Radial Dimensionality Theory I: Solar Neutrino and Helioseismic Constraints on Density-Dependent Spatial Dimensionality

This work derives solar constraints on Radial Dimensionality Theory (RDT), a phenomenological framework in which the effective spatial dimensionality decreases weakly with local density through a saturating opening law d_eff(ρ) = 3·Ω(ρ), with Ω(ρ) = 1 − A·x/(1+x) and x = (ρ/ρ₀)^α. The modification is applied to the gravitational divergence operator only, and the exact modified Lane–Emden equation is derived and solved — including a previously neglected logarithmic-derivative term and a reference-scale convention that makes the model unit-invariant. Standard and modified n=3 polytropes are calibrated to the solar mass and radius, and the predicted pp, pep, Be-7, and B-8 neutrino flux ratios are confronted with Borexino measurements normalized to the B16-GS98 Standard Solar Model, together with a conservative helioseismic bound on the induced sound-speed perturbation. The data show no preference for a dimensional perturbation: the best fit improves on the standard three-dimensional Sun by only Δχ² ≈ 0.03 with two free parameters. The two observables are complementary — neutrino fluxes exclude sharp, core-localized perturbations while the sound-speed bound excludes broad, shallow ones — yielding a closed 95% confidence upper limit A < 0.0087 for all transition exponents 0.02 ≤ α ≤ 0.5. Equivalently, the effective dimension of the solar core satisfies d_eff > 2.988, and the high-density saturated floor of the law satisfies d_eff > 2.974 at any density. The included Python code (MIT licensed) provides the exact modified Lane–Emden solver with validation gates, self-consistent solar calibration, neutrino and sound-speed observables, the (A, α) parameter scan, and the statistical analysis, and reproduces every number quoted in the paper. Version 3 Update (June 2026): This version supersedes the results of Versions 1–2. The earlier analysis contained several compounding implementation errors (a unit-dependent expansion of the modified divergence operator, an incorrect central boundary condition, an incorrect central-temperature scaling, and missing calibration prefactors in the flux ratios), which together made a strongly excluded parameter region appear favored. With the corrected, exactly derived equations, the previously reported best fit (d_eff ≈ 2.95, reduced chi-square ≈ 0.16) does not survive: that parameter point is now excluded at more than 8σ, and the analysis is reframed as a constraint. The dimensional law is also reparameterized as a function of density rather than radius, so that a single law spans the Sun, white dwarfs (Paper II), and neutron stars (Paper III) without retuning. The qualitative conclusion that solar data meaningfully constrain density-dependent dimensionality is unchanged; all quantitative results are new.