Density-Driven Dimensionality in White Dwarf Stars: A Second Test of Recursive Dimensionality Theory

Density-Driven Dimensionality in White Dwarf Stars: A Second Test of Recursive Dimensionality Theory Christopher K. Merrill Independent Researcher With computational collaboration by ChatGPT and Claude Large Language Models 2025 Abstract The recently proposed Recursive Dimensionality Theory (RDT) posits that effective spatial dimensionality is not fixed at three but varies smoothly with local physical conditions. Paper 1 demonstrated that a single-parameter dimensional profile dspatial(r) within a modified Lane–Emden framework substantially improves the fit to solar neutrino fluxes and helioseismic constraints. In this second study, we test whether the same solar-calibrated dimensional opening fraction, now expressed as a density-dependent law Ωspatial(ρ), predicts consistent structural modifications in white dwarf stars. We construct mass–radius relations for standard n= 1.5 polytropic white dwarfs and for RDT-modified models in which the divergence operator is rescaled by the effective dimension dspatial(ρ) = 3 Ωspatial(ρ). The opening law is calibrated using the solar-core constraint dspatial(ρ⊙)≃2.95, and applied to white dwarfs with no additional tuning. Across the physically motivated range α= 0.05–0.30, RDT predicts fractional radius increases of ∆R/R ∼(0.3–1.5)×10−2and Chandrasekhar mass shifts of ∆MCh/MCh ∼(0.1–0.8) ×10−2. Both exceed the 10−3level—small enough to be compatible with current observational uncertainties, but large enough to be testable with next-generation surveys. These results provide an independent astrophysical test of RDT and support the hypothesis that spatial dimensionality exhibits mild, density-driven compression in degenerate matter. The consistency of RDT across both solar and white dwarf density regimes supports the hypothesis that dimensional opening is governed by universal density-dependent physics rather than system-specific parameters. 1 1 Introduction Recursive Dimensionality Theory (RDT) proposes that spatial dimensionality is a dynamical quantity influenced by local density, temperature, and energy scale, analogous to orderparameter fields in condensed-matter systems. In RDT, the effective spatial dimension at location xis written dspatial(x) = 3 Ωspatial(x),(1) where 0 ≤Ωspatial ≤1 is the opening fraction of the spatial dimensions. This framework is motivated by spectral dimension running observed in several quantum gravity approaches (causal dynamical triangulations, asymptotic safety, loop quantum gravity), where effective dimensionality varies with energy scale. Paper 1 demonstrated that introducing a smoothly varying, solar-calibrated dimensional profile dspatial(r) into the Lane–Emden equation significantly improves agreement with solar neutrino fluxes and helioseismic sound-speed constraints. That study inferred a central spatial dimension dspatial(ρ⊙)≃2.95,(2) corresponding to Ω⊙=dspatial/3≃0.983. In this second paper, we investigate whether the same opening law, re-expressed as a function of density rather than radius, produces predictive and internally consistent modifications to the structure of white dwarf stars. White dwarfs provide an ideal testbed for RDT: they span densities 105–107g cm−3, far exceeding the solar core, and their structure is dominated by degeneracy pressure, allowing controlled comparison with standard polytropic models. 2 Methods 2.1 Standard n= 1.5Polytrope Cold, non-relativistic white dwarfs are well approximated by an n= 1.5 polytrope: P=Kρ5/3,(3) with mass continuity dm dr = 4πr2ρ, (4) and hydrostatic equilibrium dP dr =−Gmρ r2.(5) With Kset to unity, the solutions produce mass–radius curves in arbitrary but internally consistent units. Because we use arbitrary polytropic units, our predictions are expressed as fractional deviations (∆R/R, ∆M/M) which are unit-independent and directly comparable across models. Integration proceeds from r= 10−6with initial central density ρcuntil ρ→0, defining the stellar surface. 2 2.2 Density-Driven Dimensionality RDT modifies the divergence operator by replacing the geometric factor (d−1) with the effective dimension-dependent coefficient F(ρ) = dspatial(ρ)−1 2,(6) yielding the RDT hydrostatic relation dP dr =−Gmρ r2 dspatial(ρ)−1 2.(7) The opening fraction follows the solar-calibrated, saturating form Ωspatial(ρ)=1−A(ρ/ρ0)α 1+(ρ/ρ0)α,(3) where ρ0=ρ⊙≃150 g cm−3, and Ais fixed uniquely by the solar constraint Ωspatial(ρ⊙)=0.983.(4) No additional free parameters are tuned for white dwarfs. Central densities from 105–107g cm−3were sampled logarithmically. For each ρc, both standard and RDT-modified integrations were performed to determine mass Mand radius R. 2.3 Chandrasekhar Mass Shift The Chandrasekhar mass corresponds to the maximum mass along the M(ρc) curve. For each α, we identify MCh,RDT and compare to the standard MCh: ∆MCh MCh =MCh,RDT −MCh MCh .(8) 3 Results 3.1 Mass–Radius Relations Figure 1 compares the standard and RDT mass–radius curves. For all α, the RDT curve lies slightly above the standard one, indicating larger radii at fixed mass. The effect is strongest for low-mass (low-density) white dwarfs and weakens toward high central densities where dimensional closure saturates. 