Methodological Documentation: Exploratory Framework Development This work emerged from an exploratory, AI-assisted research workflow thatcombined conceptual inquiry, empirical data analysis, and classical statisticalmethods. The approach is documented here to ensure transparency regarding theorigin, scope, and limitations of the results. 1. Conceptual origin and problem framing The project was initiated by theoretical and conceptual considerationsconcerning the non-random nature of galactic structure and the observed diversityof inner rotation curves.These considerations motivated a qualitative exploration of relaxation,timescales, and cumulative dynamical processing in disk galaxies. Large language models were used as structured dialogue tools to supportbrainstorming, conceptual clarification, and the formulation of testablequestions. They served as interactive assistants in the early ideation phasebut did not replace physical reasoning or empirical validation. 2. Empirical foundation and data processing All quantitative analysis is based exclusively on publicly availableobservational data from the SPARC database, comprising 175 disk galaxies withwell-measured rotation curves and mass models. Under direct author supervision, Python-based analysis scripts were developedto extract kinematic quantities and to construct derived observables relevantto the study. No simulated or synthetic data were used at any stage. 3. Iterative derivation of empirical relations Guided by the initial conceptual framework, empirical relations were formulated and iteratively tested against the SPARC dataset.This process led to the identification of statistically significant correlationsthat had not been explicitly highlighted in prior analyses. 4. Emergence of the relaxation framework and saturation scale The detected correlations (with a Spearman rank coefficientρ≃0.63\rho \simeq 0.63ρ≃0.63) revealed a characteristic change in behaviour at rotationvelocities of approximately 120 km s−1120\,\mathrm{km\,s^{-1}}120kms−1. This observational feature motivated the introduction of an empiricalrelaxation-based framework and the definition of discrete relaxation classes(R-I to R-V), intended as a taxonomic ordering of galaxy dynamics rather than adefinitive physical model. 5. Technical implementation and consistency checks All numerical calculations, statistical evaluations, and visualisations wereperformed in Python using standard scientific libraries, including NumPy,Pandas, Matplotlib, and SciPy where required. Large language models were used to assist with code inspection, conceptualconsistency checks, and documentation refinement.They were not used for data generation, numerical optimisation, statisticalinference, or simulation. Scope and interpretation The present work should be understood as an exploratory observational study andan organisational proposal rather than a definitive physical explanation.Its primary aim is to motivate further independent investigation,reproducibility efforts, and professional follow-up by the scientific community.

In a companion study we reported the detection of a statistically robust empirical correlation between inner galactic kinematics and a proxy for the cumulative dark-matter scattering history in disk galaxies. In this work, we examine whether this relaxation signature persists uniformly across halo velocity scales. Using the same observationally derived quantities and rotation-curve data from the SPARC database, we perform a velocity-resolved analysis based on regime comparisons, transition diagnostics, smooth suppression modelling, and multiple robustness tests. We find that the relaxation signal is clearly present in low-velocity halos but becomes statistically consistent with zero in high-velocity systems. The data favour a gradual suppression rather than a sharp transition, with a characteristic velocity range of order 100–120 km s⁻¹. The result is entirely empirical and model-agnostic, does not rely on simulations or assumptions about dark-matter microphysics, and provides a kinematic constraint on the velocity domain over which dark-matter relaxation effects are observationally supported.