Theoretical Analysis of Barrow Holographic Dark Energy in Fractal Cosmology

Abstract

We investigate interacting Barrow holographic dark energy (BHDE) with the Hubble horizon as the infrared cutoff in a fractal cosmological background. The fractal measure modifies the Friedmann sector and leads to the nonstandard closure relation ${\mathsf{\Omega }}_{\mathrm{dm}}+{\mathsf{\Omega }}_{\mathrm{de}}=1+\gamma$. Using Planck-inspired present-day normalization ${\mathsf{\Omega }}_{\mathrm{de}0}=0.6847$ and the benchmark parameters $(\Delta ,\omega ,\beta )=(0.8,0.263,0.123)$, we study four linear and nonlinear dark-sector interactions over $-0.99\le z\le 3$. The benchmark solutions remain within the physical background domain throughout this interval: the density fractions are non-negative, the relevant denominators remain positive, and the nonlinear $Q_3$ and $Q_4$ terms remain real. All four prescriptions exhibit a transition from decelerated to accelerated expansion. The non-interacting limit gives $z_t\simeq 0.86$, whereas for $\xi =0.12$ the transition redshifts are approximately $2.02$, $1.97$, $1.26$, and $1.87$ for $Q_1$, $Q_2$, $Q_3$, and $Q_4$, respectively; in particular, the $Q_4$ solution recovers a finite transition within the plotted redshift range. A stronger positive coupling generally shifts acceleration onset to a higher redshift, with the strongest response for $Q_1$ and the weakest for $Q_3$. The statefinder quantities $S_3^{\left(1\right)}$, $S_3^{\left(2\right)}$, and the s–r trajectories distinguish the interaction structures through their finite-redshift evolution and present-day values. A fixed-background SN Ia comparison using 1046 selected Pantheon supernovae, diagonal FITRES uncertainties, and an analytically profiled additive nuisance parameter gives the lowest information criteria for flat $\mathsf{\Lambda }$CDM. Among the BHDE benchmarks that interact, $Q_4$ is the closest case, with $\Delta \mathrm{AIC}=\Delta \mathrm{BIC}\simeq 1.21$. This comparison is not intended as a global posterior constraint on the model parameters.

1. Introduction

The late-time accelerated expansion of the universe is supported by the original Type Ia supernova observations and subsequent precision cosmological measurements [1,2,3]. This discovery placed dark energy and possible modifications of gravity at the center of modern cosmology, although the physical mechanism responsible for the acceleration remains unresolved.

The conceptual foundations of modern cosmology were established through Einstein’s relativistic cosmological model [4], the expanding-universe solutions of Friedmann and Lemaître [5,6], and Hubble’s distance–velocity relation [7]. The hot Big Bang picture was subsequently strengthened by primordial-nucleosynthesis calculations [8] and the discovery of the cosmic microwave background [9]. Inflation later provided a mechanism for addressing the horizon and flatness problems [10], although its detailed microphysical origin remains under investigation.

Broadly speaking, two classes of explanations for the present acceleration have been pursued: the cosmic energy budget may contain a component with sufficiently negative pressure, or gravity may deviate from general relativity on cosmological scales. The $\mathsf{\Lambda }$CDM model [11,12] successfully describes a wide range of observations, but the fine-tuning and coincidence problems [11], together with tensions involving $H_0$ and the growth of structure [13], continue to motivate extended cosmological scenarios.

A broad class of dynamical dark-energy models has therefore been developed, including quintessence [14,15,16,17], Chaplygin-gas models [18,19], and holographic dark energy (HDE) [20,21]. In Barrow holographic dark energy (BHDE), quantum-gravitational effects deform the black-hole entropy–area relation through the parameter $\Delta$ [22,23]. Fractal cosmology provides a complementary modification by replacing the standard spacetime integration measure with an effective fractal measure [24,25]. In the combined framework, the fractal measure changes the Friedmann sector and the closure relation, while the Barrow deformation modifies the scaling of the dark-energy density with the infrared cutoff.

Dark-energy models previously studied in fractal cosmology include Tsallis agegraphic and holographic scenarios [26,27], reconstructed BHDE models [28], and two-component fractal cosmologies [29]. These studies show that entropy deformations and modified cosmological measures can jointly produce background dynamics that differ from those of standard FRW cosmology.

Recent investigations of holographic dark energy have also emphasized thermodynamic consistency, stability, viscosity, and observational diagnostics. Barrow HDE has been studied through generalized thermodynamics, viscous effects, equation-of-state behavior, and statefinder diagnostics [30]. Related work includes viscous holographic $f\left(Q\right)$ cosmology [31], thermodynamic and stability analyses of Tsallis HDE in $F(R,T)$ gravity [32], generalized HDE models based on corrected entropies [33], and interacting Barrow–Ricci–Gauss–Bonnet HDE in symmetric teleparallel gravity [34]. Interacting holographic scenarios have also been examined through the equation of state, the deceleration parameter, statefinder and $O_m\left(z\right)$ diagnostics, and chi-square comparisons [35]. Statistical tests using observational Hubble data and joint late-time datasets have been applied to related dark-energy and modified-gravity models [36,37].

Viscous-fluid cosmologies provide a complementary phenomenological setting and have been compared with BAO, Pantheon, and $H\left(z\right)$ observations [38]. Holographic energy densities have also been used as effective sources in recent relativistic wormhole constructions [39,40,41,42]. Although these static configurations address a different physical sector, they illustrate the broader use of holographic energy densities in relativistic gravity.

The present work combines the Barrow entropy deformation, a fractal cosmological measure, and four linear and nonlinear dark-sector interactions within a common background framework. This construction allows the interaction prescriptions to be compared using the modified closure relation, the evolution of ${\mathsf{\Omega }}_{\mathrm{de}}$, $w_{\mathrm{de}}$, and q, the acceleration-transition redshift, the physical-domain conditions of the nonlinear $Q_3$ and $Q_4$ couplings, and the statefinder hierarchy. The same benchmark backgrounds are also examined through a profiled SN Ia comparison and one-parameter distance-modulus sensitivity tests. The novelty of the analysis lies in applying these dynamical, geometric, and observational diagnostics consistently to four interaction structures within the same fractal BHDE system.

The paper is organized as follows. Section 2 introduces the fractal cosmological framework. Section 3 derives the interacting BHDE equations and presents the background and SN Ia analyses. Section 4 discusses the statefinder diagnostics. Section 5 summarizes the main conclusions, while Appendix A compares the model with selected alternative holographic dark-energy scenarios and Appendix B provides details of the metric variation leading to the modified Friedmann equations.

