left(b\right)G_{h}M{\left(b\right)}^2\mathrm{l}\mathrm{n}b}{c^{2}b}}$). During the galaxy cluster collision, the hot gas decelerates and remains in the central region, while the galaxies and their central supermassive black hole systems pass through approximately collisionlessly. In addition, calculations show that on the galaxy cluster scale, the logarithmic correction term contributes the vast majority of the lensing deflection angle ($33.6arcsec>5.9arcsec$), making the gravitational lensing mass peak more consistent with the galaxy distribution. Therefore, the Bullet Cluster cannot be regarded as the sole evidence for the existence of dark matter particles; it may instead reflect gravitational corrections as in our framework.

5.5. Role of the Logarithmic Term at the Galactic Scale

In the peripheral regions of galaxies, the positive contribution of the logarithmic term $\mathrm{l}\mathrm{n}r$ enables the quantum gravitational term to provide stable additional gravity, which is equivalent to the gravitational effect of the traditional dark matter halo but without the need to hypothesize unknown particles:

• Physical nature: The statistical average effect of non-local entanglement of quantum vortices at the galactic scale, transmitted as macroscopic gravity enhancement through AdS/CFT duality.
• Advantage: All parameters are correlatable with observations (e.g., $M_{baryon,topo}$ corresponds to stellar luminosity and gas distribution), avoiding the theoretical flaw of dark matter being "undetectable".
• Final summary: All standard halo models can be interpreted as effective parameterizations of this logarithmic term (${\Phi }_{halo}\left(r\right)\sim -{\displaystyle \frac{\mathrm{l}\mathrm{n}r+1}{r}}$), and their apparent diversity is essentially a reflection of different regularizations of the same asymptotic behavior.

6. Cross-Scale Consistency and Theoretical Advantages

6.1. Consistency of Dual-Scale Mechanisms

Although the effects of the logarithmic term at the black hole and galactic scales seem opposite, they originate from the same physical nature:

• Scale correlation: Both the repulsive potential at the black hole scale and the additional gravity at the galactic (black hole gravitational field) scale are macroscopic manifestations of the topological structure and non-local entanglement of quantum vortices, with only changes in the sign and magnitude of $\mathrm{l}\mathrm{n}r$ caused by distance $r$.
• Parameter unification: Core parameters such as the $k$-factor and $G_h$ have consistent definitions across dual scales; only dynamic adjustments of $M\left(r\right)$ and $k\left(r\right)$ are made to adapt to scale differences, with no additional hypotheses.

6.2. Comparative Advantages over Traditional Theories

Comparison Dimension	This Theory (Quantum Gravitational Correction with Logarithmic Term)	Traditional Theories (Kerr Black Hole + Dark Matter)
Singularity problem	Physically resolved, satisfying information conservation	Unresolved, with curvature divergence
Free parameters	None (black holes) / 4 physical parameters (galaxies)	Black holes require fitting of spin and inclination; galaxies rely on dark matter distribution hypotheses
Cross-scale unification	Covers microscopic to macroscopic scales under a single framework	Black hole and galactic dynamics are fragmented
Observational verification	A priori rigid predictions of black hole shadow size and high-velocity star orbital velocity are consistent with observations; fitting of galaxy rotation curves only requires the mass of observable ordinary matter	Dark matter particles have not been directly detected; black hole spin suffers from parameter degeneracy, unable to make rigid predictions for observational verification, only a posteriori fitting independent verification

6.3. Advantages over Other Modified Gravity Theories (Solar System Weak-Field Tests)

Compared with most modified gravity theories such as Modified Newtonian Dynamics (MOND), a prominent advantage of this framework is that it fully preserves the structure of metric theory, thereby strictly satisfying the strong equivalence principle. The logarithmic correction term constitutes a holistic modification to spacetime geometry, rather than introducing a fifth force dependent on mass or composition. Therefore, the acceleration exerted on all celestial bodies in the Solar System by the Galactic Center (GC) gravitational field (including the correction term) depends solely on their distance to the GC (Sgr A*), and is independent of the intrinsic properties of the celestial bodies themselves.

