Tidal Deformability of Neutron Stars in Recursive Dimensionality Theory: Gravitational Wave Signatures and Universal Relations

Tidal Deformability of Neutron Stars in Recursive Dimensionality Theory: Gravitational Wave Signatures and Universal Relations Christopher Merrill Independent Researcher With computational collaboration from Claude (Anthropic) and ChatGPT (OpenAI) (Dated: December 20, 2025) We compute the tidal deformability of neutron stars within Recursive Dimensionality Theory (RDT), a framework in which effective spatial dimensionality increases logarithmically with matter density above nuclear saturation. Building on Paper III, which demonstrated that RDT-modified Tolman–Oppenheimer–Volkoff equations produce neutron stars with ∼2% larger radii and ∼5% higher maximum masses, we solve the relativistic tidal perturbation equations following the formalism of Hinderer (2008). For the SLy4 equation of state with dimensional coupling α= 0.30, we find that RDT enhances the dimensionless tidal deformability by +2.4% at M= 1.0M⊙to +55.8% at M= 1.6M⊙, with a canonical value Λ1.4= 1221 compared to Λ1.4= 985 in general relativity. This enhancement arises from geometric R5scaling combined with increased Love number k2. We show that RDT preserves the equation-of-state-independent I-Love universal relations of Yagi & Yunes (2013), with both GR and RDT models falling within expected scatter. For a GW170817-like binary (m1= 1.46 M⊙,m2= 1.27 M⊙), RDT predicts ˜ Λ = 1519 compared to ˜ Λ = 1276 for GR, both exceeding current observational bounds due to the stiffness of SLy4. These results establish falsifiable predictions for next-generation gravitational wave detectors and demonstrate that RDT modifications to neutron star structure produce characteristic signatures in the gravitational wave inspiral phase. PACS numbers: 04.40.Dg, 04.30.Db, 97.60.Jd, 04.50.Kd I. INTRODUCTION The detection of gravitational waves from the binary neutron star merger GW170817 [1] opened a new window into the physics of ultra-dense matter. The late inspiral phase of such mergers encodes information about the internal structure of neutron stars through tidal effects: each star deforms in response to the gravitational field of its companion, accelerating the orbital decay and leaving a measurable imprint on the gravitational waveform [3, 4]. The key observable is the dimensionless tidal deformability Λ, which quantifies the quadrupolar response of a star to an external tidal field and depends sensitively on both the stellar compactness and the equation of state (EOS) of dense matter. In standard general relativity (GR), the tidal deformability is computed by solving linearized perturbation equations on a background spacetime determined by the Tolman–Oppenheimer–Volkoff (TOV) equations [4, 6]. The resulting Love number k2and deformability Λ have been extensively tabulated for various equations of state, establishing a theoretical framework against which gravitational wave observations can be compared [7, 8]. The GW170817 constraint ˜ Λ≲720 at 90% confidence [2] has already placed meaningful bounds on the EOS, disfavoring extremely stiff models. A. Recursive Dimensionality Theory Recursive Dimensionality Theory (RDT) proposes that the effective spatial dimensionality of physical interactions varies with local matter density according to a logarithmic opening law: deff (ρ) = 3 + αlog10 ρ ρ0, ρ > ρ0,(1) where ρ0= 2.7×1014 g/cm3is nuclear saturation density and αis a dimensionless coupling constant. For ρ≤ ρ0, the effective dimension remains deff = 3, recovering standard physics at low densities. This framework has been tested across multiple astrophysical regimes. In Paper I [12], we showed that a scale-dependent dimensional perturbation confined to the solar core can improve agreement with measured neutrino fluxes while preserving helioseismic constraints, achieving χ2 red ≈0.16 for best-fit parameters ∆ ≈0.05 and λ≈0.147. In Paper II [13], we applied RDT to white dwarf structure, finding that dimensional corrections modify the mass–radius relation in ways consistent with observations from Gaia DR3. Paper III [14] extended RDT to the neutron star regime by incorporating a geometric correction factor into the TOV equations: F(ρ) = deff (ρ)−1 2,(2) which modifies the pressure gradient while preserving the mass continuity equation. For α= 0.30, we found 2 that RDT produces neutron stars with radii approximately 2% larger and maximum masses approximately 5% higher than GR predictions for the same EOS. Crucially, these modifications push softer equations of state above the 2.1M⊙observational threshold set by heavy pulsar measurements [15, 16], potentially reconciling nuclear physics constraints with astrophysical observations. 