Recursive Dimensionality Theory V: Black Holes and the Vacuum Nulling Prescription

Recursive Dimensionality Theory V: Black Holes and the Vacuum Nulling Prescription Christopher K. Merrill With computational collaboration by Claude (Anthropic) and ChatGPT (OpenAI) December 23, 2025 Abstract Recursive Dimensionality Theory (RDT) proposes that effective spatial dimensionality varies with matter density, with deff <3 in high-density environments. Papers I–IV established RDT predictions across eight orders of magnitude in density, from solar cores to neutron star interiors. Here we extend RDT to black hole spacetimes and establish the vacuum nulling prescription: because RDT couples to matter density rather than spacetime curvature, dimensional modifications vanish identically in vacuum (ρ= 0 ⇒deff = 3). Consequently, all black hole observables—horizon radius, photon sphere, shadow size, quasinormal mode frequencies, and gravitational wave signatures—match general relativity exactly. We demonstrate smooth continuity across the neutron star–black hole transition, with no discontinuities or residual dimensional effects. Event Horizon Telescope shadow measurements and LIGO-Virgo-KAGRA tests of general relativity are consistent with RDT predictions. We identify neutron star–black hole binary mergers as optimal differential tests: the black hole provides a built-in vacuum control, enabling within-event isolation of matterdependent effects. The vacuum nulling prescription closes a logical loophole in the RDT program, completing the theory’s description from stellar densities to vacuum spacetimes. 1 Introduction 1.1 RDT Recap Recursive Dimensionality Theory (RDT) proposes that the effective spatial dimensionality of spacetime varies with local matter density, providing geometric corrections to gravitational dynamics in high-density environments [1, 2, 3, 4]. The theory introduces a dimensional opening law relating effective dimensionality deff to density ρ, with deff <3 in regions of extreme compression and deff →3 as density decreases. Papers I–IV systematically tested RDT predictions across astrophysical density scales: solar neutrino production (Paper I), white dwarf structure (Paper II), neutron star properties (Paper III), and neutron star tidal deformability in binary mergers (Paper IV). Throughout this progression, the theory has maintained consistency with observations while using fixed parameters: the nuclear density threshold ρnuclear = 2.7×1014 g cm−3and the dimensional closing coefficient α. Paper III adopted α= 0.20 as the fiducial value and demonstrated saturation for α= 0.20–0.30 (maximum mass variations <0.001 M⊙). We use α= 0.25 (midpoint of the saturated range) for all calculations in the present work, noting this choice has negligible impact on observables (<0.1% variation). 1.2 Extension to Black Holes Papers I–IV addressed matter-supported objects: stars, white dwarfs, and neutron stars. A natural question arises: what happens to RDT effects when matter support vanishes entirely? Black holes—defined by their vacuum exteriors and event horizons—represent the limiting case 1 of gravitational collapse. This paper establishes RDT’s prediction for black hole spacetimes: the vacuum nulling prescription. Because RDT couples to matter density ρrather than spacetime curvature, and because ρ= 0 in the vacuum exterior of any black hole, the effective dimensionality reverts exactly to deff = 3. All geometric modifications vanish. Black holes in RDT are indistinguishable from their general relativistic counterparts. This result closes a potential logical loophole in the RDT program: the theory now provides a complete, self-consistent prescription across the full density spectrum. 1.3 Observational Motivation The past decade has witnessed unprecedented observational access to black hole spacetimes. The Event Horizon Telescope (EHT) has directly imaged the shadows of the supermassive black holes M87* [5] and Sgr A* [6], providing sub-horizon-scale constraints on photon geodesics. The LIGO-Virgo-KAGRA collaboration has detected dozens of binary black hole mergers [9] (updated through GWTC-4.0 [10]), enabling precision tests of GR through inspiral-mergerringdown consistency, post-Newtonian coefficient bounds, and quasi-normal mode measurements [11]. Crucially, the detection of neutron star–black hole (NS–BH) mergers [12] opens a new window: these asymmetric systems contain one matter-supported component (the neutron star) and one vacuum component (the black hole), providing differential tests that can isolate matter-dependent effects within a single observation. Any theory that modifies gravity must confront this observational landscape. 1.4 Paper Roadmap This paper establishes three central results: 1. Vacuum nulling prescription: We demonstrate that ρ=0⇒deff = 3, so all black hole observables match GR exactly. 