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 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. 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 5% neutron-star tidal excess, 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 excess above the solar ceiling.

Version 3 Update (June 2026):

This version unifies the paper with the revised series (Papers I–IV, versions 3/5/7/5). The earlier logarithmic dimensional law — active only above nuclear saturation density, with parameters quoted from superseded versions of Papers III–IV — is replaced by the series saturating law under Paper I's solar bound. The neutron-star dimensional profiles are recomputed accordingly: core values are now uniform at d_eff ≈ 2.975 (the saturation floor) rather than deepening with central density. References to "enhanced" neutron-star tidal deformability are restated as solar-bounded ceilings (≤5%, Paper IV), and the differential-test discussion incorporates the corrected detectability analysis (population-level, not single-event). The vacuum nulling argument itself is strengthened from a limiting statement to a by-construction one. Unlike the other papers in this series, no quantitative conclusions are withdrawn here — the central thesis was insensitive to the corrected numbers and survives unification unchanged.