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. 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. Using a subset of 115 galaxies, we compute k(r) across all radial points and identify an asymptotic parameter k_inf for each galaxy. After subtracting this offset and normalizing the radial coordinate, all profiles collapse onto a single universal function: Delta k(r) = f(r) independent of galaxy properties. The function is empirically found to follow an exponential form: f(r) = A exp(-r / r0) with A ≈ 4 and r0 ≈ 0.2. This leads to a closed structural relation: v^2 r = C_inf * L^(A exp(-r / r0)) from which flat rotation curves emerge in the asymptotic regime. All results are obtained through direct transformations of observational data, without introducing dynamical models, fitting procedures on raw rotation curves, or external assumptions. The analysis demonstrates the existence of a universal radial structure in galaxy rotation curves, characterized by a single galaxy-dependent parameter and a universal function.