2. Smooth NS–BH continuity: We show that effective dimensionality varies continuously from deff <3 in neutron star cores to deff = 3 in vacuum black hole exteriors, with no discontinuities. 3. NS–BH binaries as optimal differential tests: We identify these asymmetric systems as providing built-in vacuum controls for isolating RDT effects. We also demonstrate consistency with current EHT and LVK observations, which serve as null tests of the vacuum prescription, and provide explicit falsifiability criteria for future observations. 1.5 Parameter Consistency Throughout this work, we employ the RDT parameters established in Papers III–IV: the nuclear density threshold ρnuclear = 2.7×1014 g cm−3and the dimensional closing coefficient α. Paper III adopted α= 0.20 as the fiducial value and demonstrated saturation for α= 0.20–0.30 (maximum mass variations <0.001 M⊙). We use α= 0.25 (midpoint of the saturated range) for all calculations, noting this choice has negligible impact on observables (<0.1% variation). No new parameters are introduced for black hole spacetimes. The vacuum nulling prescription— that deff = 3 when ρ= 0—is a direct consequence of the existing theory, not an additional assumption. 2 2 The Vacuum Nulling Prescription 2.1 Dimensional Opening Law and Its Vacuum Limit The RDT dimensional opening law established in Papers III–IV relates effective dimensionality to matter density: deff(ρ) =    3 if ρ≤ρnuclear 3−αlog10 ρ ρnuclear if ρ>ρnuclear (1) where α= 0.25 (within the saturated range 0.20–0.30) and ρnuclear = 2.7×1014 g cm−3. In the vacuum limit ρ→0: lim ρ→0deff(ρ) = 3 (2) This follows directly from Eq. (1): for all ρ≤ρnuclear, including ρ= 0, we have deff = 3 exactly. 2.2 Physical Interpretation The vacuum nulling prescription has a clear physical interpretation: RDT is a theory of matter-geometry coupling, not a modification of vacuum gravity. Dimensional effects arise from the compression of matter beyond nuclear densities. In the absence of matter, there is no source for dimensional modification. This distinguishes RDT from other modified gravity theories (scalar-tensor, f(R), massive gravity) that modify the vacuum Einstein equations themselves. In RDT, vacuum spacetime is exactly described by general relativity. 2.3 Black Hole Spacetimes For any black hole with vacuum exterior (ρ= 0 for r > 0): deff(r) = 3 for all r > 0 (3) Consequently, the spacetime geometry is exactly the Schwarzschild or Kerr solution of general relativity. All observables derived from the vacuum geometry match GR predictions exactly: Table 1: RDT predictions for black hole observables Observable RDT Prediction Deviation from GR Horizon radius rH2GM/c20% ISCO radius 6GM/c20% Photon sphere 3GM/c20% Shadow radius 3√3GM/c20% QNM frequency f220 fGR(M, a) 0% QNM damping τ220 τGR(M, a) 0% PN coefficients GR values 0% Tidal deformability ΛBH 0 — Hawking temperature ℏc3/(8πGMkB) 0% 3 3 Neutron Star–Black Hole Continuity 3.1 Dimensional Profiles Across the Mass Spectrum We now demonstrate that RDT provides smooth continuity across the transition from mattersupported neutron stars to vacuum black holes. Using the SLy4 equation of state, we compute deff(r) profiles for neutron stars of increasing central density: Table 2: Dimensional profiles for NS models approaching BH transition Model ρc(g/cm3)dcore eff dsurface eff dexterior eff Canonical NS (1.4 M⊙) 5.5×1014 2.92 3.00 3.00 Massive NS (2.0 M⊙) 9.0×1014 2.87 3.00 3.00 Near-maximum NS 1.8×1015 2.79 3.00 3.00 Black hole 0 (vacuum) — — 3.00 The key features are: 1. Core dimensional closing:deff <3 in NS cores, with greater closing at higher central densities 2. Surface transition:deff →3 smoothly as density drops below ρnuclear 3. Vacuum exterior:deff = 3 exactly for all r > RNS 4. BH limit: As M→Mmax and collapse occurs, the matter-supported interior vanishes, leaving only vacuum with deff = 3 3.2 No Fossil Effects A critical feature of RDT is that dimensional effects do not persist after matter vanishes. When a neutron star collapses to form a black hole: •The matter that sourced deff <3 disappears behind the horizon •The exterior spacetime becomes vacuum (ρ= 0) •Therefore deff = 3 exactly in the exterior •No “fossil” dimensional closing remains This is a direct consequence of RDT coupling to ρ, not to curvature or horizon properties. 4 NS–BH Binaries as Differential Tests 4.1 Asymmetric Signatures Neutron star–black hole binary mergers provide a unique opportunity to test RDT because they combine matter-supported and vacuum components in a single observable system: In NS–BH systems: •The NS exhibits enhanced tidal deformability: ΛRDT NS >ΛGR NS •The BH has exactly zero tidal deformability: ΛBH = 0 •The BH component serves as an internal vacuum control 4 Table 3: RDT signatures by binary type Binary Type Component 1 Component 2 RDT Signature BNS NS (RDT effects) NS (RDT effects) Symmetric BBH BH (GR exact) BH (GR exact) None NS–BH NS (RDT effects) BH (GR exact) Asymmetric 4.2 Differential Test Advantage This asymmetry enables within-event differential tests: 1. Measure tidal effects from the NS component 2. Compare to predictions assuming standard GR for both components 3. Any excess tidal signature isolates matter-dependent RDT effects 4. The BH component confirms vacuum nulling within the same observation This design is more powerful than BNS systems (where both components show RDT effects, making isolation difficult) or BBH systems (which are null tests only). 