This work presents a structural framework for understanding the emergence and persistence of circular relations in astrophysical systems. We introduce a minimal geometric hierarchy: 4 → 4/π → √π → π In this framework: - 4 defines a discrete structural skeleton, - 4/π defines a closure threshold, - √π defines an emergence scale, - π defines a sustained circular regime. Using observational galaxy data, we find that the transition radius is governed by: R₀ ≈ √π while the completion scale appears not as a single value, but as a finite-width band around π: R ~ π (1 ± Δ) We further show that the system exhibits a low effective dimension: Dₑff ≈ 2. 2–2. 3 indicating that observed high-dimensional variables collapse into a low-dimensional structural relation. A central result is the identification of a separation principle: geometry determines where structure appears, while dynamics determines how it is populated. Within this interpretation: - geometric relations define the admissible structure, - physical parameters (e. g. , disk stability, halo fraction) modulate the state within that structure. Importantly, π is reinterpreted not as a fixed constant, but as a sustained relational band arising from deviation-driven dynamics. This framework does not replace existing physical theories, but provides a structural perspective explaining why universal patterns emerge across different galaxies. The results suggest that circular structure is not imposed by specific physical mechanisms, but emerges from underlying geometric constraints and is subsequently realized through dynamics.
umimoto (2026) studied this question.