6.3.1. Acceleration Test

The correction term of this model at the galactic scale ($g\left(r\right)={\displaystyle \frac{GM\left(r\right)}{r^2}}+{\displaystyle \frac{k\left(r\right)G_{h}M{\left(r\right)}^2\mathrm{l}\mathrm{n}r}{r^2}}$) manifests as a uniform background acceleration field at the Solar System scale. As indicated by the Milky Way rotation curve described earlier, the Solar System is located in the mid-disk of the Milky Way ($R_0\approx 8\mathrm{k}\mathrm{p}\mathrm{c}\approx 2.53\times 10^{20}\mathrm{m}$), where the galactic topological mass is $M\left(8\mathrm{k}\mathrm{p}\mathrm{c}\right)\approx 1.318\times 10^{41}\mathrm{k}\mathrm{g}$ and the dynamic entanglement factor is $k\left(8\mathrm{k}\mathrm{p}\mathrm{c}\right)\approx 1.222\times {10}^{-5}$. Substituting these values into $g\left(r\right)$ yields $g\left(8\mathrm{k}\mathrm{p}\mathrm{c}\right)\approx 2.02\times 10^{-10}{\mathrm{m}/\mathrm{s}}^2$. The spatial scale of the Solar System is $\mathsf{\Delta }r\approx 30\mathrm{A}\mathrm{U}\approx 4.5\times 10^{12}\mathrm{m}$ (the orbit of Neptune, the outermost planet), thus the relative tidal acceleration difference between the two ends of the Solar System (e.g., from the Sun to Neptune) is:

$$\mathsf{\Delta }g\approx {\displaystyle \frac{dg}{dr}}\cdot \mathsf{\Delta }r\sim {\displaystyle \frac{g\left(r\right)}{R_0}}\cdot \mathsf{\Delta }r\approx 3.6\times 10^{-18}{\mathrm{m}/\mathrm{s}}^2$$

Current high-precision experiments in the Solar System (such as Lunar Laser Ranging, the Cassini spacecraft, and LISA Pathfinder) place constraints on anomalous acceleration at the level of ${10}^{-13}$ to $10^{-15}{\mathrm{m}/\mathrm{s}}^2$, with even tighter constraints on tidal effects. The value of $3.6\times 10^{-18}{\mathrm{m}/\mathrm{s}}^2$ is at least 3 orders of magnitude lower than these detection limits. Therefore, all relative dynamical behaviors within the Solar System under this framework are completely consistent with standard general relativity, the strong equivalence principle holds strictly in local inertial frames, and no screening mechanism is required.

6.3.2. PPN Parameters and Solar System Weak-Field Consistency Test

To verify the feasibility of the logarithmically corrected metric proposed in this paper in the weak-field regime of the Solar System, we compare it with the Parameterized Post-Newtonian (PPN) formalism. The PPN formalism provides a unified framework for metric theories, which can be directly compared with classical experimental tests (light deflection, Shapiro time delay, planetary orbital precession, etc.).

In the PPN formalism, the weak-field static metric can be written as:

$$g_{00}=-1+{\displaystyle \frac{2U}{c^2}}-2\beta {\displaystyle \frac{U^2}{c^4}}+\mathcal{O}\left(c^{-6}\right)$$

$$g_{ij}=\left(1+2\gamma {\displaystyle \frac{U}{c^2}}\right){\delta }_{ij}+\mathcal{O}\left(c^{-4}\right)$$

where $U$ is the Newtonian potential, $\gamma$ describes the response of spatial curvature to mass, and $\beta$ describes the nonlinear self-coupling effect of gravity.

The corrected gravitational potential obtained in this paper is:$\Phi \left(r\right)=-{\displaystyle \frac{GM}{r}}-{\displaystyle \frac{k{G}_h{M}^2\left(\mathrm{l}\mathrm{n}r+1\right)}{r}}$. We define the effective potential function:$U_{eff}\left(r\right)=-\Phi \left(r\right)={\displaystyle \frac{GM}{r}}+{\displaystyle \frac{k{G}_h{M}^2\left(\mathrm{l}\mathrm{n}r+1\right)}{r}}$. The logarithmically corrected metric in this paper satisfies the relation between gravitational potential and metric under the weak-field approximation of GR (the metric is constructed from this relation). Therefore, the metric components are:

$$g_{00}=-\left(1-{\displaystyle \frac{2U_{eff}}{c^2}}\right)+\mathcal{O}\left(c^{-4}\right),g_{rr}=1+{\displaystyle \frac{2U_{eff}}{c^2}}+\mathcal{O}\left(c^{-4}\right)$$

Comparing with the standard PPN form ($g_{ij}=\left(1+2\gamma {\displaystyle \frac{U}{c^2}}\right){\delta }_{ij}+\mathcal{O}\left(c^{-4}\right)$), we obtain $\gamma =1$. This means that the first-order effects of light deflection, Shapiro time delay, and gravitational lensing in the weak-field limit of our theory are all consistent with GR.

