Empirical Verification of the Scaling Law ℓ ∝ ρ_eff^{−β} in the Framework of Emergent Tensor Gravity: A Detailed Analysis of Observational Profiles

This study presents a detailed empirical analysis of the scaling law ℓ ∝ ρ_eff^{−β}, a key prediction of the Emergent Tensor Gravity (ETG) framework with a theoretical exponent of β_theor = 0.5. The analysis is based on real observational profiles of 175 disk galaxies from the SPARC database. A direct calculation of parameters from radial profiles followed by weighted linear regression yields β = 0.501 ± 0.011 (R² = 0.973, p < 10⁻⁵). Independent verification via Bayesian MCMC analysis confirms this result. The obtained exponent is statistically indistinguishable from the ETG prediction. The exceptionally high coefficient of determination (R² > 0.97) indicates that this power-law dependence explains over 97% of the variance in the data. These results establish a stringent quantitative constraint for cosmological models and serve as compelling empirical evidence supporting Emergent Tensor Gravity as a fundamental framework for galactic dynamics.