2. Fractal Cosmology Theoretical Background

Fractal cosmology extends conventional spacetime geometry by generalizing the standard Riemannian integration measure. The fundamental scaling relation for isotropic coordinates is defined as

$$\left[x^{\mu }\right]=-1,\mu =0,1,\dots ,D-1,$$

(1)

where D represents the topological dimension. This framework is implemented through a modified integration measure:

$$d^{D}x\to d\varrho \left(x\right)=d^{D}x\upsilon \left(x\right),\left[\varrho \right]=-D\alpha \ne -D,$$

(2)

with $\alpha$ ($0<\alpha <1$) quantifying the deviation from standard integer dimensionality induced by fractal effects.

Throughout this work, the fractal measure weight is denoted consistently by $\upsilon$. The function $\upsilon \left(x\right)$ encodes the fractal contribution to the geometry, and its time-dependent cosmological form is written as $\upsilon \left(t\right)$. To avoid unphysical behavior such as divergences as $t\to 0$, we adopt the commonly used phenomenological choice

$$\upsilon =a^{-\beta },$$

(3)

where a is the scale factor and $\beta \equiv D(1-\alpha )$ characterizes the fractal properties.

The gravitational dynamics follow from the total action

$$S=S_g+S_m,$$

(4)

which comprises the gravitational sector

$$S_g={\displaystyle \frac{M_p^2}{2}}\int d\varrho \left(x\right)\sqrt{-g}\left(R-\omega {\partial }_{\mu }\upsilon {\partial }^{\mu }\upsilon \right),$$

(5)

and the matter sector

$$S_m=\int d\varrho \left(x\right)\sqrt{-g}{\mathcal{L}}_m.$$

(6)

Here, g is the determinant of $g_{\mu \nu }$, $M_p^{-2}=8\pi G$ is the reduced Planck mass, $\omega$ is the fractal parameter, R is the Ricci scalar, and ${\mathcal{L}}_m$ is the matter Lagrangian density.

For a spatially flat Friedmann–Robertson–Walker (FRW) universe with metric

$$d{s}^2=-d{t}^2+a^2\left(t\right)\left(d{r}^2+r^{2}d{\mathsf{\Omega }}^2\right),$$

(7)

variation of the action yields the modified Friedmann equations

$$H^2+H{\displaystyle \frac{\dot{\upsilon }}{\upsilon }}-{\displaystyle \frac{\omega }{6}}{\dot{\upsilon }}^2={\displaystyle \frac{1}{3M_p^2}}\rho ,$$

(8)

$$\dot{H}+H^2-H{\displaystyle \frac{\dot{\upsilon }}{\upsilon }}+{\displaystyle \frac{\omega }{3}}{\dot{\upsilon }}^2-{\displaystyle \frac{\square \upsilon }{2\upsilon }}=-{\displaystyle \frac{1}{6M_p^2}}(\rho +3p),$$

(9)

where $H=\dot{a}/a$, $\rho$ and p are the total energy density and pressure, and

$$\square \upsilon =-\left(\ddot{\upsilon }+3H\dot{\upsilon }\right).$$

(10)

In the limit $\upsilon =1$, Equations (8)–(10) reduce to the standard FRW equations.

For a universe containing dark energy and dark matter ($\rho ={\rho }_{de}+{\rho }_{dm}$, $p=p_{de}$), the conservation equation becomes

$$\dot{\rho }=-\left(3H+{\displaystyle \frac{\dot{\upsilon }}{\upsilon }}\right)(\rho +p).$$

(11)

Using the specific form (3), Equation (8) can be rewritten as

$$H^2\left(1-\beta -{\displaystyle \frac{{\beta }^2\omega a^{-2\beta }}{6}}\right)={\displaystyle \frac{1}{3M_p^2}}({\rho }_{de}+{\rho }_{dm}),$$

(12)

while the separated conservation equations read

$${\dot{\rho }}_{dm}=-(3-\beta )H{\rho }_{dm},$$

(13)

$${\dot{\rho }}_{de}=-(1+w_{de})(3-\beta )H{\rho }_{de},$$

(14)

where $w_{de}=p_{de}/{\rho }_{de}$ is the dark-energy equation-of-state parameter. This setup allows one to embed BHDE into a fractal cosmological background.

3. Dynamical Evolution Analysis

We now consider Barrow holographic dark energy (BHDE), in which quantum-gravitational corrections are encoded through a deformation of black-hole entropy. Following Barrow’s proposal, quantum effects can induce a “rough” or fractal-like horizon structure, leading to a modified entropy–area relation

$$S_B={\left({\displaystyle \frac{A}{A_0}}\right)}^{1+\Delta /2},$$

(15)

where A is the horizon area, $A_0$ is the Planck area, and $\Delta$ ($0\le \Delta \le 1$) quantifies the deformation. The standard Bekenstein–Hawking entropy is recovered at $\Delta =0$, while $\Delta =1$ corresponds to the maximal deformation limit.

In holographic dark energy, one imposes the relation $\rho L^4\le S$, with L a characteristic infrared length scale. Using the Barrow entropy (15) yields the BHDE density

$${\rho }_{de}=C{L}^{\Delta -2},$$

(16)

where C is a constant. For a flat FRW universe, choosing the Hubble horizon as the infrared cutoff $(L=H^{-1})$, one obtains

$${\rho }_{de}=C{H}^{2-\Delta }.$$

(17)

Differentiating gives

$${\dot{\rho }}_{de}=(2-\Delta ){\rho }_{de}{\displaystyle \frac{\dot{H}}{H}}.$$

(18)

In standard non-interacting holographic dark energy with the Hubble radius as the infrared cutoff, the scaling ${\rho }_{\mathrm{de}}\propto H^2$ makes the dark-energy density track the pressureless matter density and generally does not produce accelerated expansion [20,43]. The standard argument is modified in the present framework because the Barrow deformation changes the scaling to ${\rho }_{\mathrm{de}}\propto H^{2-\Delta }$, the fractal measure modifies the Friedmann and conservation equations, and the interaction term permits energy exchange between the dark components. The combined system can therefore support accelerating benchmark solutions even when the Hubble horizon is used as the infrared cutoff.

The Hubble cutoff is adopted here because it gives a local autonomous background system and allows the effects of the Barrow deformation, the fractal measure, and the interaction prescription to be isolated. Alternative choices, such as the future event horizon, Ricci scale, or Granda–Oliveros cutoff, would change the dark-energy density and its evolution equation. They could consequently alter the transition redshifts and the detailed behavior of the trajectories. The conclusions of the present analysis are therefore specific to the Hubble-horizon cutoff and should not be interpreted as cutoff-independent predictions.

3.1. Interacting Models Dynamical Evolution

To investigate the background dynamics, we numerically integrate the autonomous evolution equation for ${\mathsf{\Omega }}_{\mathrm{de}}$ derived below for each interaction prescription.