5 Observational Consistency 5.1 Event Horizon Telescope The EHT has measured shadow angular diameters for two supermassive black holes: Table 4: EHT shadow measurements vs RDT predictions Source θobs (µas) θGR (µas) θRDT (µas) M87* 42 ±3∼40 = θGR Sgr A* 48.7±7∼50 = θGR Both observations are consistent with RDT predictions within 1σuncertainties. These serve as null tests confirming the vacuum prescription. Clarification on Non-Circularity The agreement between RDT and EHT measurements is not circular reasoning. The vacuum nulling prescription was established on physical grounds—that RDT couples to matter density, not curvature—prior to and independently of EHT data. RDT contains no free parameters in the vacuum sector. EHT observations therefore serve as falsifiable tests: a significant deviation from GR shadow sizes would directly falsify RDT’s vacuum prescription. 5.2 LIGO-Virgo-KAGRA Tests All LVK tests of GR using binary black hole observations are consistent with RDT predictions [11]: These are null tests: they confirm the vacuum prescription but cannot distinguish RDT from GR in vacuum systems. 5 Table 5: LVK tests of GR: consistency with RDT vacuum predictions Test Category LVK Result RDT Prediction IMR consistency Consistent with GR = GR PN coefficients |δˆφi|<0.1δˆφi= 0 QNM frequencies Consistent with Kerr = Kerr Remnant properties Match NR predictions = NR Polarization Tensor only Tensor only 6 Discussion 6.1 Summary of Results The central result of this work is the vacuum nulling prescription: because RDT couples to matter density rather than spacetime curvature, dimensional modifications vanish identically in vacuum regions. For black holes, this yields deff = 3 exactly, and all observables match GR. This is not a failure but a success: it demonstrates that RDT is a theory of matter-geometry coupling with clean, falsifiable predictions. The second key result is smooth NS–BH continuity. Effective dimensionality varies continuously from deff <3 in NS cores to deff = 3 in vacuum BH exteriors. There are no discontinuities or fossil effects. 6.2 Relation to Modified Gravity Landscape RDT differs fundamentally from most modified gravity theories. Scalar-tensor, f(R), massive gravity, and EdGB theories modify vacuum gravity itself, predicting deviations from GR in black hole spacetimes. RDT does not modify vacuum gravity; it couples only to matter. This means: •RDT passes all vacuum tests by construction •Matter-containing systems provide the informative tests •RDT occupies a complementary position in the modified gravity landscape 6.3 Future Observations The vacuum nulling prescription identifies where RDT can and cannot be tested: •Null tests (confirm vacuum prescription): EHT shadows, BBH mergers, X-ray timing •Informative tests (probe RDT effects): BNS tidal measurements, NS–BH differential tests Third-generation gravitational wave detectors (Einstein Telescope, Cosmic Explorer) may achieve the precision needed for definitive tests, with projected tidal precision δΛ/Λ∼1–5% (design goal). 6.4 Falsifiability RDT makes specific, falsifiable predictions: What would falsify RDT: 1. Any vacuum deviation (shadows, QNMs, BBH tests) 6 2. No NS tidal enhancement (ΛNS = ΛGR NS exactly) 3. Non-zero BH tidal deformability (ΛBH = 0) 4. Cross-system inconsistency between BNS and NS–BH tidal effects Current status: All observations are consistent with RDT. The theory remains viable but not yet confirmed. Definitive tests await third-generation detectors. 7 Conclusion We have extended Recursive Dimensionality Theory to black hole spacetimes, establishing the vacuum nulling prescription as the central result. Because RDT is fundamentally a theory of matter-geometry coupling—dimensional modifications are sourced by matter under compression, not by spacetime curvature itself—effective dimensionality reverts to exactly three in vacuum regions. Black holes are therefore indistinguishable from their general relativistic counterparts. All observables we have examined—horizon radius, ISCO, photon sphere, shadow angular diameter, quasi-normal mode frequencies, post-Newtonian coefficients, and thermodynamic properties—match GR predictions with zero deviation. This is not a failure of RDT but a success: it demonstrates that the theory makes definite, falsifiable predictions and that dimensional effects switch off cleanly when their source (matter) is absent. Paper V completes the systematic extension of RDT across the astrophysical density spectrum. Papers I–IV tested the theory