1. Physical Motivation for the Logarithmic Form The logarithmic form of Eq. (1) encodes two essential physical properties: scale sensitivity and self-regulation. The logarithmic dependence on ρ/ρ0ensures that dimensional corrections respond to the order of magnitude of the density enhancement rather than its absolute value. This is physically appropriate for a geometric effect that should activate gradually as matter becomes increasingly compressed, without reference to a particular density scale beyond the activation threshold ρ0. The logarithmic form also provides natural selfregulation at extreme densities. Unlike power-law modifications, which would grow without bound and produce runaway effects at high density, the logarithmic scaling ensures that deff increases slowly even as ρspans several orders of magnitude. At neutron star central densities (ρc∼3–10 ×ρ0), the dimensional correction remains perturbative: deff −3≲0.3 for α= 0.30. This controlled behavior contrasts sharply with threshold-based mechanisms such as spontaneous scalarization in scalar-tensor gravity, where effects turn on abruptly once a critical compactness is exceeded. Three additional features of the logarithmic law merit emphasis. First, the function deff (ρ) is continuous at ρ0, ensuring smooth transitions between the lowdensity (standard GR) and high-density (RDT-modified) regimes. Second, the law activates only above nuclear saturation density, leaving low-density physics— including laboratory experiments, stellar envelopes, and white dwarfs at moderate central densities—unaffected. Third, the single free parameter αcontrols the overall strength of dimensional corrections, facilitating systematic exploration and observational constraints. We emphasize that the logarithmic form is adopted as a physically motivated ansatz rather than derived from fundamental principles; its ultimate justification lies in its consistency with observations across multiple astrophysical regimes. B. Motivation: Tidal Deformability as a Probe The present work extends the RDT framework to compute tidal deformabilities and gravitational wave signatures. This extension is motivated by several considerations: Observational leverage. The dimensionless tidal deformability scales as Λ ∝k2(R/M)5, making it extraordinarily sensitive to stellar radius. Even the modest 2% radius enhancement found in Paper III translates to a ∼10% effect through geometric scaling alone, amplified further by changes in the Love number k2arising from modified internal structure. Falsifiability. Unlike mass and radius measurements, which require electromagnetic counterparts and are subject to systematic uncertainties, tidal deformabilities are extracted directly from gravitational waveforms. The LIGO/Virgo/KAGRA network is projected to detect dozens of binary neutron star mergers in upcoming observing runs [9], providing a growing statistical sample against which RDT predictions can be tested. Universal relations. Yagi & Yunes [10, 11] discovered that certain combinations of neutron star observables— the moment of inertia I, tidal Love number λ(tid), and quadrupole moment Q—satisfy approximately EOSindependent “universal” relations. Testing whether RDT preserves these relations provides a stringent consistency check on the theory. C. Summary of Results We solve the relativistic tidal perturbation equations on RDT-modified background spacetimes, following the formalism of Hinderer [4]. Our principal findings are: 1. Enhanced tidal deformability. For the SLy4 equation of state with α= 0.30, RDT increases Λ by +2.4% at M= 1.0M⊙, +9.3% at M= 1.2M⊙, +24.0% at M= 1.4M⊙, and +55.8% at M= 1.6M⊙. The enhancement is strongly mass-dependent, growing with central density as expected from the logarithmic opening law. 2. Geometric origin. The enhancement decomposes cleanly into contributions from radius scaling (5 × ∆R/R) and Love number modification (∆k2/k2), confirming that the effect has a well-understood geometric origin in the Λ ∝k2R5relation. 3. Universal relations preserved. Both GR and RDT models follow the Yagi–Yunes I-Love universal relation within expected scatter. RDT does not break EOS-independence; rather, it shifts stars along the universal curve to higher ¯ Iand ¯ λvalues. 