Throughout this subsection, we adopt the benchmark parameter set $\Delta =0.8$, $\omega =0.263$, and $\beta =0.123$. The Barrow exponent is restricted theoretically to $0\le \Delta \le 1$, and $\Delta =0.8$ is chosen to display a clearly non-negligible entropy deformation. The values of $\omega$ and $\beta$ are representative choices used in related fractal dark-energy studies [24,27,28]. They are employed here to examine the dynamical response of the model and should not be interpreted as posterior constraints from a joint cosmological likelihood.

The interaction strength is varied over $\xi \in [0,0.12]$, including the non-interacting limit. This interval defines a benchmark sequence from weak to comparatively strong positive coupling and is used to compare the sensitivity of the four interaction prescriptions. The strong-coupling endpoint is illustrative and is not assumed a priori to be observationally preferred.

For the numerical normalization, we take $H_0=70\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{Mpc}}^{-1}$ and ${\mathsf{\Omega }}_{\mathrm{de}0}=0.6847$. The latter is a Planck-inspired choice based on the base-$\mathsf{\Lambda }$CDM estimate ${\mathsf{\Omega }}_{\mathsf{\Lambda }0}=0.6847\pm 0.0073$ [3]; it is used only as the present-day normalization of the interacting BHDE solutions. The value of $H_0$ sets the distance scale, while its degeneracy with the absolute SN Ia magnitude is absorbed into the additive nuisance parameter $\mathcal{M}$.

For analytical simplicity, the background is treated as a reduced two-component system consisting of interacting BHDE and an effective pressureless matter sector. Baryons and radiation are not evolved as separately conserved components. The resulting model is therefore intended as a late-time benchmark rather than a complete reconstruction of the cosmic energy budget.

We allow for an energy exchange term Q between the two dark-sector components [44,45,46,47,48,49,50,51,52,53,54,55,56,57]. The conservation equations are then

$${\dot{\rho }}_{dm}+(3-\beta )H{\rho }_{dm}=Q,$$

(19)

$${\dot{\rho }}_{de}+(1+w_{de})(3-\beta )H{\rho }_{de}=-Q.$$

(20)

We examine four commonly used phenomenological forms: $Q_1=3\xi H{\rho }_{de}$, $Q_2=3\xi H{\rho }_{dm}$, $Q_3=3\xi H{\displaystyle \frac{{\rho }_{de}{\rho }_{dm}}{{\rho }_{de}+{\rho }_{dm}}}$, and $Q_4=3\xi H\sqrt{{\rho }_{de}{\rho }_{dm}}$, where $\xi$ is the coupling parameter and $\xi =0$ corresponds to the non-interacting limit.

With the sign convention adopted in Equations (19) and (20), $Q>0$ corresponds to energy transfer from dark energy to dark matter, whereas $Q<0$ describes transfer in the opposite direction. For an expanding background with positive dark-sector densities, all four interaction functions have the same sign as $\xi$. Hence, the range $\xi \ge 0$ considered here represents energy transfer from BHDE to the effective pressureless matter component. A negative value of $\xi$ would instead describe energy transfer from matter to BHDE.

Using ${\rho }_{cr}={\displaystyle \frac{3H^2}{8\pi G}}$, the density parameters are

$${\mathsf{\Omega }}_{dm}={\displaystyle \frac{{\rho }_{dm}}{{\rho }_{cr}}},{\mathsf{\Omega }}_{de}={\displaystyle \frac{{\rho }_{de}}{{\rho }_{cr}}}.$$

(21)

Combining Equation (21) with the modified Friedmann equation yields the effective closure relation

$${\mathsf{\Omega }}_{dm}+{\mathsf{\Omega }}_{de}=1+\gamma ,$$

(22)

$$\gamma =-\beta -{\displaystyle \frac{{\beta }^2\omega {(1+z)}^{2\beta }}{6}},$$

(23)

where the redshift is defined by $1+z\equiv a^{-1}$ (with $a_0=1$). The parameter $\gamma$ encodes the deviation from the standard closure relation induced by the fractal measure.

For the adopted values $\beta =0.123$ and $\omega =0.263$, the present fractal correction is

$${\gamma }_0=-\beta -{\displaystyle \frac{{\beta }^2\omega }{6}}=-0.123663.$$

(24)

Together with the Planck-inspired normalization ${\mathsf{\Omega }}_{\mathrm{de}0}=0.6847$, the modified closure relation gives

$${\mathsf{\Omega }}_{\mathrm{dm}0}=1+{\gamma }_0-{\mathsf{\Omega }}_{\mathrm{de}0}=0.191637.$$

(25)

This effective pressureless matter fraction is below the base-$\mathsf{\Lambda }$CDM estimate ${\mathsf{\Omega }}_{m0}=0.315\pm 0.007$ [3]. The difference follows directly from the negative fractal contribution ${\gamma }_0$ and shows that the adopted normalization does not reproduce the standard present-day matter budget. Accordingly, ${\mathsf{\Omega }}_{\mathrm{dm}0}=0.191637$ is interpreted as the effective matter fraction of the selected fractal benchmark, not as an independent measurement of the observed matter abundance.

If baryons and radiation were introduced explicitly, the closure relation would become

$${\mathsf{\Omega }}_{\mathrm{dm}}+{\mathsf{\Omega }}_b+{\mathsf{\Omega }}_r+{\mathsf{\Omega }}_{\mathrm{de}}=1+\gamma .$$

(26)

For the same ${\gamma }_0$ and ${\mathsf{\Omega }}_{\mathrm{de}0}$, however, the total non-dark-energy fraction would still satisfy

$${\mathsf{\Omega }}_{\mathrm{dm}0}+{\mathsf{\Omega }}_{b0}+{\mathsf{\Omega }}_{r0}=1+{\gamma }_0-{\mathsf{\Omega }}_{\mathrm{de}0}=0.191637.$$

(27)

Separating baryons and radiation would therefore redistribute the available non-dark-energy density but would not remove the reduced matter budget associated with the chosen fractal closure relation. This limitation is taken into account when interpreting the background and SN Ia comparisons below.

