Projected Kinematic Anisotropy and Velocity-Dispersion Gradients in Galactic Globular Clusters: An Interval-Defined Framework from HST Proper Motions

We present an interval-defined framework for summarising projected internal kinematics in globular clusters (GCs) using the Global Kinematic Anisotropy Parameter (KAP), the Global Kinematic Index (GKI), and the raw and noise-aware profile-scatter diagnostics Ds and Ds, corr. KAP is a variance-weighted ratio of projected radial–tangential squared-dispersion contrasts, whereas GKI is the empirical outward logarithmic slope of the combined velocity-dispersion profile over a specified radial interval. We apply the framework to homogeneous HST/HACKS proper-motion profiles for 20 Galactic GCs. Within the primary normalized interval R ≤ Rh, l, five clusters satisfy the radial-KAP criterion, fifteen are consistent with zero net projected KAP contrast, and none satisfies the tangential criterion. Over the full available HST range, seven clusters satisfy the radial-KAP criterion and thirteen are consistent with zero net projected KAP contrast; NGC 6388 and NGC 6441 account for the two interval-dependent class changes. Eighteen clusters have significantly negative GKI within R ≤ Rh, l, while NGC 5927 and NGC 6441 are consistent with flat empirical profiles. Eleven clusters have Ds, corr=0, and nine retain non-zero corrected scatter. KAP and GKI are radial-interval-dependent observational summaries, while Ds and Ds, corr only identify profiles requiring bin-level inspection. A negative GKI over the broader HST interval is not specific to an intermediate-mass black hole, and unresolved spheres of influence cannot be excluded by these binned profiles. Analytic uncertainty propagation, parametric resampling, leave-one-bin-out tests, and controlled synthetic experiments support the internal consistency of the framework under the conditions examined.