Universal Collapse of Galaxy Rotation Curves from a Logarithmic Structural Variable

We analyze galaxy rotation curves using observational data from the SPARC database (Lelli, McGaugh & Schombert, 2016), based exclusively on the measured radial distance r and rotation velocity v, without introducing dynamical models or external assumptions. A derived observable is constructed as C(r) = v(r)^2 r, and a logarithmic structural variable is introduced: k(r) = log(C(r)/E*) / log(L) with L = 0.25 and E* a global scale. Across a dataset of 115 galaxies, we compute k(r) and identify a galaxy-dependent asymptotic parameter k_inf. After subtracting this offset and normalizing the radial coordinate, all profiles collapse onto a single universal function: Delta k(r) = k(r) − k_inf independent of galaxy properties. The resulting function follows an exponential law: Delta k(r) = A exp(-r / r0) with A ≈ 4 and r0 ≈ 0.2, leading to a closed structural relation for rotation curves: v^2 r = C_inf * L^(A exp(-r / r0)) Flat rotation curves emerge as a direct consequence of this structure. The analysis demonstrates that galaxy rotation curves admit a universal description in terms of a single parameter and a universal radial function, suggesting the existence of an underlying structural law independent of standard dynamical modeling.