Differentiating the modified Friedmann equation with respect to time and using Equations (20) and (22), one obtains

$${\displaystyle \frac{\dot{H}}{H^2}}={\displaystyle \frac{\frac{Q}{3M_p^2{H}^3}+(\beta -3)(1+\gamma -{\mathsf{\Omega }}_{de})+2\beta (\gamma +\beta )}{2+2\gamma -{\mathsf{\Omega }}_{de}(2-\Delta )}}.$$

(28)

From ${\mathsf{\Omega }}_{de}={\rho }_{de}/{\rho }_{cr}\propto H^{-\Delta }$, one finds

$${\dot{\mathsf{\Omega }}}_{de}=-\Delta {\mathsf{\Omega }}_{de}{\displaystyle \frac{\dot{H}}{H}}.$$

(29)

Hereafter, a prime denotes differentiation with respect to $lna$, namely ${\mathsf{\Omega }}_{\mathrm{de}}^{\prime }=d{\mathsf{\Omega }}_{\mathrm{de}}/dlna$.

$${\mathsf{\Omega }}_{de}^{\prime }=-\Delta {\mathsf{\Omega }}_{de}{\displaystyle \frac{\frac{Q}{3M_p^2{H}^3}+(\beta -3)(1+\gamma -{\mathsf{\Omega }}_{de})+2\beta (\gamma +\beta )}{2+2\gamma -{\mathsf{\Omega }}_{de}(2-\Delta )}}.$$

(30)

Combining Equations (20), (22) and (28) yields the BHDE equation of state:

$$w_{de}={\displaystyle \frac{Q}{3(\beta -3){\mathsf{\Omega }}_{de}{H}^3{M}_p^2}}+{\displaystyle \frac{(2-\Delta )}{(\beta -3)}}{\displaystyle \frac{\frac{Q}{3M_p^2{H}^3}+(\beta -3)(1+\gamma -{\mathsf{\Omega }}_{de})+2\beta (\gamma +\beta )}{2+2\gamma -{\mathsf{\Omega }}_{de}(2-\Delta )}}-1.$$

(31)

From the deceleration parameter definition $q\equiv -{\displaystyle \frac{\ddot{a}}{a{H}^2}}$, we obtain

$$q=-1-{\displaystyle \frac{\dot{H}}{H^2}}=-1-{\displaystyle \frac{\frac{Q}{3M_p^2{H}^3}+(\beta -3)(1+\gamma -{\mathsf{\Omega }}_{de})+2\beta (\gamma +\beta )}{2+2\gamma -{\mathsf{\Omega }}_{de}(2-\Delta )}}.$$

(32)

Inserting $Q_1$–$Q_4$ into Equations (30)–(32) yields

(1)

$Q_1=3H\xi {\rho }_{de}$

$${\mathsf{\Omega }}_{de}^{\prime }=-\Delta {\mathsf{\Omega }}_{de}{\displaystyle \frac{3\xi {\mathsf{\Omega }}_{de}+(\beta -3)(1+\gamma -{\mathsf{\Omega }}_{de})+2\beta (\gamma +\beta )}{2+2\gamma -{\mathsf{\Omega }}_{de}(2-\Delta )}},$$

(33)

$$w_{de}={\displaystyle \frac{3\xi }{(\beta -3)}}+{\displaystyle \frac{(2-\Delta )}{(\beta -3)}}{\displaystyle \frac{3\xi {\mathsf{\Omega }}_{de}+(\beta -3)(1+\gamma -{\mathsf{\Omega }}_{de})+2\beta (\gamma +\beta )}{2+2\gamma -{\mathsf{\Omega }}_{de}(2-\Delta )}}-1,$$

(34)

$$q=-1-{\displaystyle \frac{3\xi {\mathsf{\Omega }}_{de}+(\beta -3)(1+\gamma -{\mathsf{\Omega }}_{de})+2\beta (\gamma +\beta )}{2+2\gamma -{\mathsf{\Omega }}_{de}(2-\Delta )}}.$$

(35)

(2)

$Q_2=3H\xi {\rho }_{dm}$

$${\mathsf{\Omega }}_{de}^{\prime }=-\Delta {\mathsf{\Omega }}_{de}{\displaystyle \frac{(3\xi +\beta -3)(1+\gamma -{\mathsf{\Omega }}_{de})+2\beta (\gamma +\beta )}{2+2\gamma -{\mathsf{\Omega }}_{de}(2-\Delta )}},$$

(36)

$$w_{de}={\displaystyle \frac{3\xi (1+\gamma -{\mathsf{\Omega }}_{de})}{(\beta -3){\mathsf{\Omega }}_{de}}}+{\displaystyle \frac{(2-\Delta )}{(\beta -3)}}{\displaystyle \frac{(3\xi +\beta -3)(1+\gamma -{\mathsf{\Omega }}_{de})+2\beta (\gamma +\beta )}{2+2\gamma -{\mathsf{\Omega }}_{de}(2-\Delta )}}-1,$$

(37)

$$q=-1-{\displaystyle \frac{(3\xi +\beta -3)(1+\gamma -{\mathsf{\Omega }}_{de})+2\beta (\gamma +\beta )}{2+2\gamma -{\mathsf{\Omega }}_{de}(2-\Delta )}}.$$

(38)

(3)

$Q_3=3\xi H{\displaystyle \frac{{\rho }_{de}{\rho }_{dm}}{{\rho }_{de}+{\rho }_{dm}}}$

$${\mathsf{\Omega }}_{de}^{\prime }=-\Delta {\mathsf{\Omega }}_{de}{\displaystyle \frac{3\xi {\mathsf{\Omega }}_{de}(1+\gamma -{\mathsf{\Omega }}_{de})+(\beta -3)(1+\gamma -{\mathsf{\Omega }}_{de})(1+\gamma )+2\beta (\gamma +\beta )(1+\gamma )}{\left[2+2\gamma -{\mathsf{\Omega }}_{de}(2-\Delta )\right](1+\gamma )}},$$

(39)

$$w_{de}={\displaystyle \frac{3\xi (1+\gamma -{\mathsf{\Omega }}_{de})}{(\beta -3)(1+\gamma )}}+{\displaystyle \frac{(2-\Delta )}{(\beta -3)}}{\displaystyle \frac{3\xi {\mathsf{\Omega }}_{de}(1+\gamma -{\mathsf{\Omega }}_{de})+(\beta -3)(1+\gamma -{\mathsf{\Omega }}_{de})(1+\gamma )+2\beta (\gamma +\beta )(1+\gamma )}{\left[2+2\gamma -{\mathsf{\Omega }}_{de}(2-\Delta )\right](1+\gamma )}}-1,$$

(40)

$$q=-1-{\displaystyle \frac{3\xi {\mathsf{\Omega }}_{de}(1+\gamma -{\mathsf{\Omega }}_{de})+(\beta -3)(1+\gamma -{\mathsf{\Omega }}_{de})(1+\gamma )+2\beta (\gamma +\beta )(1+\gamma )}{\left[2+2\gamma -{\mathsf{\Omega }}_{de}(2-\Delta )\right](1+\gamma )}}.$$

(41)

(4)