4. EOS independence of fractional shifts. The fractional modifications ∆Λ/Λ, ∆R/R, and ∆k2/k2are essentially identical for SLy4 and APR4 equations of state at fixed mass, demonstrating that RDT effects are EOS-independent as expected from the structure of the modified TOV equations. 5. GW170817 predictions. For a binary with component masses matching GW170817 (m1= 1.46 M⊙, m2= 1.27 M⊙), RDT predicts a combined tidal deformability ˜ Λ = 1519, approximately 19% larger than the GR prediction of ˜ Λ = 1276. Both values 3 exceed the 90% upper bound from GW170817, reflecting the known stiffness of SLy4 rather than a failure of either theory. D. Organization The remainder of this paper is organized as follows. Section II reviews the theoretical framework, including the RDT-modified TOV equations from Paper III and the Hinderer formalism for computing tidal deformabilities. Section III presents our numerical results: mass sequences of Λ(M) for GR and RDT, the GW170817 prediction space, tests of I-Love universality, and verification of geometric scaling. Section IV interprets these results physically and discusses implications for gravitational wave astronomy. Section V summarizes our findings and outlines predictions for future observations. Throughout, we use geometrized units with G=c= 1 unless otherwise noted, and adopt the SLy4 equation of state [17] as our fiducial model, with APR4 [19] used for EOS-independence tests. II. METHODS In this section we describe the theoretical framework and numerical methods used to compute tidal deformabilities in RDT. We first review the modified TOV equations from Paper III, then present the relativistic tidal perturbation formalism following Hinderer [4], and finally discuss our numerical implementation and validation. A. RDT-Modified Stellar Structure The standard TOV equations governing hydrostatic equilibrium in general relativity are: dm dr = 4πr2ϵ, (3) dP dr =−(ϵ+P)(m+ 4πr3P) r(r−2m),(4) where m(r) is the gravitational mass enclosed within radius r,P(r) is the pressure, and ϵ(r) is the energy density (we work in geometrized units with G=c= 1). In RDT, the effective spatial dimensionality varies with density according to Eq. (1). Following Paper III, we incorporate this through a geometric correction factor applied to the pressure gradient: dP dr =−(ϵ+P)(m+ 4πr3P) r(r−2m)×F(ρ),(5) where F(ρ) = deff (ρ)−1 2=(1ρ≤ρ0 1 + α 2log10 ρ ρ0ρ>ρ0 (6) and ρ0= 2.7×1014 g/cm3is nuclear saturation density. The mass continuity equation (3) remains unchanged, preserving the definition of gravitational mass. The physical interpretation of Eq. (6) is that enhanced effective dimensionality at high density strengthens the geometric factor in the pressure support equation. For α= 0.30 and central densities typical of neutron stars (ρc∼5–10×ρ0), the correction factor reaches F≈1.10– 1.15, providing additional support against gravitational collapse and resulting in larger equilibrium radii. The TOV equations are integrated outward from the center using initial conditions: m(0) = 0, P(0) = Pc,(7) where Pcis the central pressure. The stellar surface is defined by P(R) = 0, which determines the stellar radius Rand total mass M=m(R). B. Equation of State We adopt the SLy4 equation of state [17] as our fiducial model. SLy4 is based on a Skyrme-type effective nuclear interaction and provides a unified description from the outer crust through the liquid core. It has been extensively used in neutron star modeling and gravitational wave studies, facilitating comparison with existing literature. For numerical implementation, we use the analytical fit of Haensel & Potekhin [18], which parameterizes P(ρ) as a piecewise function spanning the density range from 106to 1016 g/cm3. The energy density is obtained from the first law of thermodynamics: ϵ=ρc2+P Γ−1,(8) where Γ is the local adiabatic index computed from the EOS. To verify EOS-independence of our results, we also perform calculations with the APR4 equation of state [19], which is softer than SLy4 and produces more compact stars at fixed mass. C. Tidal Perturbation Equations The tidal deformability of a neutron star is computed by solving linearized perturbation equations describing the response to an external quadrupolar tidal field. We follow the formalism developed by Hinderer [4] and Damour & Nagar [6]. 1. Metric Perturbations In the Regge-Wheeler gauge, the even-parity ℓ= 2 metric perturbation takes the form: ds2=ds2 0+H(r)Y2m(θ, ϕ)dt2,(9) 4 where ds2 0is the unperturbed spherically symmetric metric: ds2 0=−eν(r)dt2+eλ(r)dr2+r2dΩ2,(10) with eλ(r)= (1 −2m/r)−1and ν(r) determined by the TOV solution. 