The PPN parameter $\beta$ is determined by the second-order term $U^2/c^4$ of the metric component $g_{00}$. Since the metric in this paper is directly constructed from the gravitational potential and only retained up to $\mathcal{O}\left(c^{-2}\right)$ order, $\mathsf{\beta }$ cannot be uniquely determined from the existing expressions. If the metric is completed to the standard first post-Newtonian (1PN) form:

$$g_{00}=-\left(1-{\displaystyle \frac{2U_{eff}}{c^2}}+2{\displaystyle \frac{U_{eff}^2}{c^4}}\right)+\mathcal{O}\left(c^{-6}\right)$$

the corresponding nonlinear parameter is $\beta =1$. Nevertheless, the current Solar System experimental constraint $\gamma -1=\left(2.1\pm 2.3\right)\times {10}^{-5}$ (from the Cassini Shapiro time delay experiment) is fully satisfied by our result $\gamma =1$, confirming its consistency with GR.

7. Problems and Motivation

In the current formulation of our theoretical framework, the non-local entanglement strength parameter $k$ takes different forms across various scales:

• In the strong-field regime (near the black hole): $k=M_{SgrA*}/M$, treated as a coefficient related to the mass of the central black hole.
• On galactic scales: $k\left(r\right)=k_0{\left(r_{peak}/r\right)}^{\alpha }$, manifesting as a power-law function evolving with radial distance $r$.

While these two forms can well describe the observational phenomena at their corresponding scales (black hole shadows and rotation curves, respectively) and are self-consistent in the physical picture, we provide a unified description of the $k$ parameter that seamlessly connects the physics across different scales in Appendix D to improve the completeness of the theory.

8. Conclusions and Outlook

In this paper, through a single effective theoretical framework, we reveal the cross-scale universality of gravitational correction with a logarithmic term. Its minimal mathematical form not only prevents collapse into a singularity via a repulsive potential at the core of black holes and offers a possible solution to the information paradox, but also maintains the velocity of high-velocity stars and the flattening of rotation curves through additional gravity in the black hole gravitational field and the outer regions of galaxies. This provides a possible solution to the two major cross-scale challenges (singularity resolution and flattening of rotation curves) without the need for hypotheses such as extra dimensions or dark matter particles. Starting from the analysis of the mathematical asymptotic behavior of dark matter halo dynamics, we conduct a preliminary cross-scale multiple verification of the logarithmically corrected gravity through black hole shadows (EHT observations), high-velocity stars, rotation curves of multiple galaxies (astronomical measurements), combined with the mathematical asymptotics of gravitational lensing. This provides a possible scheme for a unified description of gravity from the microscopic to the macroscopic scale (no longer separated), and also offers an observable, repeatedly verifiable empirical framework for quantum gravity theory that is different from the current mainstream approaches (such as string theory, loop quantum gravity, etc.). Furthermore, the six rigid predictions for black hole shadows presented in this paper make this theory one of the very few theoretical frameworks at present that can provide clear, specific, and falsifiable targets (without adjusting the spin α and inclination i) for the next generation of EHT observations.

Future research will focus on the following aspects: 1) Further improving the microscopic derivation through more rigorous field theory calculations, numerical simulations, and other methods; 2) Attempting to apply the theoretical framework to a priori predictions in multi-messenger astronomical research, including black hole thermodynamics, fast radio bursts, and astroparticle physics, and using telescopes such as JWST, CTA, H.E.S.S., Fermi-LAT, IceCube, and KM3NeT to test the universal boundaries of the application of logarithmically corrected gravity (quantum gravity); 3) Studying the impact of this correction term on the Friedmann equations, and exploring potential insights into cosmological puzzles such as dark energy dynamics and the Hubble tension; 4) Directly verifying the non-local entanglement (quantum entanglement) effect corresponding to the logarithmic term through laboratory simulations (e.g., superfluid helium quantum vortex systems), to provide a more solid microscopic experimental foundation for the theory.

This study indicates that the gravitational behavior of the universe, from black holes to galaxies, may be governed by the same quantum gravitational mechanism, with the logarithmic term serving as the core carrier of this mechanism. It also strongly suggests that black holes and galaxies may share a common topological origin, which we interpret as follows: the overall dynamics of galaxy disks may be the holographic manifestation of the quantum topological structure of their central black holes on the macrocosmic scale through hierarchical nesting ($AdS₂/CFT₁\subseteq AdS₃/CFT₂\subseteq AdS₄/CFT₃$). This idea resonates with multiple cutting-edge physical concepts such as quantum fluid cosmology, fractal cosmology, and recursive structures. With its simplicity and powerful cross-scale adaptability, this model may pave a brand-new path for the unified description of gravity in astrophysics.

