A RTLI Non-Parametric Estimator of Inner Rotation-Curve Slope from Velocity Distributions

We investigate whether higher-order power-mean transitions within the Heuristic Calculus (HC) framework provide novel structural information about galaxy dynamics beyond established rotation-curve descriptors. Focusing on the quartic transition index S_{3\to4} = [g(4)-g(3)] - [g(-3)-g(-4)], we perform a rigorous, preregistered evaluation across multiple datasets, including SPARC integrated properties, SPARC mass models, and MaxDisk radial rotation curves. We demonstrate that S_{3\to4} is redundant for integrated galaxy properties, showing strong correlations with kurtosis and the cubic depth ratio M_3/M_1. However, for non-integrated radial velocity profiles, S_{3\to4} exhibits conditional non-redundancy and a robust monotonic correlation with the inner logarithmic slope of rotation curves (r = -0.83). Controlling for standard variability metrics confirms that this behavior reflects sensitivity to floor-dominated velocity structure (inner rise from zero) rather than tail-dominated dispersion. Crucially, we show that S_{3\to4} does not encode independent physical information beyond the inner slope s_0. Instead, it functions as a non-parametric, fitting-free surrogate estimator of inner rotation-curve slope, computable directly from velocity distributions without explicit mass modeling or curve fitting. This work establishes a clear domain boundary for the HC framework: higher-order power-mean extensions do not provide new physics for integrated galaxy properties, but can offer computationally efficient screening diagnostics for large rotation-curve datasets. The result is a scientifically valuable null-plus outcome—demonstrating both where the framework adds utility and where it must stop.