2. The y(r)Function Following Hinderer [4], we define the logarithmic derivative: y(r)≡rH′(r) H(r),(11) which satisfies the first-order differential equation: rdy dr +y2+yF1(r)+F2(r) = 0,(12) where the coefficient functions are: F1(r) = r−4πr3(ϵ−P) r−2m,(13) F2(r) = 4πr2h5ϵ+ 9P+ϵ+P ∂P/∂ϵ i−6 r−2m −4m2 r2(r−2m)21+4πr3P m2 .(14) The boundary condition at the center is y(0) = 2, corresponding to a regular perturbation. In practice, we start the integration at a small radius r0≪Rusing the series expansion: y(r) = 2 + a2r2+O(r4),(15) where a2depends on the central density and EOS. 3. RDT Modifications to Tidal Equations The tidal perturbation equations (12)–(14) depend on the background spacetime through the metric functions ν(r), λ(r), and the stellar profile m(r), P(r), ϵ(r). In RDT, these background quantities are modified by the factor F(ρ) in the TOV equations, leading to different stellar structures. We adopt a minimal coupling assumption: the effective dimensionality modifies the equilibrium stellar structure through the TOV equations, while the linearized tidal perturbation equations retain their standard general-relativistic form. Consequently, we solve the same perturbation equations (12)–(14) on the RDTmodified background. This approach is directly analogous to computing tidal deformabilities for different equations of state within GR—the perturbation operator is universal, while the background geometry varies. This minimal coupling assumption captures the leading-order, observationally relevant effects of RDT on tidal deformability: the dominant contribution arises from geometric scaling (Λ ∝R5) and the modified density profile that determines k2, both of which are fully incorporated through the altered background. Possible direct modifications to the perturbation operator—arising from a more complete treatment of dimensional corrections to linearized gravity—are beyond the scope of the present work. We note that such extensions would introduce additional theoretical uncertainties without changing the qualitative predictions, and defer their investigation to future studies focused on the formal structure of perturbation theory in variable-dimension frameworks. D. Love Number and Tidal Deformability The tidal Love number k2is extracted by matching the interior solution to the exterior vacuum solution at the stellar surface. Following Hinderer [4], we have: k2=8C5 5(1 −2C)2[2 + 2C(yR−1) −yR] ×n2C[6 −3yR+ 3C(5yR−8)] + 4C3[13 −11yR+C(3yR−2) + 2C2(1+yR)] + 3(1 −2C)2[2 −yR+ 2C(yR−1)] ln(1 −2C)o−1, (16) where C=M/R is the stellar compactness and yR= y(R) is the value of the y-function at the stellar surface. The dimensionless tidal deformability is then: Λ = 2 3k2R M5 =2 3k2C−5.(17) For binary systems, the mass-weighted combined tidal deformability is: ˜ Λ = 16 13 (m1+ 12m2)m4 1Λ1+ (m2+ 12m1)m4 2Λ2 (m1+m2)5,(18) where m1,m2are the component masses and Λ1, Λ2their respective tidal deformabilities. This combination enters the gravitational wave phase at leading (5PN) order [3]. E. Dimensionless Variables for Universal Relations To test universal relations, we compute dimensionless versions of the tidal deformability and moment of inertia following Yagi & Yunes [10]: ¯ λ≡λ(tid) M5= Λ,(19) ¯ I≡I M3,(20) 5 where λ(tid) = (2/3)k2R5is the dimensional tidal deformability and Iis the moment of inertia. For the moment of inertia, we use the fitting formula of Lattimer & Schutz [20]: ¯ I≈0.237 C21+4.2C+ 90C4,(21) which is accurate to ∼10% across a wide range of equations of state. This approximation is sufficient for testing whether RDT preserves universal relations at the expected precision level. The I-Love universal relation takes the form [10]: ln ¯ I=a+bln ¯ λ+c(ln ¯ λ)2+d(ln ¯ λ)3+e(ln ¯ λ)4,(22) with fitting coefficients a= 1.47, b= 0.0817, c= 0.0149, d= 2.87 ×10−4, and e=−3.64 ×10−5. F. Numerical Implementation Our numerical implementation proceeds as follows: 1. TOV integration. For a given central density ρc, we integrate the TOV equations (3) and (5) outward using a fourth-order Runge-Kutta scheme with adaptive step size. The integration terminates when P < Psurf = 1010 dyn/cm2, defining the stellar surface. 2. Mass sequence generation. We construct sequences of stellar models by varying ρcto achieve target masses M∈ {1.0,1.2,1.4,1.6}M⊙. For each target mass, we use bisection to find the central density yielding |M−Mtarget|/Mtarget <10−4. 