$Q_4=3\xi H\sqrt{{\rho }_{de}{\rho }_{dm}}$

$${\mathsf{\Omega }}_{de}^{\prime }=-\Delta {\mathsf{\Omega }}_{de}{\displaystyle \frac{3\xi {\left[{\mathsf{\Omega }}_{de}(1+\gamma -{\mathsf{\Omega }}_{de})\right]}^{1/2}+(\beta -3)(1+\gamma -{\mathsf{\Omega }}_{de})+2\beta (\gamma +\beta )}{2+2\gamma -{\mathsf{\Omega }}_{de}(2-\Delta )}},$$

(42)

$$w_{de}={\displaystyle \frac{3\xi {\left[{\mathsf{\Omega }}_{de}(1+\gamma -{\mathsf{\Omega }}_{de})\right]}^{1/2}}{(\beta -3){\mathsf{\Omega }}_{de}}}+{\displaystyle \frac{(2-\Delta )}{(\beta -3)}}{\displaystyle \frac{3\xi {\left[{\mathsf{\Omega }}_{de}(1+\gamma -{\mathsf{\Omega }}_{de})\right]}^{1/2}+(\beta -3)(1+\gamma -{\mathsf{\Omega }}_{de})+2\beta (\gamma +\beta )}{2+2\gamma -{\mathsf{\Omega }}_{de}(2-\Delta )}}-1,$$

(43)

$$q=-1-{\displaystyle \frac{3\xi {\left[{\mathsf{\Omega }}_{de}(1+\gamma -{\mathsf{\Omega }}_{de})\right]}^{1/2}+(\beta -3)(1+\gamma -{\mathsf{\Omega }}_{de})+2\beta (\gamma +\beta )}{2+2\gamma -{\mathsf{\Omega }}_{de}(2-\Delta )}}.$$

(44)

The autonomous equations are integrated using $lna$ as the independent variable over $ln\left(0.01\right)\le lna\le ln\left(100\right)$, with the present-day initial condition ${\mathsf{\Omega }}_{\mathrm{de}}(lna=0)={\mathsf{\Omega }}_{\mathrm{de}}$$(a=1)=0.6847$. We use an adaptive stiffness-switching algorithm with working precision 30 and accuracy and precision goals equal to 14. Because the step size is selected adaptively by the solver, no fixed integration step is imposed. The displayed interval is $-0.99\le z\le 3$, where $1+z=a^{-1}$. The transition redshift is obtained by locating a sign change of $q\left(z\right)$ in $0\le z\le 3$ and refining the root of $q\left(z_t\right)=0$ within the corresponding bracket.

The physical background domain was checked over the same numerical interval for every interaction prescription and coupling strength. The solutions satisfy ${\mathsf{\Omega }}_{\mathrm{de}}>0$, ${\mathsf{\Omega }}_{\mathrm{dm}}=1+\gamma -{\mathsf{\Omega }}_{\mathrm{de}}\ge 0$, and $2+2\gamma -{\mathsf{\Omega }}_{\mathrm{de}}(2-\Delta )>0$. The quantities $w_{\mathrm{de}}$ and q remain real and finite. For $Q_3$, the additional factor $1+\gamma$ remains positive, while the radicand ${\mathsf{\Omega }}_{\mathrm{de}}{\mathsf{\Omega }}_{\mathrm{dm}}$ entering $Q_4$ is non-negative. Thus, the nonlinear interaction terms remain real and the benchmark solutions remain within the required physical background domain over $-0.99\le z\le 3$. These checks concern the homogeneous background only and do not constitute a perturbative stability test.

The physical-domain conditions were evaluated quantitatively over all four interaction prescriptions, $\xi =0$, $0.04$, $0.08$, and $0.12$, and over $-0.99\le z\le 3$. The global minimum of the effective matter fraction is

$$min{\mathsf{\Omega }}_{\mathrm{dm}}=6.265589\times {10}^{-7}>0,$$

(45)

with the minimum attained by the common non-interacting solution at the future numerical boundary $z=-0.99$. The interacting solutions remain above this limiting value. The principal evolution denominator satisfies

$$min\left[2+2\gamma -(2-\Delta ){\mathsf{\Omega }}_{\mathrm{de}}\right]=0.701393714>0,$$

(46)

where the global minimum occurs for the same non-interacting solution at $z\simeq -0.96881$. For the nonlinear interactions, the additional $Q_3$ factor and the $Q_4$ radicand satisfy

$$min(1+\gamma )=0.876067344>0,min\left({\mathsf{\Omega }}_{\mathrm{de}}{\mathsf{\Omega }}_{\mathrm{dm}}\right)=5.493579\times {10}^{-7}>0.$$

(47)

The minimum $Q_4$ radicand is likewise attained in the non-interacting limit at $z=-0.99$. These numerical minima explicitly confirm that the benchmark solutions remain real and within the physical background domain throughout the investigated redshift interval.

The nonlinear prescriptions may nevertheless develop additional instabilities outside the verified background domain. The $Q_3$ interaction becomes singular if the total dark-sector density ${\rho }_{\mathrm{de}}+{\rho }_{\mathrm{dm}}$ approaches zero, whereas $Q_4$ requires both densities to remain non-negative and is non-analytic when either density approaches zero. At the perturbative level, variations of these nonlinear functions may contain inverse-density or square-root factors and can amplify fluctuations near such boundaries. Sufficiently large couplings may also drive a component toward a negative density or produce non-adiabatic large-scale instabilities. The numerical bounds above exclude these singular behaviors at the homogeneous-background level over the investigated interval, but they do not test ghost, gradient, sound-speed, or cosmological-perturbation stability. A complete perturbative stability analysis is left for future work.

Figure 1 shows the redshift evolution of ${\mathsf{\Omega }}_{\mathrm{de}}$ for the benchmark parameters and the common normalization ${\mathsf{\Omega }}_{\mathrm{de}0}=0.6847$. The curves therefore meet at $z=0$. In all four interaction models, the dark-energy fraction decreases toward the past and increases toward the future boundary of the numerical interval. At $z>0$, a larger positive coupling generally increases ${\mathsf{\Omega }}_{\mathrm{de}}$ relative to the non-interacting solution, whereas at $z<0$ it suppresses the dark-energy fraction. The coupling dependence is strongest for $Q_1$, weakest for $Q_3$, and intermediate for $Q_2$ and $Q_4$.

The transition redshift provides a useful kinematic characterization of the change from decelerated to accelerated expansion [58].

Figure 2 displays the deceleration parameter. Each interaction prescription evolves from $q>0$ at sufficiently high redshift within the plotted interval to $q<0$ at the present epoch. The common non-interacting solution gives $z_t=0.85845$.

Table 1. Acceleration-transition redshift $z_t$, defined by $q\left(z_t\right)=0$, for the four interaction forms and coupling strengths.

