The Golden-Ratio Dark Halo: Testing a Geometry-Motivated Rotation Curve Profile Against the Full SPARC Catalogue

We present the φ-core dark halo profile, v(r) = v₀ r / √(φ²r₀² + r²), where φ = (1+√5)/2 ≈ 1.618 is the golden ratio fixed by geometry and not fitted. We test this profile against the complete SPARC rotation curve catalogue (175 galaxies; Lelli, McGaugh & Schombert 2016), fitting dark-matter-only residuals after subtracting baryonic contributions with a fixed stellar mass-to-light ratio Y★ = 0.5 M☉/L☉. Across 172 successfully fitted galaxies, the φ-core profile achieves a median reduced χ² of 1.221, compared to 1.293 for the pseudo-isothermal sphere (ISO, 2 params) and 2.439 for the NFW profile with freely fitted concentration (3 params). It outperforms ISO on 108/172 (62.8%) of galaxies and NFW with freely fitted concentration (3 free parameters) on 131/172 (76.2%) galaxies. A functional-form consistency test—confirming that v(φr₀)/v₀ = 1/√2 = 0.7071 holds across 140 galaxies with sufficient radial coverage—yields an observed mean of 0.7005 ± 0.054 (+0.07σ). We discuss the theoretical motivation for the profile shape, its open derivations, and the honest epistemic boundaries between prediction and fit.