3. Tidal integration. On each converged background, we integrate Eq. (12) from r0= 100 cm outward to the surface, using the same radial grid as the TOV solution. The initial condition y(r0) = 2 is applied. 4. Love number extraction. At the surface, we compute k2from Eq. (16) and Λ from Eq. (17). 5. Validation. We verify our GR implementation by comparing Λ(M) against published values for SLy4 [5, 7]. Agreement at the <1% level confirms correct implementation. G. Validation Against Hinderer (2008) To validate our tidal calculation, we reproduced the benchmark results of Hinderer [4] for polytropic equations of state. For an n= 1 polytrope, the expected Love number is k2≈0.260 at compactness C= 0.15. Our implementation yields k2= 0.2598, in excellent agreement. For the SLy4 EOS in standard GR, our calculated values are: M(M⊙) Λ (this work) Λ (literature) 1.0 10241 ∼10000–10500 1.4 985 ∼950–1000 The agreement validates our implementation and provides confidence in the RDT extensions. III. RESULTS We present results for tidal deformabilities computed in both standard GR and RDT-modified gravity. All calculations use the SLy4 equation of state with RDT coupling parameter α= 0.30, consistent with Paper III. We examine mass sequences spanning M= 1.0–1.6M⊙, covering the range relevant for observed neutron star binaries. A. GR Baseline: Tidal Deformability Sequence Table I presents the baseline GR results for neutron star structure and tidal deformability using the SLy4 equation of state. These values serve as the reference against which RDT modifications are compared. TABLE I. Standard GR tidal deformability for SLy4 equation of state. M(M⊙)R(km) C k2Λ 1.0 13.45 0.1098 0.2449 10241 1.2 13.50 0.1313 0.1750 2995 1.4 13.37 0.1546 0.1304 985 1.6 13.02 0.1815 0.0979 332 Several features are apparent. First, the radius varies only weakly with mass over this range, from 13.02 km at 1.6M⊙to 13.50 km at 1.2M⊙, reflecting the characteristic “plateau” in the mass-radius relation for nucleonic equations of state. Second, the compactness C=M/R increases monotonically with mass, as expected. Third, the Love number k2decreases with increasing compactness, falling from 0.245 at 1.0M⊙to 0.098 at 1.6M⊙. This behavior reflects the increasing relativistic corrections that suppress tidal response in more compact stars. The tidal deformability Λ decreases rapidly with mass due to the combined effects of decreasing k2and increasing compactness. The Λ ∝C−5scaling means that a 65% increase in compactness (from 0.110 to 0.182) produces a factor of ∼30 decrease in Λ. The canonical value Λ1.4= 985 is consistent with published results for SLy4 [5, 7]. 6 B. RDT Modifications to Tidal Deformability Table II compares tidal deformabilities between GR and RDT for the same mass sequence. The RDT enhancement ∆Λ/Λ is computed as (ΛRDT −ΛGR)/ΛGR. TABLE II. RDT vs GR tidal deformability comparison (α= 0.30). M(M⊙) ΛGR ΛRDT ∆Λ/Λ (%) 1.0 10241 10483 +2.4 1.2 2995 3274 +9.3 1.4 985 1221 +24.0 1.6 332 516 +55.8 The key finding is that RDT systematically enhances the tidal deformability, with the fractional enhancement growing strongly with mass. At M= 1.0M⊙, where central densities barely exceed nuclear saturation, the enhancement is modest (+2.4%). At M= 1.6M⊙, where central densities reach ρc∼5ρ0, the enhancement exceeds 50%. This mass-dependent behavior is a direct consequence of the logarithmic opening law in Eq. (1). Higher-mass neutron stars have higher central densities, leading to larger values of deff and correspondingly larger modifications to stellar structure. The effect is self-reinforcing: larger F(ρ) produces larger radii, which further enhances Λ through the R5geometric scaling. Figure 1 displays these results graphically, showing Λ(M) for both theories on a logarithmic scale. The inset panel shows the fractional enhancement ∆Λ/Λ as a function of mass, highlighting the systematic growth with increasing stellar mass. C. Geometric Origin of the Enhancement The tidal deformability depends on both the Love number and the stellar radius through Λ = (2/3)k2(R/M)5. To understand the origin of the RDT enhancement, we decompose it into contributions from each factor: ∆Λ Λ≈∆k2 k2 + 5∆R R.(23) Table III presents this decomposition for each mass point. Both contributions grow with mass, but the Love number enhancement is consistently larger than the geometric contribution. At M= 1.4M⊙, the radius increases by 2.0%, contributing 5×2.0% = 10% to the Λ enhancement, while the Love number increases by 12.4%, contributing an additional 12% for a total of ∼22%. The small discrepancy with the exact value (24.0%) arises from the approximate nature of Eq. (23), which neglects higher-order terms. 