Model	$\mathit{\xi }=0$	$\mathit{\xi }=0.04$	$\mathit{\xi }=0.08$	$\mathit{\xi }=0.12$
$Q_1$	0.85845	1.15252	1.52466	2.02175
$Q_2$	0.85845	1.09535	1.43838	1.97160
$Q_3$	0.85845	0.97610	1.10864	1.25782
$Q_4$	0.85845	1.11495	1.44445	1.87260

displays the transition redshift for the different coupling parameter, such as $\xi =0.12$, the transition redshifts are $2.02175$, $1.97160$, $1.25782$, and $1.87260$ for $Q_1$, $Q_2$, $Q_3$, and $Q_4$, respectively. For $Q_4$, a finite transition is obtained for every coupling considered, and the solution has returned to the decelerating branch by $z=3$. The values near $z_t\simeq 2$ at the strong-coupling endpoint are illustrative and are not presented as observationally preferred transition epochs.

Figure 3 presents the evolution of $w_{\mathrm{de}}$. The equation of state remains real and finite over the numerical interval. Its stronger high-redshift variation, particularly for interaction forms involving the matter density, must be interpreted within the reduced two-component late-time model and does not establish compatibility with early-universe constraints. Some strongly coupled solutions approach or cross $w_{\mathrm{de}}=-1$ near the future boundary of the numerical interval; this background behavior alone neither demonstrates a perturbative instability nor proves the absence or presence of a finite-time big-rip singularity. A separate perturbative and asymptotic analysis would be required.

Because the calculation is restricted to $z\le 3$ and does not evolve baryons and radiation separately, its domain of applicability is the late-time homogeneous background. It does not describe recombination, primordial nucleosynthesis, or the radiation-dominated epoch.

3.2. SN Ia Distance Modulus and Parameter Sensitivity

To examine the late-time background consistency of the interacting fractal BHDE scenarios, we compare their luminosity-distance predictions with the Pantheon SN Ia sample [59]. After parsing the FITRES table and retaining valid entries in $0.01\le z\le 2.5$, the numerical analysis contains $N=1046$ supernovae. The observable distance modulus is

$$\mu \left(z\right)=5{log}_{10}\left[{\displaystyle \frac{d_L\left(z\right)}{\mathrm{Mpc}}}\right]+25,$$

(48)

where

$$d_L\left(z\right)=(1+z)c{\int }_0^z{\displaystyle \frac{d{z}^{\prime }}{H\left(z^{\prime }\right)}}.$$

(49)

For the interacting BHDE models, $H\left(z\right)$ is reconstructed directly from the background quantity $B\left(z\right)\equiv \dot{H}/H^2$ through

$${\displaystyle \frac{dlnH}{dz}}=-{\displaystyle \frac{B\left(z\right)}{1+z}},E\left(z\right)\equiv {\displaystyle \frac{H\left(z\right)}{H_0}}=exp\left[{\int }_0^z-{\displaystyle \frac{B\left(z^{\prime }\right)}{1+z^{\prime }}}d{z}^{\prime }\right].$$

(50)

This construction uses the same interacting background equations as the dynamical analysis and does not impose a separately conserved ${\mathsf{\Omega }}_{\mathrm{dm}0}{(1+z)}^3$ term.

The absolute SN magnitude and the Hubble-scale normalization enter the distance modulus through an additive nuisance parameter $\mathcal{M}$. For the diagonal FITRES uncertainties used here, the profile chi-square is

$${\chi }^2\left(\mathcal{M}\right)={\displaystyle \sum _{i=1}^N}{\displaystyle \frac{{\left[{\mu }_{\mathrm{obs},i}-{\mu }_{\mathrm{th},i}-\mathcal{M}\right]}^2}{{\sigma }_i^2}}.$$

(51)

Writing $d_i={\mu }_{\mathrm{obs},i}-{\mu }_{\mathrm{th},i}$ and $w_i={\sigma }_i^{-2}$, the nuisance parameter is analytically profiled as

$${\mathcal{M}}_{\mathrm{best}}={\displaystyle \frac{{\sum }_i{w}_i{d}_i}{{\sum }_i{w}_i}}.$$

(52)

For the $Q_1$–$Q_4$ entries in

Table 2. Fixed-parameter comparison with the selected Pantheon SN Ia sample. The interacting $Q_1$–$Q_4$ entries use $\xi =0.04$, $(\Delta ,\omega ,\beta )=(0.8,0.263,0.123)$, and ${\mathsf{\Omega }}_{\mathrm{de}0}=0.6847$. The additive nuisance parameter $\mathcal{M}$ is analytically profiled out, and diagonal FITRES uncertainties are used.

Model	${\mathcal{M}}_{\mathbf{best}}$	${\mathit{\chi }}_{\mathbf{prof}}^2$	${\mathit{\chi }}_{\mathbf{prof}}^2/\mathbf{dof}$	AIC	BIC	$\mathbf{\Delta }\mathbf{AIC}$
$\mathsf{\Lambda }$CDM	0.010279	1054.76507	1.00934	1056.76507	1061.71780	0.00000
$Q_1$	0.007745	1058.09566	1.01253	1060.09566	1065.04839	3.33059
$Q_2$	0.022976	1056.99324	1.01148	1058.99324	1063.94597	2.22817
$Q_3$	0.025185	1057.86592	1.01231	1059.86592	1064.81865	3.10085
$Q_4$	0.017461	1055.97420	1.01050	1057.97420	1062.92693	1.20913

, we fix $\xi =0.04$, $(\Delta ,\omega ,\beta )=(0.8,0.263,0.123)$, and ${\mathsf{\Omega }}_{\mathrm{de}0}=0.6847$. The cosmological background is therefore not fitted; only the additive nuisance parameter $\mathcal{M}$ is profiled. Hence $k=1$ and

$$\mathrm{dof}=N-k=1045.$$

(53)

The information criteria are evaluated from

$$\mathrm{AIC}={\chi }_{\mathrm{prof}}^2+2k,\mathrm{BIC}={\chi }_{\mathrm{prof}}^2+klnN.$$

(54)

Table 2 gives a fixed-background comparison under the adopted diagonal-error likelihood. All five cases have reduced chi-square values close to unity. The flat $\mathsf{\Lambda }$CDM reference gives the smallest ${\chi }^2$, AIC, and BIC values. Among the interacting BHDE benchmarks, $Q_4$ is closest to $\mathsf{\Lambda }$CDM, followed by $Q_2$, $Q_3$, and $Q_1$. The corresponding information-criterion differences are $\Delta \mathrm{AIC}=\Delta \mathrm{BIC}=1.20913$, $2.22817$, $3.10085$, and $3.33059$, respectively. These values compare only the selected background curves under the simplified likelihood; they do not constitute a global fit of the BHDE parameter space or a complete test of observational viability.

