Closure Fraction, Loop Time, and Cosmic Composition: A Minimal Structural Interpretation of Dark Matter

We present Elastic Spacetime with Scale-Dependent Coupling (ESSC) as a minimal structural framework in which the observed cosmic mass composition emerges from the balance between relational closure and cosmological expansion. In this approach, physical systems are described in terms of a dimensionless closure fraction f, defined as a structural descriptor of the balance between inner (closed) and outer (open) relational contributions. Empirical analysis of galaxy rotation curves indicates that the dominant structural interaction follows a two-component form f(1−f), representing the competition between closure and openness. We introduce a minimal dynamical representation, df/dt = (1−f)(Af−H), where A characterizes local closure strength and H represents cosmic expansion. This expression is not interpreted as a fundamental dynamical law, but as a compact representation of the structural balance between closure and expansion. A key result of this work is the geometric interpretation of closure. The requirement that a relation becomes self-consistent only after completing a full cycle leads to a loop timescale τ_loop = 2πr/V. Comparing this with the Hubble time τ_H = 1/H yields a scaling relation: f ≈ 2π H r0 / V_b(r0), evaluated at the radius r0 where baryonic and non-baryonic contributions are comparable. Applying this relation to a sample of 124 galaxies, we obtain a median value f ≈ 0.143 ≈ 0.15, corresponding to a dark-to-baryon ratio Ω_DM/Ω_b ≈ 5.5, without introducing free parameters. These results suggest that the observed cosmic mass composition may be understood as a structural consequence of incomplete relational closure in an expanding spacetime. In this framework, dark matter is not treated as an additional substance, but as the observable manifestation of a closure deficit, and cosmic composition is determined by the ratio between loop-closure time and the Hubble time. The ESSC framework is not intended as a replacement for existing cosmological models, but as a complementary structural perspective that links local dynamics, geometric closure, and global expansion within a unified description.