Constrained Low-Dimensional Structure in Normalized SPARC Rotation Curves

This version includes substantial improvements to the interpretation, methodological clarity, and presentation of results. Key updates: Expanded the Introduction to position the analysis relative to the Universal Rotation Curve (Persic, Salucci & Stel 1996), the rotation-curve diversity results of Oman et al. (2015), and the radial acceleration relation (McGaugh et al. 2016). Clarified that the paper addresses the intrinsic dimensionality of the normalized rotation-curve family rather than fitting a specific dynamical model. Added an explicit definition of “low-dimensional” and clarified that the term “manifold” is used informally to describe an empirically restricted region of function space. Added a sample-summary section and tables describing the galaxy sample and key structural parameters. Added outlier characterization for the high-residual tail, including discussion of likely contributions from low surface brightness systems, sparse sampling, irregular morphology, and normalization sensitivity. Expanded the Discussion section to strengthen the interpretation of the result in the context of both dark matter and modified-dynamics frameworks. Added a bootstrap stability analysis of the PCA decomposition together with a new figure showing the bootstrap distributions of explained variance for the first three principal components. Corrected the PCA analysis to use a consistent galaxy subset across sampling resolutions, ensuring a fair comparison of dimensional structure. Updated the Data Availability statement to reflect public archiving of derived products and analysis materials. The core empirical result is refined: the normalized SPARC rotation-curve family exhibits constrained structure under parameter-based descriptions, while its variance is distributed across multiple modes under linear decomposition. This version improves the framing, robustness checks, and interpretive clarity of that result.