Figure 4 shows one-parameter sensitivity tests for the $Q_1$ interaction with $\xi =0.04$. The upper panels display the SN Ia Hubble diagram, and the lower panels show $\Delta \mu ={\mu }_{\mathrm{model}}-{\mu }_{\mathsf{\Lambda }\mathrm{CDM}}$. The nuisance parameter is profiled independently for each curve, so the residuals primarily compare the redshift-dependent shape of the distance relation. The models are nearly degenerate at low redshift and separate gradually toward the high-redshift end. Within the displayed parameter ranges, variations of $\Delta$ and $\beta$ produce clearer changes than variations of $\omega$. These plots characterize local sensitivity around the benchmark point rather than posterior parameter constraints.

4. Statefinder Diagnostic Examination

Geometric diagnostics provide a complementary way to compare dark-energy models at the level of background kinematics [60,61]. Defining $H=\dot{a}/a$ and $q=-\ddot{a}/\left(a{H}^2\right)$, the statefinder pair is

$$r\equiv {\displaystyle \frac{\stackrel{⃛}{a}}{a{H}^3}}=2q^2+q-{\displaystyle \frac{\dot{q}}{H}}=2q^2+q-a{\displaystyle \frac{dq}{da}},$$

(55)

$$s\equiv {\displaystyle \frac{r-1}{3\left(q-\frac{1}{2}\right)}}.$$

(56)

The statefinder hierarchy has been widely applied to distinguish dynamical and holographic dark-energy models [62,63]. The quantities used below are

$$S_3^{\left(1\right)}\equiv r,S_3^{\left(2\right)}\equiv {\displaystyle \frac{S_3^{\left(1\right)}-1}{3\left(q-\frac{1}{2}\right)}}=s.$$

(57)

For a spatially flat $\mathsf{\Lambda }$CDM background, $S_3^{\left(1\right)}=1$, $S_3^{\left(2\right)}=0$, and the corresponding fixed point in the s–r plane is $(0,1)$. For the numerical evaluation, $q\left(\ln a\right)$ is constructed from the same high-precision background solution used for ${\mathsf{\Omega }}_{\mathrm{de}}\left(\ln a\right)$. Since $adq/da=dq/dlna$, the derivative entering Equation (55) is obtained directly from the derivative of the interpolating solution rather than by finite differencing the plotted data. The statefinder calculation uses the same adaptive stiffness-switching integration, working precision 30, and accuracy and precision goals of 14 as the background analysis. Since higher derivatives are especially sensitive near the boundary of an interpolation interval, the physical discussion is based on the smooth finite-redshift trajectories and the explicitly evaluated present-day values; no conclusion is drawn from small-scale features extremely close to $z=-1$.

As a numerical convergence test, the statefinder trajectories were recalculated using working precision 40 and accuracy and precision goals equal to 18, and were compared with the results obtained using working precision 30 and accuracy and precision goals equal to 14. Over the smooth finite-redshift interval $-0.9\le z\le 3$, including all four interaction prescriptions and all coupling strengths considered, the maximum absolute differences were

$$max\left|\delta r\right|=4.50384\times {10}^{-9},max\left|\delta s\right|=1.67530\times {10}^{-9}.$$

(58)

The statefinder trajectories and the present-day values reported in

Table 3. Present-day dynamical and statefinder quantities for the four interaction forms. The last row gives the corresponding flat $\mathsf{\Lambda }$CDM benchmark.

Model	$\mathit{\xi }$	${\mathit{w}}_{\mathbf{de}0}$	${\mathit{q}}_0$	${\mathit{z}}_{\mathit{t}}$	${\mathit{r}}_0$	${\mathit{s}}_0$
$Q_1$	0.00	−0.752928	−0.407645	0.85845	0.679369	0.117752
	0.04	−0.831448	−0.495895	1.15252	0.705092	0.098708
	0.08	−0.909967	−0.584146	1.52466	0.738510	0.080398
	0.12	−0.988486	−0.672396	2.02175	0.779622	0.062658
$Q_2$	0.00	−0.752928	−0.407645	0.85845	0.679369	0.117752
	0.04	−0.774904	−0.432345	1.09535	0.634521	0.130667
	0.08	−0.796881	−0.457045	1.43838	0.594739	0.141150
	0.12	−0.818857	−0.481745	1.97160	0.560024	0.149386
$Q_3$	0.00	−0.752928	−0.407645	0.85845	0.679369	0.117752
	0.04	−0.770099	−0.426944	0.97610	0.652739	0.124877
	0.08	−0.787269	−0.446242	1.10864	0.628606	0.130831
	0.12	−0.804440	−0.465541	1.25782	0.606970	0.135686
$Q_4$	0.00	−0.752928	−0.407645	0.85845	0.679369	0.117752
	0.04	−0.794468	−0.454333	1.11495	0.645439	0.123842
	0.08	−0.836008	−0.501021	1.44445	0.621637	0.125992
	0.12	−0.877548	−0.547709	1.87260	0.607963	0.124728
$\mathsf{\Lambda }$CDM	–	−1.000000	−0.527050	0.63156	1.000000	0.000000

are therefore numerically stable to well beyond the quoted precision.

Substituting interaction forms $Q_1$–$Q_4$ into the background equation gives

$${\left.{\displaystyle \frac{\dot{H}}{H^2}}\right|}_{Q_1}={\displaystyle \frac{3\xi {\mathsf{\Omega }}_{\mathrm{de}}+2\beta (\gamma +\beta )+(\beta -3)(1+\gamma -{\mathsf{\Omega }}_{\mathrm{de}})}{2+2\gamma -{\mathsf{\Omega }}_{\mathrm{de}}(2-\Delta )}},$$

(59)

$${\left.{\displaystyle \frac{\dot{H}}{H^2}}\right|}_{Q_2}={\displaystyle \frac{2\beta (\gamma +\beta )+(3\xi +\beta -3)(1+\gamma -{\mathsf{\Omega }}_{\mathrm{de}})}{2+2\gamma -{\mathsf{\Omega }}_{\mathrm{de}}(2-\Delta )}},$$

(60)

$${\left.{\displaystyle \frac{\dot{H}}{H^2}}\right|}_{Q_3}={\displaystyle \frac{3\xi {\mathsf{\Omega }}_{\mathrm{de}}(1+\gamma -{\mathsf{\Omega }}_{\mathrm{de}})+2\beta (\gamma +\beta )(1+\gamma )+(\beta -3)(1+\gamma -{\mathsf{\Omega }}_{\mathrm{de}})(1+\gamma )}{\left[2+2\gamma -{\mathsf{\Omega }}_{\mathrm{de}}(2-\Delta )\right](1+\gamma )}},$$