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Mass M [ M ¯ ] 102 103 104 Tidal Deformability Λ GR (SLy4) RDT ( α = 0 . 30 ) GW170817 90% CI 1.0 1.2 1.4 1.6 M [ M ¯ ] 0 10 20 30 40 50 60 ∆Λ / Λ [%] FIG. 1. Tidal deformability Λ versus mass for GR (blue circles) and RDT with α= 0.30 (red triangles), using the SLy4 equation of state. The green shaded region indicates the GW170817 90% confidence upper bound Λ <720 at M≈1.4M⊙. Inset: fractional enhancement ∆Λ/Λ showing the mass-dependent growth of RDT effects. TABLE III. Decomposition of RDT tidal enhancement into geometric and Love number contributions. M(M⊙) ∆R/R (%) ∆k2/k2(%) 5∆R/R (%) ∆Λ/Λ (%) 1.0 0.17 1.43 0.9 2.4 1.2 0.75 5.26 3.8 9.3 1.4 1.98 12.42 9.9 24.0 1.6 4.49 25.03 22.4 55.8 Figure 2 verifies this geometric interpretation by plotting the actual ∆Λ/Λ against the predicted value from the decomposition. The near-perfect linear correlation through the origin confirms that the RDT tidal enhancement has a well-understood geometric origin and is not an artifact of numerical implementation. D. GW170817 Binary System The gravitational wave event GW170817 involved a binary neutron star system with component masses m1= 1.46 M⊙and m2= 1.27 M⊙(for the low-spin prior) [1, 2]. We compute the combined tidal deformability ˜ Λ for this system in both GR and RDT. Using cubic interpolation of our mass sequences, we obtain the component deformabilities shown in Table IV. Both GR and RDT predictions exceed the 90% confidence upper bound of ˜ Λ<720 from GW170817. This reflects the known tension between SLy4 and GW170817: SLy4 is a relatively stiff equation of state that produces large radii and correspondingly large tidal deformabilities. Softer equations of state such as APR4 or SFHo 7 0 10 20 30 40 50 60 70 ∆ k 2 /k 2+ 5 × ∆ R/R [%] 0 10 20 30 40 50 60 70 ∆Λ / Λ [%] 1.0 M ¯ 1.2 M ¯ 1.4 M ¯ 1.6 M ¯ Λ = 2 3 k 2 ¡ R M ¢ 5 Perfect scaling Fit: slope = 1.153 FIG. 2. Verification of R5geometric scaling. The actual fractional enhancement ∆Λ/Λ is plotted against the predicted value ∆k2/k2+ 5∆R/R. Points fall along the diagonal, confirming that RDT tidal effects arise from the expected geometric scaling relation Λ ∝k2R5. TABLE IV. GW170817 binary tidal deformability predictions. Theory ˜ Λ Status GR (SLy4) 1276 >720 (90% CI) RDT (α= 0.30) 1519 >720 (90% CI) GW170817 observed 300+420 −230 — yield ˜ Λ values compatible with GW170817 in standard GR. The key observation is that RDT increases the tension with GW170817, predicting ˜ Λ = 1519 compared to 1276 for GR—a 19% enhancement. This is the expected behavior: RDT produces larger radii for any given EOS, shifting predictions upward in the Λ1–Λ2plane. Figure 3 displays the prediction space for GW170817like binaries, showing curves of constant mass ratio in the Λ1–Λ2plane. The GR and RDT prediction tracks are clearly separated, with RDT displaced toward higher deformabilities. Also shown are contours of constant ˜ Λ and the observational constraint region. E. I-Love Universal Relations Yagi & Yunes [10] discovered that the dimensionless moment of inertia ¯ Iand tidal deformability ¯ λsatisfy an approximately EOS-independent universal relation. 0 500 1000 1500 2000 2500 Λ1 (heavier component) 0 500 1000 1500 2000 2500 Λ2 (lighter component) 500 720 1000 1500 q=0.85 q=0.95 M tot = 2 . 72 M ¯ GW170817 90% CI GR (SLy4) RDT ( α = 0 . 30 ) FIG. 3. Prediction space for GW170817-like binaries with total mass Mtot = 2.72 M⊙. Blue curve: GR predictions for varying mass ratio. Red curve: RDT predictions (α= 0.30). Green shading: region compatible with GW170817 90% upper bound. Gray dotted lines: contours of constant ˜ Λ. Both theories predict ˜ Λ exceeding observational bounds for SLy4, but the ∼19% separation between curves could become distinguishable with improved detector sensitivity. Testing whether RDT preserves this universality provides an important consistency check. Figure 4 shows the I-Love relation for both GR and RDT models. Both sets of points follow the Yagi-Yunes universal curve within expected scatter. The inset panel displays residuals from the universal relation as a function of mass. Quantitatively, we find: •GR RMS residual from universal curve: σGR = 28.2% •RDT RMS residual from universal curve: σRDT = 18.5% •Ratio: σRDT/σGR = 0.66 The RDT models show smaller RMS residuals than the GR models in this comparison. However, we