from solar core densities (ρ∼150 g/cm3) through white dwarf interiors (ρ∼106g/cm3) to neutron star cores (ρ∼1015 g/cm3)—spanning eight orders of magnitude. The present work extends this to the conceptual endpoint: vacuum black holes (ρ= 0). Throughout this progression, RDT has maintained internal consistency using fixed parameters. The smooth continuity we have demonstrated across the NS–BH transition closes a potential logical loophole. The theory now provides a complete prescription: dimensional closing occurs where matter is compressed beyond nuclear density, and dimensional effects vanish precisely where matter vanishes. The vacuum nulling prescription identifies where RDT can and cannot be tested. Black hole observations serve as null tests that confirm the vacuum prescription but cannot positively distinguish RDT from GR. The informative tests occur in matter-containing systems. NS– BH mergers offer a unique advantage: the black hole component provides an internal vacuum control, enabling differential tests within a single observation. Third-generation detectors may achieve the sensitivity required for definitive tests. If RDT is correct, these observations should reveal systematic tidal enhancements in neutron stars while black holes remain exactly as GR predicts. If instead future precision measurements find no enhancement, or detect vacuum deviations, RDT would be falsified. Either outcome advances our understanding of gravity in extreme environments. Acknowledgments This work was developed with computational collaboration from Claude (Anthropic), with project management by Claude Sonnet and scientific review by ChatGPT (OpenAI). The author thanks all contributors to the RDT research program. References [1] C. K. Merrill, “Recursive Dimensionality Theory I: Solar Neutrino Production,” (2025). [2] C. K. Merrill, “Recursive Dimensionality Theory II: White Dwarf Structure,” (2025). 7 [3] C. K. Merrill, “Recursive Dimensionality Theory III: Neutron Star Structure and the Maximum Mass Problem,” (2025). [4] C. K. Merrill, “Recursive Dimensionality Theory IV: Tidal Deformability in Binary Neutron Star Mergers,” (2025). [5] Event Horizon Telescope Collaboration, “First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole,” Astrophys. J. Lett. 875, L1 (2019). [6] Event Horizon Telescope Collaboration, “First Sagittarius A* Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole in the Center of the Milky Way,” Astrophys. J. Lett. 930, L12 (2022). [7] Event Horizon Telescope Collaboration, “First M87 Event Horizon Telescope Results. VI. The Shadow and Mass of the Central Black Hole,” Astrophys. J. Lett. 875, L6 (2019). [8] D. Psaltis et al., “Gravitational Test Beyond the First Post-Newtonian Order with the Shadow of the M87 Black Hole,” Phys. Rev. Lett. 125, 141104 (2020). [9] R. Abbott et al. (LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration), “GWTC-3: Compact Binary Coalescences Observed by LIGO and Virgo During the Second Part of the Third Observing Run,” Phys. Rev. X 13, 041039 (2023). [10] LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration, “GWTC4.0: Updating the Gravitational-Wave Transient Catalog with Observations from the First Part of the Fourth LIGO-Virgo-KAGRA Observing Run,” arXiv:2508.18082 (2025). [11] R. Abbott et al. (LIGO Scientific Collaboration, Virgo Collaboration, and KAGRA Collaboration), “Tests of General Relativity with GWTC-3,” Phys. Rev. D 112, 084080 (2025). [12] R. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), “Observation of Gravitational Waves from Two Neutron Star–Black Hole Coalescences,” Astrophys. J. Lett. 915, L5 (2021). [13] B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), “GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral,” Phys. Rev. Lett. 119, 161101 (2017). [14] E. Berti et al., “Testing General Relativity with Present and Future Astrophysical Observations,” Class. Quantum Grav. 32, 243001 (2015). [15] N. Yunes and F. Pretorius, “Fundamental Theoretical Bias in Gravitational Wave Astrophysics and the Parameterized Post-Einsteinian Framework,” Phys. Rev. D 80, 122003 (2009). [16] K. Schwarzschild, “¨ Uber das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie,” Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) 1916, 189 (1916). [17] R. P. Kerr, “Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics,” Phys. Rev. Lett. 11, 237 (1963). [18] J. M. Bardeen, W. H. Press, and S. A. Teukolsky, “Rotating Black Holes: Locally Nonrotating Frames, Energy Extraction, and Scalar Synchrotron Radiation,” Astrophys. J. 178, 347 (1972). [19] E. Berti, V. Cardoso, and A. O. Starinets, “Quasinormal Modes of Black Holes and Black Branes,” Class. Quantum Grav. 26, 163001 (2009). 8 [20] T. Hinderer, “Tidal Love Numbers of Neutron Stars,” Astrophys. J. 677, 1216 (2008). [21] F. Douchin and P. Haensel, “A Unified Equation of State of Dense Matter and Neutron Star Structure,” Astron. Astrophys. 380, 151 (2001). 9