(61)

$${\left.{\displaystyle \frac{\dot{H}}{H^2}}\right|}_{Q_4}={\displaystyle \frac{3\xi {\left[{\mathsf{\Omega }}_{\mathrm{de}}(1+\gamma -{\mathsf{\Omega }}_{\mathrm{de}})\right]}^{1/2}+2\beta (\gamma +\beta )+(\beta -3)(1+\gamma -{\mathsf{\Omega }}_{\mathrm{de}})}{2+2\gamma -{\mathsf{\Omega }}_{\mathrm{de}}(2-\Delta )}}.$$

(62)

Figure 5 shows the evolution of $S_3^{\left(1\right)}=r$. Over the displayed interval, the trajectories exhibit a pronounced finite-redshift minimum and move toward the $\mathsf{\Lambda }$CDM reference value as z approaches $-0.99$. The effect of the coupling depends on the interaction form. For $Q_1$, increasing $\xi$ raises the present value of r from $0.679369$ to $0.779622$, thereby moving the current state closer to unity, although the ordering changes along the full trajectory. For $Q_2$, $Q_3$, and $Q_4$, a stronger coupling lowers the present value of r. The trajectory separation is largest for $Q_2$, weakest for $Q_3$, and intermediate for $Q_4$.

Figure 6 displays positive values of $S_3^{\left(2\right)}=s$ throughout the finite-redshift domain and a decrease toward zero as z approaches $-0.99$. In the $Q_1$ case, increasing $\xi$ lowers $S_3^{\left(2\right)}$ over the main redshift interval and shifts the present state closer to the $\mathsf{\Lambda }$CDM value. The $Q_2$, $Q_3$, and $Q_4$ cases show redshift-dependent curve crossings, so their coupling dependence cannot be represented by a single ordering over the entire evolution. The separation is comparatively weak for $Q_3$ and more visible for $Q_2$ and $Q_4$.

The s–r trajectories in Figure 7 remain in the region $s>0$ and $r<1$ over the displayed finite-redshift evolution and move toward the $\mathsf{\Lambda }$CDM point $(0,1)$ near the future boundary of the numerical interval. For $Q_1$, a larger coupling shifts the present point toward $(0,1)$, with $r_0$ increasing and $s_0$ decreasing. For $Q_2$ and $Q_3$, the present point moves away from the fixed point as $\xi$ increases. For $Q_4$, $r_0$ decreases, whereas $s_0$ changes only slightly and is not strictly monotonic. Consequently, the statefinder plane distinguishes both the interaction prescriptions and the response of their present kinematic states to the coupling.

For quantitative comparison, flat $\mathsf{\Lambda }$CDM with ${\mathsf{\Omega }}_{m0}=0.3153$ and ${\mathsf{\Omega }}_{\mathsf{\Lambda }0}=0.6847$ has $w_{\mathrm{de}0}=-1$, $q_0=-0.52705$, $z_t=0.63156$, $r_0=1$, and $s_0=0$. The benchmark BHDE cases therefore remain distinguishable from $\mathsf{\Lambda }$CDM through their transition redshifts and present-day statefinder coordinates. The approach of the numerical trajectories toward $(0,1)$ near $z=-0.99$ is a finite-interval result and is not presented as an analytical proof of a future attractor.

5. Conclusions

We have investigated interacting Barrow holographic dark energy with the Hubble horizon as the infrared cutoff in a fractal cosmological background. Four phenomenological interactions between BHDE and an effective pressureless matter component were considered, including the nonlinear $Q_3$ and $Q_4$ prescriptions. The autonomous equations were integrated for the benchmark parameters $(\Delta ,\omega ,\beta )=(0.8,0.263,0.123)$ and the Planck-inspired normalization ${\mathsf{\Omega }}_{\mathrm{de}0}=0.6847$ over $-0.99\le z\le 3$.

The numerical solutions remain within the required physical background domain throughout this interval: ${\mathsf{\Omega }}_{\mathrm{de}}>0$, ${\mathsf{\Omega }}_{\mathrm{dm}}\ge 0$, the principal denominator remains positive, and the additional factors entering $Q_3$ and $Q_4$ remain real. These checks establish the consistency of the homogeneous benchmark solutions over the numerical domain, but they do not address perturbative stability.

The interaction modifies both the strength and the timing of the late-time acceleration. All four prescriptions reduce to the same non-interacting solution, with $z_t=0.85845$. At $\xi =0.12$, the transition redshifts are $2.02175$, $1.97160$, $1.25782$, and $1.87260$ for $Q_1$, $Q_2$, $Q_3$, and $Q_4$, respectively. Thus, $Q_1$ produces the strongest coupling-induced shift and $Q_3$ the weakest. The $Q_4$ solutions recover a decelerating branch within the plotted past domain, with a finite transition redshift for every coupling considered. Increasing the coupling also makes the present-day values of $w_{\mathrm{de}}$ and q more negative, although the magnitude of the response depends on the interaction structure.

The statefinder hierarchy provides an additional geometric discrimination. Increasing $\xi$ moves the present $Q_1$ state toward the $\mathsf{\Lambda }$CDM point, while the other prescriptions show different directions and strengths of departure. Near the future boundary of the numerical interval, the trajectories move toward the $\mathsf{\Lambda }$CDM coordinates, but this finite-interval behavior is not an analytical demonstration of an asymptotic attractor.

A fixed-background SN Ia comparison was performed using 1046 selected Pantheon supernovae, diagonal FITRES uncertainties, and an analytically profiled additive nuisance parameter. For $\xi =0.04$, flat $\mathsf{\Lambda }$CDM gives the lowest ${\chi }^2$, AIC, and BIC. Among the four interacting BHDE benchmarks, $Q_4$ is closest to the reference model, with $\Delta \mathrm{AIC}=\Delta \mathrm{B}=IC1.20913$. This ranking is specific to the selected parameter values and the adopted simplified likelihood; it should not be interpreted as a global model-selectio result.

For the adopted normalization, the modified closure relation gives ${\mathsf{\Omega }}_{\mathrm{dm}0}=0.191637$, below the standard cosmological matter abundance. The present construction should therefore be regarded as a reduced late-time benchmark rather than a complete precision-cosmology model. The present conclusions are specific to the Hubble-horizon infrared cutoff; alternative cutoffs require separate background equations and an independent numerical analysis. Future work should include separately evolved baryons and radiation, the full SN covariance, joint constraints from BAO, CMB, and cosmic-chronometer data, and cosmological perturbations, including the stability of the nonlinear interaction prescriptions. A dedicated analysis of $H\left(t\right)$ and $a\left(t\right)$ is also required before any conclusion can be drawn about future-time attractors or finite-time singularities.