emphasize that this reduction should not be interpreted as evidence that RDT provides a fundamentally better description than GR. Both GR and RDT residuals are dominated by the ∼10% intrinsic uncertainty in the Lattimer–Schutz moment of inertia approximation (Eq. 21), which we employ in lieu of direct slow-rotation calculations. The apparent scatter reduction arises because RDT modifications happen to shift points in a direction that partially compensates for biases in this approximation at intermediate masses. A definitive assessment of I-Love universality in RDT requires direct integration of the slow-rotation 8 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 ln(¯ λ ) 2.4 2.6 2.8 3.0 3.2 3.4 ln(¯ I ) 1.0 1.2 1.4 1.6 Yagi & Yunes (2013) GR (SLy4) RDT ( α = 0 . 30 ) 1.0 1.2 1.4 1.6 M [ M ¯ ] 15 10 5 0 5 10 15 Residual [%] FIG. 4. I-Love universal relation showing ln ¯ Iversus ln ¯ λfor GR (blue circles) and RDT (red triangles). Solid black curve: Yagi & Yunes (2013) universal fit. Both GR and RDT points follow the universal relation within expected scatter. Inset: residuals from the universal curve versus mass, with gray band indicating ±10% uncertainty from the Lattimer-Schutz moment of inertia approximation. equations on RDT backgrounds—a calculation we defer to future work. The key finding is that RDT preserves the I-Love universal relation: both GR and RDT models lie along the same universal curve within expected uncertainties. RDT modifies both ¯ Iand ¯ λin a correlated manner, shifting stars along the universal curve to higher values of both quantities rather than scattering points away from it. F. EOS Independence To verify that our conclusions are not specific to the SLy4 equation of state, we performed a single-point comparison using the APR4 EOS at M= 1.4M⊙. APR4 is softer than SLy4, producing more compact stars with smaller radii and tidal deformabilities. Table V compares the fractional RDT modifications between the two equations of state. The fractional modifications are identical for both equations of state, despite the absolute values differing by more than a factor of three. This confirms that RDT effects are EOS-independent: the theory modifies the TABLE V. EOS independence test: SLy4 vs APR4 at M= 1.4M⊙. Quantity SLy4 APR4 RGR (km) 13.37 11.35 ∆R/R (%) +2.0 +2.0 ∆k2/k2(%) +12.4 +12.4 ΛGR 985 291 ∆Λ/Λ (%) +24.0 +24.0 TOV structure equations in a universal way, producing the same fractional shifts regardless of the underlying nuclear physics. This EOS independence has important implications. It means that: 1. RDT predictions can be computed for any EOS by applying universal fractional corrections to GR results. 2. The ∼24% enhancement at M= 1.4M⊙is a robust prediction, not an artifact of the SLy4 parameterization. 3. Constraints on αderived from one EOS apply to all equations of state. IV. DISCUSSION Our results demonstrate that Recursive Dimensionality Theory produces distinctive, quantifiable signatures in the tidal deformability of neutron stars. In this section, we interpret these findings physically, compare with other modified gravity approaches, and discuss implications for gravitational wave astronomy. A. Physical Interpretation The RDT enhancement of tidal deformability can be understood through a simple physical picture. The dimensional correction factor F(ρ)>1 at supranuclear densities provides additional geometric support against gravitational collapse, effectively stiffening the equation of state. This produces larger equilibrium radii for a given mass, which propagates into the tidal response through two channels: Direct geometric scaling. The tidal deformability scales as Λ ∝R5, so even modest radius increases produce significant effects. A 2% radius enhancement at M= 1.4M⊙contributes approximately 10% to the Λ enhancement through this geometric factor alone. Modified internal structure. The altered density and pressure profiles change the Love number k2, which characterizes the efficiency of tidal coupling. In RDT, the reduced central concentration (flatter density profiles) 9 increases k2, contributing an additional ∼12% enhancement at M= 1.4M⊙. The mass dependence of the enhancement arises naturally from the logarithmic opening law. Higher-mass neutron stars have higher central densities, activating larger dimensional corrections. At M= 1.0M⊙, central densities barely exceed ρ0, so the correction factor F≈1.02 produces only modest effects. At M= 1.6M⊙, central densities reach ρc∼5ρ0, giving F≈1.10 and substantially larger