This work presents Version 21 of the Elastic Spacetime with Scale-Dependent Coupling (ESSC) framework, focusing on the structural interpretation of the appearance of π in galaxy rotation curve data. Previous analyses suggested a relation of the form ω ≈ π, which could be interpreted as a potential invariant. In this study, we revisit this result through a systematic analysis across coordinate representations, fitting models, and galaxy subsamples. We find that the characteristic frequency ω does not converge to a single fixed value. Instead, it forms a distribution organized around a scale that appears near π under specific representational conditions. This π-scale is most clearly observed in the normalized radial coordinate R/Rd, while it weakens or disappears under logarithmic and min–max normalized coordinates. Model comparison shows that cosine-based representations capture this structure more effectively than polynomial models. This does not imply that the system is inherently sinusoidal, but rather that the cosine basis aligns with the dominant structural mode of the data. Within the ESSC framework, these results lead to a reinterpretation of π: not as a universal invariant appearing directly in observational data, but as a preferred closure-like scale that emerges under specific coordinate and modeling conditions. In this context, π reflects a structural boundary associated with translation closure rather than a fixed numerical constant. This work does not introduce new physical laws, forces, or predictive equations. Instead, it identifies the structural conditions under which a familiar mathematical constant emerges as a feature of observational data. The results emphasize the role of representation in revealing or obscuring structural patterns, and position π as a representation-dependent signature of closure in galactic dynamics. All analyses are based on galaxy rotation curve data (SPARC). The conclusions are empirical and structural, and future work will test the persistence of the π-scale across independent datasets and representations.
umimoto (Sat,) studied this question.