The vortex field and non-local entanglement picture presented in the Appendix, although not a premise of this effective theory, can naturally derive the modified Poisson equation and the dimensional structure of $G_h$ used in the main text within the same logical framework. It is also consistent with the mathematical structure of the nested AdS/CFT correspondence, indicating the potential embeddability of this effective theory into a more fundamental theoretical framework.

B.2.4 Spherical Integration, Logarithmic Factor, and Synthesis of ${\mathit{G}}_{\mathit{h}}$

We calculate the integral on the two-dimensional curved $CF{T}_2$ boundary (approximated as a spherical surface with radius $r_{h,in}$ (see Appendix C)):

$$I=\int d^2\mathbf{y}\sqrt{-g\left(\mathbf{y}\right)}{\displaystyle \frac{e^{i8\theta (\mathbf{x},\mathbf{y})}}{|\mathbf{x}-\mathbf{y}{|}^3}}$$

Expanding this integral in spherical coordinates and dualizing it to $Ad{S}_2/CF{T}_1$, its angular integral (${\int }_0^{2\pi }{e}^{i8\varphi \prime }d{\varphi }^{\prime }$) yields a constant factor of 1/8 (the oscillation factor $e^{i8\theta (\mathbf{x},\mathbf{y})}$ is zero under continuous integration: ${\int }_0^{2\pi }{e}^{i8\varphi \prime }d{\varphi }^{\prime }=0$, hence we adopt discrete statistical averaging: ${⟨e^{i8\theta }⟩}_{stat}\approx 1/C=1/8$, which further gives the discrete statistical average of the integral: ${\int }_0^{2\pi }{e}^{i8\varphi \prime }d{\varphi }^{\prime }\approx {⟨e^{i8\theta }⟩}_{stat}\approx 1/8$). The radial integral is then reduced to:

$${\displaystyle \frac{1}{r}}{\int }_{r_{*}}^r{\displaystyle \frac{d{r}^{\prime }}{r^{\prime }}}={\displaystyle \frac{1}{r}}\ln (r/r_{*})$$

Normalizing it (as described in Section 3.2 of the main text), we obtain the integral result:

$$I={\displaystyle \frac{\ln r}{8r}}$$

Finally:

$${\varphi }_{vortex}\left(r\right)\approx {\mathcal{O}}_{vortex,0}\int d^2\mathit{y}\sqrt{-g\left(\mathit{y}\right)}{\displaystyle \frac{e^{i8\theta (\mathit{x},\mathit{y})}}{|\mathit{x}-\mathit{y}{|}^3}}\approx {\displaystyle \frac{k\hslash }{4\pi }}{\left(GMc\right)}^2{\displaystyle \frac{\ln r}{8r}}$$

(28)

We define: $G_h=\hslash c^2{G}^3/8$, and the above equation becomes:

$${\varphi }_{vortex}\left(r\right)\approx {\displaystyle \frac{k{G}_h{M}^2\ln r}{4\pi Gr}}$$

On macroscopic scales, the combination of $\mathrm{l}\mathrm{n}r$ and the constant 1, "$\mathrm{l}\mathrm{n}r+1$", is almost equivalent to "$\mathrm{l}\mathrm{n}r$" (negligible observational deviation). To match the logarithmically corrected gravitational potential, we adopt:

$${\varphi }_{vortex}\left(r\right)\approx {\displaystyle \frac{k{G}_h{M}^2\left(\ln r+1\right)}{4\pi Gr}}$$

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C.4.2.1 Tensor Self-Consistency: Definition and Conservation

For the metric:

$$d{s}^2=-A\left(r\right)c^{2}d{t}^2+B\left(r\right)d{r}^2+r^{2}d{\Omega }^2$$

where $A\left(r\right)=1-u\left(r\right)$, $B\left(r\right)=1+u\left(r\right)$, and $u\left(r\right)={\displaystyle \frac{2GM}{c^{2}r}}+{\displaystyle \frac{2k{G}_h{M}^2\left(\mathrm{l}\mathrm{n}r+1\right)}{c^{2}r}}$, the Einstein tensor $G^{\mu } {}_{\nu }$ is a closed-form function $G^{\mu } {}_{\nu }[u,u^{\prime },u"]$. By the Bianchi identity ${\nabla }_{\mu }{G}^{\mu } {}_{\nu }=0$, we define$T^{eff\mu } {}_{\nu }={\displaystyle \frac{c^4}{8\pi G}}{G}^{\mu } {}_{\nu }$, which satisfies ${\nabla }_{\mu }{T}^{eff\mu } {}_{\nu }=0$. Therefore, this model is a strictly self-consistent theory with conserved quantities at the tensor level, requiring no additional constraints.