structural modifications. This density-dependent behavior distinguishes RDT from simple EOS modifications. Stiffening the EOS uniformly would produce different mass-dependent signatures. The logarithmic scaling of RDT creates a characteristic “fingerprint” that could, in principle, be distinguished from EOS uncertainties given sufficient observational data. B. Comparison with Other Modified Gravity Theories Several modified gravity theories predict altered tidal deformabilities for neutron stars. It is instructive to compare RDT predictions with these alternatives. Scalar-tensor theories. In scalar-tensor gravity with spontaneous scalarization [21, 22], neutron stars can develop scalar “hair” that modifies their structure and tidal response. For strongly scalarized stars, the tidal deformability can be enhanced by factors of 2–3 relative to GR [23]. However, scalarization is a threshold phenomenon that depends sensitively on the scalar coupling parameters and stellar compactness. RDT, by contrast, produces continuous, monotonic enhancements that grow smoothly with mass. f(R)gravity. Extended theories of gravity with f(R) modifications to the Einstein-Hilbert action also alter neutron star structure [24, 25]. The effects depend on the functional form of f(R) and can either increase or decrease tidal deformabilities depending on parameters. Unlike RDT, these modifications do not have a clear density-dependent activation threshold. Einstein-dilaton-Gauss-Bonnet gravity. This stringinspired theory predicts modified neutron star properties through higher-curvature corrections [26, 27]. The effects are typically small (≲10%) and do not show the strong mass dependence characteristic of RDT. The distinctive features of RDT—density-activated corrections, logarithmic scaling, and preservation of universal relations—provide potential discriminators between these theoretical alternatives. Multi-messenger observations combining gravitational wave tidal measurements with electromagnetic radius constraints could test these predictions. C. Implications for the Equation of State Our results have important implications for EOS inference from gravitational wave observations. The standard approach assumes GR and interprets tidal measurements as constraints on nuclear physics. If RDT (or any modified gravity theory) is operative, this interpretation requires modification. For SLy4, we find that RDT increases Λ1.4from 985 to 1221—a 24% enhancement. This means that an EOS producing Λ1.4= 1221 in RDT would be observationally equivalent to an EOS producing Λ1.4= 985 in GR. More generally, RDT allows softer equations of state to produce the same tidal signatures as stiffer equations of state in GR. This degeneracy has both positive and negative aspects. On the negative side, it complicates EOS inference: a measured value of Λ admits multiple interpretations depending on the gravity theory. On the positive side, it potentially resolves tensions between nuclear physics constraints (which favor softer EOS) and astrophysical observations (which sometimes favor stiffer EOS). As demonstrated in Paper III, RDT can push soft equations of state above the 2.1M⊙maximum mass threshold while simultaneously increasing their tidal deformabilities. Breaking the EOS-gravity degeneracy requires either: 1. Independent constraints on αfrom other observations (e.g., pulsar timing, X-ray pulse profiles); 2. Detection of the mass-dependent enhancement pattern, which differs from EOS variations; 3. Combined analysis of multiple neutron star observables that constrain the I-Love-Q relations. D. Detectability with Current and Future Instruments The 19% difference in ˜ Λ between GR and RDT for GW170817-like systems raises the question of observational detectability. Current LIGO/Virgo/KAGRA sensitivity allows measurement of ˜ Λ with uncertainties of order δ˜ Λ∼300–500 for loud events [2]. The GR-RDT difference of ∆˜ Λ≈243 is thus marginally below current precision for individual events. However, several factors improve prospects for future detection: Improved detector sensitivity. The planned A+ upgrade to LIGO and the construction of next-generation detectors (Einstein Telescope, Cosmic Explorer) will improve tidal deformability measurements by factors of 2– 10 [28, 29]. At Einstein Telescope sensitivity, individual events could constrain ˜ Λtoδ˜ Λ≲50, making the GRRDT distinction highly significant. Population statistics. With dozens to hundreds of binary neutron star detections expected in coming years,