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

This paper extends Radial Dimensionality Theory (RDT) to black hole spacetimes, establishing the vacuum nulling prescription: black holes in RDT are exactly the black holes of general relativity. RDT proposes that effective spatial dimensionality decreases weakly with matter density through a saturating opening law whose amplitude is bounded — not detected — by solar data (A < 0.0087 at 95% confidence, Paper I). Papers II–IV propagated that bound to white dwarfs, neutron stars, and tidal deformability, yielding percent-level contraction ceilings on all matter-sector signatures. The present work completes the density spectrum at its vacuum endpoint. The vacuum result follows doubly. First, the opening law satisfies Ω(0) = 1 identically and is smooth in density, with no activation threshold. Second — and more strongly — the series' relativistic implementation modifies only the matter sector: the Einstein field equations are untouched, so by Birkhoff's theorem every vacuum exterior is the Schwarzschild or Kerr geometry by construction, independent of the law's form or amplitude. This holds for both the canonical divergence-operator mechanism now used in Papers III–IV and the earlier force-rescaling ansatz, so the conclusion is robust to the sign correction that distinguishes them — and it is now demonstrated rather than asserted, since Papers III and IV exhibit the operator form measurably contracting matter while every vacuum observable stays exactly that of general relativity. All black hole observables (horizon, photon sphere, shadow size, quasi-normal modes, tidal deformability, thermodynamics) match general relativity exactly, with no free parameters in the vacuum sector. Effective dimensionality runs smoothly from the solar-bounded saturation floor (d_eff ≥ 2.974 in neutron-star cores, uniform across masses) to exactly 3 in vacuum, with no fossil effects surviving collapse. Event Horizon Telescope shadow measurements and LIGO-Virgo-KAGRA tests of general relativity are consistent null tests. Neutron star–black hole mergers remain the optimal differential probe — the black hole provides a built-in vacuum control — now quantified by Paper IV's ceiling: at most a ~4% neutron-star tidal reduction, testable at population level rather than in single events. Falsifiability criteria are explicit: any confirmed vacuum deviation from GR rules the theory out, as would a matter-sector tidal deviation beyond the solar ceiling. Version 3 Update (June 2026): Erratum on versioning: the previous synopsis was labeled "Version 3 Update" in error — it in fact described the Version 2 manuscript. The present synopsis describes and supersedes that as the actual Version 3. This version aligns the paper's matter-sector references with the canonical divergence-operator mechanism now adopted in Papers III (Version 8) and IV (Version 6), under which RDT contracts compact objects. Neutron-star and tidal signatures, previously described as a maximum-mass increase (≲0.9%) and a tidal-deformability enhancement (≲5%), are restated as the corrected contractions: a maximum-mass and radius decrease (≲0.7%) and a tidal-deformability reduction (≲4%) — all the same sign as the solar and white-dwarf deficits. That earlier "enhancement/increase" language had followed the superseded force-rescaling (F-multiplier) sign of Papers III V7 / IV V5. The proof that vacuum nulling holds for the divergence-operator form as well as the F-multiplier is promoted from a footnote to the main text and strengthened: it is now demonstrated through the matter-sector papers (the operator form measurably contracts matter while the vacuum sector remains exactly GR), not merely asserted as a ρ → 0 limit. The series bibliography is updated to Papers III V8 and IV V6 (series versions now 3/5/8/6). The single quantitative element — the dimensional-profile table and figure — is unchanged: d_eff(ρ) follows from the opening law alone and is invariant under the sign correction (core values uniform at d_eff ≈ 2.975, the saturation floor). As with the prior revision, no quantitative conclusions are withdrawn: the central thesis was insensitive to the corrected signs and survives unchanged — indeed strengthened, since it is now exhibited by a mechanism with a definite, measurable matter-sector effect.