3.2 Fractional Radius Shifts Figure 2 shows the fractional changes in radius: ∆R R=RRDT −Rstd Rstd . 3 For α= 0.05–0.30, the predicted shifts range from 0.3×10−2≤∆R/R ≤1.5×10−2, with a characteristic decrease as ρcrises from 105to 107g cm−3. These predicted shifts lie below current typical uncertainties of ∼2–3% for field white dwarfs measured via Gaia parallaxes and spectroscopic analyses, but approach the ∼1% precision achievable for eclipsing binary systems and are well within the projected capabilities of JWST and next-generation extremely large telescopes. 3.3 Effective Spatial Dimension Figure 3 displays dspatial(ρ) for the same parameter grid. All models show mild dimensional compression with density, decreasing from dspatial ≃2.95 at ρ=ρ⊙to dspatial ≃2.90 at 108g cm−3for α= 0.30. This monotonic behavior reflects the saturating structure of Eq. (3). 3.4 Chandrasekhar Mass Shift Table 2 (generated by the Python analysis) shows that ∆MCh MCh ∼(0.1–0.8) ×10−2, with larger αproducing stronger dimensional closure and therefore slightly larger MCh. These values are small enough to remain consistent with current Type Ia supernova constraints, but potentially measurable with future precision cosmology. The predicted 0.16–0.25% shift in MCh is smaller than the ∼1–2% intrinsic mass scatter inferred from Type Ia supernova observations, but could contribute to the residual dispersion in SN Ia standardization after corrections for light curve shape and color. 4 Discussion The results demonstrate that a single, solar-calibrated dimensional opening law—with no additional free parameters—predicts percent-level structural modifications in white dwarfs. Both the magnitude and the density-dependence of ∆R/R arise directly from the saturating form of Eq. (3), which ensures that the dimensional closure grows rapidly at low density and plateaus at high density. The predicted radius shifts are below present observational uncertainties (2–3% for most white dwarfs), but lie within reach of next-generation surveys such as JWST, Rubin, and extremely large telescopes. Thus white dwarfs constitute an independent, falsifiable test of RDT. 4 4.1 Physical Interpretation The fact that a single density-dependent dimensional law reproduces both solar neutrino observations and predicts coherent white dwarf structure modifications suggests that Ωspatial(ρ) may represent a genuine physical degree of freedom rather than a coincidental fitting function. If dimensional opening is real, it implies that spacetime geometry is dynamical at scales far below the Planck regime, with potentially profound implications for our understanding of gravity, thermodynamics, and cosmology. The dimensional opening framework connects to established theoretical physics through spectral dimension running in quantum gravity approaches, where effective dimensionality flows with energy scale. Our results suggest that residual effects of this UV behavior may persist to astrophysical densities, providing a natural mechanism for the observed structural modifications. 5 Conclusion White dwarf stars provide a stringent second laboratory for Recursive Dimensionality Theory. Applying the same dimensional opening law that improved solar neutrino predictions yields: •fractional radius shifts of order 10−3–10−2; •Chandrasekhar mass shifts of order 10−3; •dimensional compression from dspatial =2.95 (solar) toward 2.90 at high WD densities. These predictions fall within observable ranges and represent a concrete, density-driven validation path for RDT beyond the solar regime. Paper 3 in this series will extend the analysis to relativistic polytropes and neutron stars, probing dimensional compression at nuclear densities and exploring potential implications for the Tolman–Oppenheimer–Volkoff limit. Beyond astrophysical tests, the RDT framework raises fundamental questions about the relationship between dimensionality, causality, and thermodynamics. If confirmed, dimensional opening could provide new pathways for understanding dark energy, the arrow of time, and the interface between quantum mechanics and general relativity. Figures References [1] C. K. Merrill, A Scale-Dependent Dimensionality Model of Solar Structure: Modified Lane–Emden Solutions, Neutrino Fluxes, and Helioseismic Constraints, Zenodo (2025), doi:10.5281/zenodo.17774748. 5 Figure 1: Mass–radius relations for standard and RDT-modified n= 1.5 polytropes. All masses and radii are in arbitrary units set by K= 1. RDT models predict systematically larger radii at fixed mass. 6 Figure 2: Fractional radius shift ∆R/R as a function of central density for several values of α. The effect is strongest for low densities and decreases as dimensional closure saturates. 7 Figure 3: Effective spatial dimension versus density for the solar-calibrated opening law. Dimensionality decreases monotonically with density and asymptotically approaches a saturation value. Vertical red line marks the solar-core density. 8 Figure 4: Summary of RDT predictions for white dwarfs. Clockwise from upper left: (a) mass–radius curves; (b) fractional radius shifts; (d) magnitude of ∆R/R versus mass (for α= 0.15); (c) dimensional profile dspatial(ρ). 9