Logarithmic Velocity Correction in Galaxy Rotation Curves: An Empirical Test with 151 SPARC Galaxies

We present an empirical test of a logarithmic velocity correction model using 151 galaxy rotation curves from the SPARC database (Lelli, McGaugh & Schombert 2016). Model. The observed rotation velocity is modeled as V²(r) = V²_bar(r) + γ·ln(r/r₀), where V_bar is the baryonic contribution computed from 3.6 μm photometry, γ is a coupling parameter, and r₀ is a characteristic scale length. This velocity ansatz corresponds to an effective potential Φ_eff ∝ [ln(r/r₀)]² and a modified isothermal density profile ρ ∝ [ln(r/r₀)+1]/r². The model is motivated by the Topological Knot-Web Cosmology (TKWC) framework (Feshter 2025, doi:10.5281/zenodo.18650668). Principal results: 1. Statistical performance. Both models achieve excellent fits to the rotation curve data (median data residuals χ²_data = 0.54 for the logarithmic model and 0.14 for MOND-RAR). The models differ in their treatment of stellar mass-to-light ratios: MOND achieves lower data residuals but requires larger deviations from population synthesis predictions (median M/L prior penalty 1.88 vs 0.19). On data residuals alone, MOND is preferred in 60% of the sample; by AIC (ΔAIC = +2.6), MOND is preferred in 85%. The logarithmic model's contribution is not superior goodness-of-fit but a confirmed scaling prediction and consistency with stellar population constraints. 2. Scaling relation. Using baryonic masses computed with fixed mass-to-light ratios (Υ_disk = 0.5, Υ_bulge = 0.7) independent of the fitting procedure, the coupling parameter scales as γ ∝ M_bar^α with α = 0.485 ± 0.030, consistent with the geometric prediction α = 0.50 within 0.5σ. Bootstrap resampling (10⁴ iterations) yields 68% CI = [0.457, 0.512] and 95% CI = [0.431, 0.540], both containing the predicted value. The result is stable across mass-to-light ratio assumptions (α ∈ [0.467, 0.504] for Υ_disk ∈ [0.3, 0.8]). 3. Scale length. The fitted r₀ ≈ 0.6 kpc shows no significant correlation with baryonic mass (Spearman ρ = 0.149), galaxy extent (ρ = 0.108), or outer acceleration (ρ = −0.083), suggesting it represents a structural property of the velocity correction rather than a scaling of the baryonic distribution. 4. Limitations. Individual gas-dominated dwarf galaxies (e.g., DDO 154) remain challenging. The high-mass regime shows a flattened local slope (α_local ≈ 0.37), consistent with a mass-dependent deviation from the global scaling. No comparison with NFW halo fits is included. Methodology. Fitting uses Nelder-Mead minimization with Gaussian priors on stellar mass-to-light ratios. Quality cuts require ≥5 data points, successful convergence, and γ > 10 km²/s². MOND comparison uses the RAR interpolation function with free a₀. All analysis code is in Python (NumPy, SciPy). Multi-agent analysis. The analysis was independently cross-checked using multiple large language model systems to detect logical inconsistencies, pipeline errors, and methodological issues. Related work: - TKWC Nodes 734–735: doi:10.5281/zenodo.18650668 - Topological Screening of Dark Energy: doi:10.5281/zenodo.18422868 The Entropy Sink: doi:10. 5281/zenodo.18479788